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Research Article The Lagrangian, Self-Adjointness, and...
6
Research Article The Lagrangian, Self-Adjointness, and Conserved Quantities for a Generalized Regularized Long-Wave Equation Long Wei and Yang Wang Department of Mathematics, Hangzhou Dianzi University, Zhejiang 310018, China Correspondence should be addressed to Long Wei; [email protected] Received 9 January 2014; Accepted 2 May 2014; Published 8 May 2014 Academic Editor: Tiecheng Xia Copyright © 2014 L. Wei and Y. Wang. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the Lagrangian and the self-adjointness of a generalized regularized long-wave equation and its transformed equation. We show that the third-order equation has a nonlocal Lagrangian with an auxiliary function and is strictly self-adjoint; its transformed equation is nonlinearly self-adjoint and the minimal order of the differential substitution is equal to one. en by Ibragimov’s theorem on conservation laws we obtain some conserved qualities of the generalized regularized long-wave equation. 1. Introduction e generalized regularized long-wave (GRLW) equation + + + =0 (1) was first put forward as a model for small-amplitude long waves on the surface of water in a channel by Peregrine [1, 2], where is a positive integer, and and are two constants. It is an alternative description of nonlinear dispersive waves to the Korteweg de Vries equation [1] and has been used to describe phenomena with weak nonlinearity and dispersion waves, including nonlinear transverse waves in shallow water, ion-acoustic and magnetohydrodynamic waves in plasma and phonon packets in nonlinear crystals, pressure waves in liquid-gas bubble mixture, rotating flow down a tube, and lossless propagation of shallow water waves; see [3] and the references therein. Equation (1) has been studied extensively in the past both analytically and numerically by means of, for example, the Adomian decomposition method, the Fourier method, the finite difference method, the finite element method, the variational iteration method, the mesh-free method, the cubic B-spline collocation procedure, the sine-cosine method, and the double reduction method [4–11], and so forth. A few of the conserved quantities by using the 1- soliton solution and solitary-wave solution of the (, ) equation were calculated in [12]. In [11], the authors employed the Lie symmetry method and double reduction theory to obtain some exact solutions and derived the conservation laws for (1) via the so-called “partial Noether approach” aſter increasing its order. Noting that the GRLW equation is a third-order partial differential equation and it is difficult to obtain a variational formulation/Lagrangian for (1) by a classical approach, the authors converted the equation to the following fourth-order equation by assuming new dependent variable V to be the derivative of original dependent variable by setting = V as follows: V + V +V V +V = 0. (2) e authors claimed that the conserved vectors constructed by partial Noether’s theorem failed to satisfy the divergence relation and need to be adjusted to satisfy the divergence relationship. For the “partial Noether approach,” in [13] Sarlet had given some negative comments (see Section 4 in [13]). So we will give some conservation laws of (1) by other methods. In this paper, we investigate the Lagrangian and the self- adjointness and construct some new conservation laws for the generalized regularized long-wave equation by Ibragimov’s theorem [14]. We should point out that the exact Lagrangian of (1) obtained in Section 2 is a nonlocal Lagrangian with an auxiliary function Ψ. In Section 3, we show that (1) is strictly self-adjoint, (2) is nonlinearly self-adjoint, and the minimal order of the differential substitution of (2) is equal to one. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 173192, 5 pages http://dx.doi.org/10.1155/2014/173192
Transcript of Research Article The Lagrangian, Self-Adjointness, and...
Research ArticleThe Lagrangian Self-Adjointness and Conserved Quantities fora Generalized Regularized Long-Wave Equation
Long Wei and Yang Wang
Department of Mathematics Hangzhou Dianzi University Zhejiang 310018 China
Correspondence should be addressed to Long Wei alongweigmailcom
Received 9 January 2014 Accepted 2 May 2014 Published 8 May 2014
Academic Editor Tiecheng Xia
Copyright copy 2014 L Wei and Y WangThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We consider the Lagrangian and the self-adjointness of a generalized regularized long-wave equation and its transformed equationWe show that the third-order equation has a nonlocal Lagrangian with an auxiliary function and is strictly self-adjoint itstransformed equation is nonlinearly self-adjoint and the minimal order of the differential substitution is equal to one Then byIbragimovrsquos theorem on conservation laws we obtain some conserved qualities of the generalized regularized long-wave equation
1 Introduction
The generalized regularized long-wave (GRLW) equation
119906119905+ 119906119909+ 119886119906119898119906119909+ 119887119906119909119909119905= 0 (1)
was first put forward as a model for small-amplitude longwaves on the surface of water in a channel by Peregrine [1 2]where 119898 is a positive integer and 119886 and 119887 are two constantsIt is an alternative description of nonlinear dispersive wavesto the Korteweg de Vries equation [1] and has been used todescribe phenomena with weak nonlinearity and dispersionwaves including nonlinear transverse waves in shallowwaterion-acoustic and magnetohydrodynamic waves in plasmaand phonon packets in nonlinear crystals pressure waves inliquid-gas bubble mixture rotating flow down a tube andlossless propagation of shallow water waves see [3] and thereferences therein
Equation (1) has been studied extensively in the pastboth analytically and numerically by means of for examplethe Adomian decomposition method the Fourier methodthe finite difference method the finite element methodthe variational iteration method the mesh-free methodthe cubic B-spline collocation procedure the sine-cosinemethod and the double reduction method [4ndash11] and soforth A few of the conserved quantities by using the 1-soliton solution and solitary-wave solution of the 119877(119898 119899)equationwere calculated in [12] In [11] the authors employed
the Lie symmetry method and double reduction theory toobtain some exact solutions and derived the conservationlaws for (1) via the so-called ldquopartial Noether approachrdquo afterincreasing its order Noting that the GRLW equation is athird-order partial differential equation and it is difficultto obtain a variational formulationLagrangian for (1) by aclassical approach the authors converted the equation to thefollowing fourth-order equation by assuming new dependentvariable V to be the derivative of original dependent variable119906 by setting 119906 = V
119909as follows
V119909119905+ V119909119909+ 119886V119898119909V119909119909+ 119887V119909119909119909119905
= 0 (2)
The authors claimed that the conserved vectors constructedby partial Noetherrsquos theorem failed to satisfy the divergencerelation and need to be adjusted to satisfy the divergencerelationship For the ldquopartial Noether approachrdquo in [13] Sarlethad given some negative comments (see Section 4 in [13]) Sowe will give some conservation laws of (1) by other methods
In this paper we investigate the Lagrangian and the self-adjointness and construct somenew conservation laws for thegeneralized regularized long-wave equation by Ibragimovrsquostheorem [14] We should point out that the exact Lagrangianof (1) obtained in Section 2 is a nonlocal Lagrangian with anauxiliary function Ψ In Section 3 we show that (1) is strictlyself-adjoint (2) is nonlinearly self-adjoint and the minimalorder of the differential substitution of (2) is equal to one
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 173192 5 pageshttpdxdoiorg1011552014173192
2 Abstract and Applied Analysis
Based on these facts some new conservation laws for theGRLW equation (1) are derived
2 The Auxiliary Lagrangian
In this section we will study the Lagrangian for the gener-alized regularized long-wave equation (1) First we brieflypresent some notations to be used in what follows Let 119909 =(1199091 1199092 119909
119899) be 119899 independent variables let 119906 = 119906(119909) be
dependent variables let
119863119894=
120597
120597119909119894
+ 119906119894
120597
120597119906
+ 119906119894119895
120597
120597119906119894
+ sdot sdot sdot (3)
be the total differentiation operator and let
120575
120575119906
=
120597
120597119906
+ sum
119898ge1
(minus1)1198981198631198941
sdot sdot sdot 119863119894119898
120597
1205971199061198941sdotsdotsdot119894119898
(4)
denote the Euler-Lagrange operatorIt is well known that a partial differential equation with
odd order does not often admit a Lagrangian Howevernoting the structure of (1) we can rewrite the equation as
[119906 + 119887119906119909119909]119905+ [119906 +
119886
119898 + 1
119906119898+1]
119909
= 0 (5)
thus we find an auxiliary Lagrangian of (1) by the semi-inverse method [15] Consequently some conserved quanti-ties of (1) can be derived from Noetherrsquos theorem
According to (5) we introduce an auxiliary function Ψdefined as
Ψ119909= 119906 + 119887119906
119909119909 (6)
Ψ119905= minus(119906 +
119886
119898 + 1
119906119898+1) (7)
so that (5) is automatically satisfiedBy the semi-inverse method [15] we construct a trial
Lagrangian in the form
119871 (119906 Ψ) = (119906 + 119887119906119909119909) Ψ119905minus (119906 +
119886
119898 + 1
119906119898+1)Ψ119909+ 119865 (8)
where119865 is an unknown function to be determinedTheEuler-Lagrange equations of (8) are
minus119863119905(119906 + 119887119906
119909119909) + 119863119909(119906 +
119886
119898 + 1
119906119898+1) +
120575119865
120575Ψ
= 0
Ψ119905+ 1198871198632
119909Ψ119905minus Ψ119909(1 + 119886119906
119898) +
120575119865
120575119906
= 0
(9)
Substituting (6) and (7) into the above equations we obtainthat
120575119865
120575Ψ
= 2Ψ119909119905
120575119865
120575119906
= 2119906 +
119886 (119898 + 2)
119898 + 1
119906119898+1
+ 2119887 (1 + 119886119906119898) 119906119909119909
+ 119886119887119898119906119898minus11199062
119909
(10)
Thus we can take 119865 as
119865 = minusΨ119909Ψ119905+ 1199062+
119886
119898 + 1
119906119898+2
minus 119887 (1 + 119886119906119898) 1199062
119909 (11)
which implies that the Lagrangian of (1) reads
119871 = (119906 + 119887119906119909119909) Ψ119905minus (119906 +
119886119906119898+1
119898 + 1
)Ψ119909minus Ψ119909Ψ119905
+ 1199062(1 +
119886119906119898
119898 + 1
) minus 119887 (1 + 119886119906119898) 1199062
119909
(12)
which is subjected to (6) Therefore we demonstrate thefollowing result
Theorem 1 Equation (1) admits a Lagrangian as follows
119871 = (119906 + 119887119906119909119909) Ψ119905minus (119906 +
119886119906119898+1
119898 + 1
)Ψ119909minus Ψ119909Ψ119905
+ 1199062(1 +
119886119906119898
119898 + 1
) minus 119887 (1 + 119886119906119898) 1199062
119909
(13)
with constraint Ψ119909= 119906 + 119887119906
119909119909
Proof A direct calculation yields the Euler-Lagrangian equa-tions of 119871(119906 Ψ) as follows
minus119863119905(119906 + 119887119906
119909119909) + 119863119909(119906 +
119886
119898 + 1
119906119898+1) + 2Ψ
119909119905= 0 (14)
Ψ119905minus (1 + 119886119906
119898) Ψ119909+ 2119906 +
119886 (119898 + 2)
119898 + 1
119906119898+1
+ 119898119886119887119906119898minus11199062
119909+ 119887Ψ119905119909119909
+ 2119887 (1 + 119886119906119898) 119906119909119909= 0
(15)
Considering the constraint equation (6) it is obvious that (14)is equivalent to (5) and (15) can be rewritten in the form of
(Ψ119905+ 119906 +
119886
119898 + 1
119906119898+1) + 119887(Ψ
119905+ 119906 +
119886
119898 + 1
119906119898+1)
119909119909
= 0
(16)
Solving the last equation we obtain that
Ψ119905= minus119906 minus
119886
119898 + 1
119906119898+1
+ 119875 (17)
where 119875 = 119875(119909 119905) satisfies
119887119875119909119909+ 119875 = 0 (18)
Noting that Ψ119909= 119906 + 119887119906
119909119909 we deduce from Ψ
119909119905= Ψ119905119909that
119906119905+ 119887119906119909119909119905+ 119906119909+ 119886119906119898119906119909minus 119875119909= 0 (19)
which combining with (5) yields 119875119909= 0 Thus (18) implies
119875 = 0 and (17) implies that (7) holds
Remark 2 The Lagrangian (13) of (1) with constraint (6)implies that auxiliary equation (7) holds
Abstract and Applied Analysis 3
3 Self-Adjointness and Conservation Laws
In this section we will investigate the self-adjointness andconservation laws of (1) by Ibragimovrsquos theorem
31 Self-Adjointness for (1) and (2) Let
119867 = 119906119905+ 119906119909+ 119886119906119898119906119909+ 119887119906119909119909119905 (20)
then we have the following formal Lagrangian for (1)
L1= 119908 (119906
119905+ 119906119909+ 119886119906119898119906119909+ 119887119906119909119909119905) (21)
where 119908 = 119908(119909 119905) is a new function Computing the vari-ational derivative of this formal Lagrangian we obtain thefollowing adjoint equation of (1)
119867lowast= minus119908119905minus 119908119909minus 119886119908119909119906119898minus 119887119908119909119909119905= 0 (22)
Assume that119867lowast|119908=120601(119905119909119906)
= 120582119867 for a certain function 120582Then it is easy to obtain that 120601 = 119888
1119906 + 1198882and 120582 = minus120601
119906= minus1198881
where 1198881 1198882are two arbitrary constantsThis implies that (1) is
quasiself-adjoint especially if 1198881= 1 and 119888
2= 0 that is 120601 = 119906
then (1) is strictly self-adjoint (see [16] for the definitions)Thus we have demonstrated the following statement
Theorem3 Equation (1) is quasiself-adjoint with the substitu-tion 119908 = 119888
1119906 + 1198882 where 119888
1= 0 1198882are two arbitrary constants
Especially (1) is strictly self-adjoint with the substitution119908 = 119906
Next we discuss the self-adjointness of (2) and then bythe nonlinear self-adjointness and Ibragimovrsquos theorem onconservation laws we construct some conserved quantities of(1)
Similar to the above we set
119864 = V119909119905+ V119909119909+ 119886V119898119909V119909119909+ 119887V119909119909119909119905
(23)
then the formal Lagrangian for (2) is
L2= 119908 (V
119909119905+ V119909119909+ 119886V119898119909V119909119909+ 119887V119909119909119909119905
) (24)
and the adjoint equation of (1) is
119864lowast= 119908119909119905+ 119908119909119909+ 119886119908119909119909V119898119909+ 119886119898119908
119909V119898minus1119909
V119909119909+ 119887119908119909119909119909119905
= 0
(25)
According to [16] (2) will be nonlinearly self-adjoint ifthere exists a differential substitution
119908 = ℎ (119909 119905 V V119909 V119905 V119909119909 ) ℎ = 0 (26)
involving a finite number of partial derivatives of V withrespect to 119909 and 119905 such that the equation
119864lowast1003816100381610038161003816119908=ℎ
= 1205830119864 + 1205831119863119909119864 + 1205832119863119905119864 + 12058331198632
119909119864 + sdot sdot sdot (27)
holds identically in the variables 119905 119909 119906 119906119909 119906119905 where
1205830 1205831 are undetermined variable coefficients different
from infin on the solutions of (2) The highest order ofderivatives involved in 119906 is called the order of the differentialsubstitution (26) The calculation provides the followingresult
Theorem 4 Equation (2) is nonlinearly self-adjoint Theminimal order of the differential substitution (26) satisfying(27) is equal to one and is given by the function ℎ = 119888
1V119909+
1198882V119905+ 119892(119905) where 119888
1and 1198882are two constants and 119892(119905) is a
differentiable function of 119905
32 Conservation Laws of (1) Now we study the conserva-tion laws of (1) We will first construct the conservation lawsof (2) and then reduce them to the ones of (1) From theclassical Lie group theory [17] we assume that a Lie pointsymmetry of (2) is a vector field
119883 = 120585119909(119905 119909 V)
120597
120597119909
+ 120585119905(119905 119909 V)
120597
120597119905
+ 120588 (119905 119909 V)120597
120597V(28)
on R+ times R times R such that 119883(4)119864 = 0 when 119864 = 0 where 119864is given by (23) Taking into account (2) the operator 119883(4) isgiven as follows
119883(4)= 119883 + 120588
(1)
119909
120597
120597V119909
+ 120588(2)
119909119905
120597
120597V119909119905
+ 120588(2)
119909119909
120597
120597V119909119909
+ 120588(4)
119909119909119909119905
120597
120597V119909119909119909119905
(29)where
120588(1)
119909= 119863119909120588 minus (119863
119909120585119909) V119909minus (119863119909120585119905) V119905
120588(2)
119909119905= 119863119905120588(1)
119909minus (119863119905120585119909) V119909119909minus (119863119905120585119905) V119909119905
120588(2)
119909119909= 119863119909(120588(1)
119909) minus (119863
119909120585119909) V119909119909minus (119863119909120585119905) V119909119905
120588(4)
119909119909119909119905= 119863119909(120588(3)
119909119909119909) minus (119863
119909120585119909) V119909119909119909119909
minus (119863119909120585119905) V119909119909119909119905
(30)
The condition 119883(4)119864|119864=0
= 0 yields determining equationsSolving these determining equations we can obtain thesymmetries of (5) as follows
1198831=
120597
120597119909
1198832=
120597
120597119905
1198833= 119891 (119905)
120597
120597V(31)
with 119891(119905) as an arbitrary function Ibragimovrsquos theorem onconservation laws yields the following conserved quantitiesof (2) for a general Lie symmetry 119883 = 120585
119909(120597120597119909) + 120585
119905(120597120597119905) +
120588(120597120597V) Consider the following
119862119905= 120585119905L2minus119882(
119908119909
2
+
119887119908119909119909119909
4
) +119882119909(
119908
2
+
119887119908119909119909
4
)
minus119882119909119909(
119887119908119909
4
) +119882119909119909119909(
119887119908
4
)
119862119909= 120585119909L2+119882(119886119898V119898minus1
119909V119909119909119908 minus 119863
119909[119908 (1 + 119886V119898
119909)]
minus
119908119905
2
minus
3119887119908119909119909119905
4
)
+119882119909((1 + 119886V119898
119909) 119908 +
119887119908119909119905
2
) +119882119905(
119908
2
+
119887119908119909119909
4
)
minus119882119909119909(
119887119908119905
4
)
minus119882119909119905
119887119908119909
2
+119882119909119909119905(
3119887119908
4
)
(32)
4 Abstract and Applied Analysis
where119882 = 120588 minus 120585119909V119909minus 120585119905V119905is the Lie characteristic function
and119908 is given by the differential substitution119908 = 1198881V119909+1198882V119905+
119892(119905) Let us construct the conserved vector corresponding tothe space translation group with the generator
1198831=
120597
120597119909
(33)
In this case 119882 = minusV119909 Taking account of the differential
substitution 119908 = 1198881V119909+ 1198882V119905+ 119892(119905) and V
119909= 119906 from
(32) we obtain the conserved quantities of (2) which can betransformed to the ones of (1) as follows
119862119905
1=
1
4
1198882(2119906119906119905+ 119887119906119906
119909119909119905minus 2119906119909int119906119905119889119909
minus 119887119906119909119906119909119905+ 119887119906119905119906119909119909minus 119887119906119909119909119909int119906119905119889119909)
minus
1
4
119887119892 (119905) 119906119909119909119909
minus
1
2
119892 (119905) 119906119909
=
1
2
1198882(2119906119906119905+ 119887119906119906
119909119909119905+ 119887119906119905119906119909119909)
minus
1
4
1198882119863119909(119887119906119906119909119905+ 2119906int119906
119905119889119909 + 119887119906
119909119909int119906119905119889119909)
minus
1
4
119863119909[119887119892 (119905) 119906
119909119909+ 2119892 (119905) 119906]
119862119909
1=
1
4
1198882(3119887119906119906
119909119905119905minus 2119887119906
119909119906119905119905+ 119887119906119905119906119909119905+ 119887119906119909119909int119906119905119905119889119909
+ 2119906int119906119905119905119889119909 + 4119886119906
119898+1119906119905
+4119906119906119905+ 2119906119905int119906119905119889119909 + 119887119906
119909119909119905int119906119905119889119909)
+
119887
4
[119892 (119905) 119906119909119909119905+ 1198921015840(119905) 119906119909119909]
+ 1198881119906 (119906119909+ 119906119905+ 119886119906119898119906119909+ 119887119906119909119909119905) +
1
2
[1199061198921015840(119905) + 119892 (119905) 119906
119905]
= 1198881119906 (119906119909+ 119906119905+ 119886119906119898119906119909+ 119887119906119909119909119905)
+
1
2
1198882(119887119906119906119909119905119905minus 119887119906119909119906119905119905+ 2119886119906
119898+1119906119905+ 2119906119906
119905)
+
1
4
1198882119863119905(119887119906119906119909119905+ 2119906int119906
119905119889119909 + 119887119906
119909119909int119906119905119889119909)
+
1
4
119863119905[119887119892 (119905) 119906
119909119909+ 2119892 (119905) 119906]
(34)
which can be reduced to the form
119862119905
1=
1
2
1198882(2119906119906119905+ 119887119906119906
119909119909119905+ 119887119906119905119906119909119909)
119862119909
1=
1198882
2
(119887119906119906119909119905119905minus 119887119906119909119906119905119905+ 2119886119906
119898+1119906119905+ 2119906119906
119905)
(35)
Here we have used (1)
By the procedure analogous to that used above weobtain the conserved quantities of (2) corresponding tothe generators 119883
2and 119883
3 which can be reduced by the
differential substitutions119908 = 1198881V119909+ 1198882V119905+ 119892(119905) and V
119909= 119906 to
the conserved quantities of (1) as follows
119862119905
2=
1198881119887
2
(119906119906119909119909119905minus 119906119905119906119909119909) + 119886119892 (119905) 119906
119898119906119909
119862119909
2=
1198881119887
2
(119906119909119906119905119905minus 119906119906119909119905119905) minus 119886119892 (119905) 119906
119898119906119905
+ 1198921015840(119905) (int 119906
119905119889119909 + 119906 + 119887119906
119909119905)
119862119905
3= 119891 (119905) (119888
1119906119909+ 1198882119906119905)
119862119909
3= 119891 (119905) (119888
1119906119909+ 1198882119906119905) (1 + 119886119906
119898)
+ 119887119891 (119905) (1198881119906119909119909119905+ 1198882119906119909119905119905)
minus 1198911015840(119905) (1198881119906 + 1198882int119906119905119889119909 + 119892 (119905))
(36)
Here we have used (1)
Remark 5 In fact (2) admits a Lagrangian
1198712= minus
1
2
V119909V119905minus
1
2
V2119909minus
119886
(119898 + 1) (119898 + 2)
V119898+2119909
+
119887
2
V119909119905V119909119909
(37)
and has the following Noether symmetry generators
1198835=
120597
120597119909
1198836=
120597
120597119905
1198837= 119891 (119905)
120597
120597Vwith gangue function 119860119905 = 0
119860119909= minus
1
2
1198911015840(119905) V
(38)
where119891(119905) is a differentiable function in 119905 Noetherrsquos theoremon conservation laws [18] can yield the conserved quantitiesof (2) which are the same as the ones in [11] Those wereconstructed by the so-called ldquopartial Noether approachrdquo
4 Conclusions
In this paper we investigate the Lagrangians and self-adjointnesses of the GRLW equation (1) and its transformedequation (2) We show that (1) has a nonlocal Lagrangianwith an auxiliary function and is strictly self-adjoint itstransformed equation is nonlinearly self-adjoint and theminimal order of the differential substitution is equal to oneThen by Ibragimovrsquos theorem on conservation laws we obtainsome conserved qualities of the generalized regularized long-wave equation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 5
Acknowledgments
The first author is partly supported by Zhejiang Provin-cial Natural Science Foundation of China under Grantno LY12A01003 and Subjects Research and DevelopmentFoundation of Hangzhou Dianzi University under Grantno ZX100204004-6 The second author is partly supportedby NSFC under Grant no 11101111 and Zhejiang Provin-cial Natural Science Foundation of China under Grant noLY14A010029
References
[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966
[2] D H Peregrine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967
[3] M Dehghan and R Salehi ldquoThe solitary wave solution of thetwo-dimensional regularized long-wave equation in fluids andplasmasrdquoComputer Physics Communications vol 182 no 12 pp2540ndash2549 2011
[4] J C Eilbeck and G R McGuire ldquoNumerical study of the reg-ularized long-wave equation II interaction of solitary wavesrdquoJournal of Computational Physics vol 23 no 1 pp 63ndash73 1977
[5] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979
[6] Q Chang ldquoConservative scheme for a model of nonlineardispersive waves and its solitary waves induced by boundarymotionrdquo Journal of Computational Physics vol 93 no 2 pp360ndash375 1991
[7] D Bhardwaj and R Shankar ldquoComputational method forregularized long wave equationrdquo Computers and Mathematicswith Applications vol 40 no 12 pp 1397ndash1404 2000
[8] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004
[9] A-M Wazwaz ldquoA sine-cosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004
[10] A Mohebbi ldquoSolitary wave solutions of the nonlinear general-ized Pochhammer-Chree and regularized long wave equationsrdquoNonlinear Dynamics vol 70 no 4 pp 2463ndash2474 2012
[11] R Naz M D Khan and I Naeem ldquoConservation laws andexact solutions of a class of non linear regularized long waveequations via double reduction theory and Lie symmetriesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 4 pp 826ndash834 2013
[12] A Biswas ldquoSolitary waves for power-law regularized long-waveequation and 119877(119898 119899) equationrdquo Nonlinear Dynamics vol 59no 3 pp 423ndash426 2010
[13] W Sarlet ldquoComment on lsquoconservation laws of higher ordernonlinear PDEs and the variational conservation laws in theclass withmixed derivativesrsquordquo Journal of Physics AMathematicaland Theoretical vol 43 no 45 Article ID 458001 2010
[14] N H Ibragimov ldquoA new conservation theoremrdquo Journal ofMathematical Analysis and Applications vol 333 no 1 pp 311ndash328 2007
[15] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsand Fractals vol 19 no 4 pp 847ndash851 2004
[16] N H Ibragimov ldquoNonlinear self-adjointness and conservationlawsrdquo Journal of Physics A Mathematical and Theoretical vol44 no 43 Article ID 432002 2011
[17] P J Olver Application of Lie Groups To Differential EquationsSpringer Berlin Germany 1986
[18] E Noether ldquoInvariante variationsproblemerdquo Nachrichten vonder Gesellschaft derWissenschaften zuGottingenMathematisch-Physikalische Klasse vol 2 pp 235ndash257 1918
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
Based on these facts some new conservation laws for theGRLW equation (1) are derived
2 The Auxiliary Lagrangian
In this section we will study the Lagrangian for the gener-alized regularized long-wave equation (1) First we brieflypresent some notations to be used in what follows Let 119909 =(1199091 1199092 119909
119899) be 119899 independent variables let 119906 = 119906(119909) be
dependent variables let
119863119894=
120597
120597119909119894
+ 119906119894
120597
120597119906
+ 119906119894119895
120597
120597119906119894
+ sdot sdot sdot (3)
be the total differentiation operator and let
120575
120575119906
=
120597
120597119906
+ sum
119898ge1
(minus1)1198981198631198941
sdot sdot sdot 119863119894119898
120597
1205971199061198941sdotsdotsdot119894119898
(4)
denote the Euler-Lagrange operatorIt is well known that a partial differential equation with
odd order does not often admit a Lagrangian Howevernoting the structure of (1) we can rewrite the equation as
[119906 + 119887119906119909119909]119905+ [119906 +
119886
119898 + 1
119906119898+1]
119909
= 0 (5)
thus we find an auxiliary Lagrangian of (1) by the semi-inverse method [15] Consequently some conserved quanti-ties of (1) can be derived from Noetherrsquos theorem
According to (5) we introduce an auxiliary function Ψdefined as
Ψ119909= 119906 + 119887119906
119909119909 (6)
Ψ119905= minus(119906 +
119886
119898 + 1
119906119898+1) (7)
so that (5) is automatically satisfiedBy the semi-inverse method [15] we construct a trial
Lagrangian in the form
119871 (119906 Ψ) = (119906 + 119887119906119909119909) Ψ119905minus (119906 +
119886
119898 + 1
119906119898+1)Ψ119909+ 119865 (8)
where119865 is an unknown function to be determinedTheEuler-Lagrange equations of (8) are
minus119863119905(119906 + 119887119906
119909119909) + 119863119909(119906 +
119886
119898 + 1
119906119898+1) +
120575119865
120575Ψ
= 0
Ψ119905+ 1198871198632
119909Ψ119905minus Ψ119909(1 + 119886119906
119898) +
120575119865
120575119906
= 0
(9)
Substituting (6) and (7) into the above equations we obtainthat
120575119865
120575Ψ
= 2Ψ119909119905
120575119865
120575119906
= 2119906 +
119886 (119898 + 2)
119898 + 1
119906119898+1
+ 2119887 (1 + 119886119906119898) 119906119909119909
+ 119886119887119898119906119898minus11199062
119909
(10)
Thus we can take 119865 as
119865 = minusΨ119909Ψ119905+ 1199062+
119886
119898 + 1
119906119898+2
minus 119887 (1 + 119886119906119898) 1199062
119909 (11)
which implies that the Lagrangian of (1) reads
119871 = (119906 + 119887119906119909119909) Ψ119905minus (119906 +
119886119906119898+1
119898 + 1
)Ψ119909minus Ψ119909Ψ119905
+ 1199062(1 +
119886119906119898
119898 + 1
) minus 119887 (1 + 119886119906119898) 1199062
119909
(12)
which is subjected to (6) Therefore we demonstrate thefollowing result
Theorem 1 Equation (1) admits a Lagrangian as follows
119871 = (119906 + 119887119906119909119909) Ψ119905minus (119906 +
119886119906119898+1
119898 + 1
)Ψ119909minus Ψ119909Ψ119905
+ 1199062(1 +
119886119906119898
119898 + 1
) minus 119887 (1 + 119886119906119898) 1199062
119909
(13)
with constraint Ψ119909= 119906 + 119887119906
119909119909
Proof A direct calculation yields the Euler-Lagrangian equa-tions of 119871(119906 Ψ) as follows
minus119863119905(119906 + 119887119906
119909119909) + 119863119909(119906 +
119886
119898 + 1
119906119898+1) + 2Ψ
119909119905= 0 (14)
Ψ119905minus (1 + 119886119906
119898) Ψ119909+ 2119906 +
119886 (119898 + 2)
119898 + 1
119906119898+1
+ 119898119886119887119906119898minus11199062
119909+ 119887Ψ119905119909119909
+ 2119887 (1 + 119886119906119898) 119906119909119909= 0
(15)
Considering the constraint equation (6) it is obvious that (14)is equivalent to (5) and (15) can be rewritten in the form of
(Ψ119905+ 119906 +
119886
119898 + 1
119906119898+1) + 119887(Ψ
119905+ 119906 +
119886
119898 + 1
119906119898+1)
119909119909
= 0
(16)
Solving the last equation we obtain that
Ψ119905= minus119906 minus
119886
119898 + 1
119906119898+1
+ 119875 (17)
where 119875 = 119875(119909 119905) satisfies
119887119875119909119909+ 119875 = 0 (18)
Noting that Ψ119909= 119906 + 119887119906
119909119909 we deduce from Ψ
119909119905= Ψ119905119909that
119906119905+ 119887119906119909119909119905+ 119906119909+ 119886119906119898119906119909minus 119875119909= 0 (19)
which combining with (5) yields 119875119909= 0 Thus (18) implies
119875 = 0 and (17) implies that (7) holds
Remark 2 The Lagrangian (13) of (1) with constraint (6)implies that auxiliary equation (7) holds
Abstract and Applied Analysis 3
3 Self-Adjointness and Conservation Laws
In this section we will investigate the self-adjointness andconservation laws of (1) by Ibragimovrsquos theorem
31 Self-Adjointness for (1) and (2) Let
119867 = 119906119905+ 119906119909+ 119886119906119898119906119909+ 119887119906119909119909119905 (20)
then we have the following formal Lagrangian for (1)
L1= 119908 (119906
119905+ 119906119909+ 119886119906119898119906119909+ 119887119906119909119909119905) (21)
where 119908 = 119908(119909 119905) is a new function Computing the vari-ational derivative of this formal Lagrangian we obtain thefollowing adjoint equation of (1)
119867lowast= minus119908119905minus 119908119909minus 119886119908119909119906119898minus 119887119908119909119909119905= 0 (22)
Assume that119867lowast|119908=120601(119905119909119906)
= 120582119867 for a certain function 120582Then it is easy to obtain that 120601 = 119888
1119906 + 1198882and 120582 = minus120601
119906= minus1198881
where 1198881 1198882are two arbitrary constantsThis implies that (1) is
quasiself-adjoint especially if 1198881= 1 and 119888
2= 0 that is 120601 = 119906
then (1) is strictly self-adjoint (see [16] for the definitions)Thus we have demonstrated the following statement
Theorem3 Equation (1) is quasiself-adjoint with the substitu-tion 119908 = 119888
1119906 + 1198882 where 119888
1= 0 1198882are two arbitrary constants
Especially (1) is strictly self-adjoint with the substitution119908 = 119906
Next we discuss the self-adjointness of (2) and then bythe nonlinear self-adjointness and Ibragimovrsquos theorem onconservation laws we construct some conserved quantities of(1)
Similar to the above we set
119864 = V119909119905+ V119909119909+ 119886V119898119909V119909119909+ 119887V119909119909119909119905
(23)
then the formal Lagrangian for (2) is
L2= 119908 (V
119909119905+ V119909119909+ 119886V119898119909V119909119909+ 119887V119909119909119909119905
) (24)
and the adjoint equation of (1) is
119864lowast= 119908119909119905+ 119908119909119909+ 119886119908119909119909V119898119909+ 119886119898119908
119909V119898minus1119909
V119909119909+ 119887119908119909119909119909119905
= 0
(25)
According to [16] (2) will be nonlinearly self-adjoint ifthere exists a differential substitution
119908 = ℎ (119909 119905 V V119909 V119905 V119909119909 ) ℎ = 0 (26)
involving a finite number of partial derivatives of V withrespect to 119909 and 119905 such that the equation
119864lowast1003816100381610038161003816119908=ℎ
= 1205830119864 + 1205831119863119909119864 + 1205832119863119905119864 + 12058331198632
119909119864 + sdot sdot sdot (27)
holds identically in the variables 119905 119909 119906 119906119909 119906119905 where
1205830 1205831 are undetermined variable coefficients different
from infin on the solutions of (2) The highest order ofderivatives involved in 119906 is called the order of the differentialsubstitution (26) The calculation provides the followingresult
Theorem 4 Equation (2) is nonlinearly self-adjoint Theminimal order of the differential substitution (26) satisfying(27) is equal to one and is given by the function ℎ = 119888
1V119909+
1198882V119905+ 119892(119905) where 119888
1and 1198882are two constants and 119892(119905) is a
differentiable function of 119905
32 Conservation Laws of (1) Now we study the conserva-tion laws of (1) We will first construct the conservation lawsof (2) and then reduce them to the ones of (1) From theclassical Lie group theory [17] we assume that a Lie pointsymmetry of (2) is a vector field
119883 = 120585119909(119905 119909 V)
120597
120597119909
+ 120585119905(119905 119909 V)
120597
120597119905
+ 120588 (119905 119909 V)120597
120597V(28)
on R+ times R times R such that 119883(4)119864 = 0 when 119864 = 0 where 119864is given by (23) Taking into account (2) the operator 119883(4) isgiven as follows
119883(4)= 119883 + 120588
(1)
119909
120597
120597V119909
+ 120588(2)
119909119905
120597
120597V119909119905
+ 120588(2)
119909119909
120597
120597V119909119909
+ 120588(4)
119909119909119909119905
120597
120597V119909119909119909119905
(29)where
120588(1)
119909= 119863119909120588 minus (119863
119909120585119909) V119909minus (119863119909120585119905) V119905
120588(2)
119909119905= 119863119905120588(1)
119909minus (119863119905120585119909) V119909119909minus (119863119905120585119905) V119909119905
120588(2)
119909119909= 119863119909(120588(1)
119909) minus (119863
119909120585119909) V119909119909minus (119863119909120585119905) V119909119905
120588(4)
119909119909119909119905= 119863119909(120588(3)
119909119909119909) minus (119863
119909120585119909) V119909119909119909119909
minus (119863119909120585119905) V119909119909119909119905
(30)
The condition 119883(4)119864|119864=0
= 0 yields determining equationsSolving these determining equations we can obtain thesymmetries of (5) as follows
1198831=
120597
120597119909
1198832=
120597
120597119905
1198833= 119891 (119905)
120597
120597V(31)
with 119891(119905) as an arbitrary function Ibragimovrsquos theorem onconservation laws yields the following conserved quantitiesof (2) for a general Lie symmetry 119883 = 120585
119909(120597120597119909) + 120585
119905(120597120597119905) +
120588(120597120597V) Consider the following
119862119905= 120585119905L2minus119882(
119908119909
2
+
119887119908119909119909119909
4
) +119882119909(
119908
2
+
119887119908119909119909
4
)
minus119882119909119909(
119887119908119909
4
) +119882119909119909119909(
119887119908
4
)
119862119909= 120585119909L2+119882(119886119898V119898minus1
119909V119909119909119908 minus 119863
119909[119908 (1 + 119886V119898
119909)]
minus
119908119905
2
minus
3119887119908119909119909119905
4
)
+119882119909((1 + 119886V119898
119909) 119908 +
119887119908119909119905
2
) +119882119905(
119908
2
+
119887119908119909119909
4
)
minus119882119909119909(
119887119908119905
4
)
minus119882119909119905
119887119908119909
2
+119882119909119909119905(
3119887119908
4
)
(32)
4 Abstract and Applied Analysis
where119882 = 120588 minus 120585119909V119909minus 120585119905V119905is the Lie characteristic function
and119908 is given by the differential substitution119908 = 1198881V119909+1198882V119905+
119892(119905) Let us construct the conserved vector corresponding tothe space translation group with the generator
1198831=
120597
120597119909
(33)
In this case 119882 = minusV119909 Taking account of the differential
substitution 119908 = 1198881V119909+ 1198882V119905+ 119892(119905) and V
119909= 119906 from
(32) we obtain the conserved quantities of (2) which can betransformed to the ones of (1) as follows
119862119905
1=
1
4
1198882(2119906119906119905+ 119887119906119906
119909119909119905minus 2119906119909int119906119905119889119909
minus 119887119906119909119906119909119905+ 119887119906119905119906119909119909minus 119887119906119909119909119909int119906119905119889119909)
minus
1
4
119887119892 (119905) 119906119909119909119909
minus
1
2
119892 (119905) 119906119909
=
1
2
1198882(2119906119906119905+ 119887119906119906
119909119909119905+ 119887119906119905119906119909119909)
minus
1
4
1198882119863119909(119887119906119906119909119905+ 2119906int119906
119905119889119909 + 119887119906
119909119909int119906119905119889119909)
minus
1
4
119863119909[119887119892 (119905) 119906
119909119909+ 2119892 (119905) 119906]
119862119909
1=
1
4
1198882(3119887119906119906
119909119905119905minus 2119887119906
119909119906119905119905+ 119887119906119905119906119909119905+ 119887119906119909119909int119906119905119905119889119909
+ 2119906int119906119905119905119889119909 + 4119886119906
119898+1119906119905
+4119906119906119905+ 2119906119905int119906119905119889119909 + 119887119906
119909119909119905int119906119905119889119909)
+
119887
4
[119892 (119905) 119906119909119909119905+ 1198921015840(119905) 119906119909119909]
+ 1198881119906 (119906119909+ 119906119905+ 119886119906119898119906119909+ 119887119906119909119909119905) +
1
2
[1199061198921015840(119905) + 119892 (119905) 119906
119905]
= 1198881119906 (119906119909+ 119906119905+ 119886119906119898119906119909+ 119887119906119909119909119905)
+
1
2
1198882(119887119906119906119909119905119905minus 119887119906119909119906119905119905+ 2119886119906
119898+1119906119905+ 2119906119906
119905)
+
1
4
1198882119863119905(119887119906119906119909119905+ 2119906int119906
119905119889119909 + 119887119906
119909119909int119906119905119889119909)
+
1
4
119863119905[119887119892 (119905) 119906
119909119909+ 2119892 (119905) 119906]
(34)
which can be reduced to the form
119862119905
1=
1
2
1198882(2119906119906119905+ 119887119906119906
119909119909119905+ 119887119906119905119906119909119909)
119862119909
1=
1198882
2
(119887119906119906119909119905119905minus 119887119906119909119906119905119905+ 2119886119906
119898+1119906119905+ 2119906119906
119905)
(35)
Here we have used (1)
By the procedure analogous to that used above weobtain the conserved quantities of (2) corresponding tothe generators 119883
2and 119883
3 which can be reduced by the
differential substitutions119908 = 1198881V119909+ 1198882V119905+ 119892(119905) and V
119909= 119906 to
the conserved quantities of (1) as follows
119862119905
2=
1198881119887
2
(119906119906119909119909119905minus 119906119905119906119909119909) + 119886119892 (119905) 119906
119898119906119909
119862119909
2=
1198881119887
2
(119906119909119906119905119905minus 119906119906119909119905119905) minus 119886119892 (119905) 119906
119898119906119905
+ 1198921015840(119905) (int 119906
119905119889119909 + 119906 + 119887119906
119909119905)
119862119905
3= 119891 (119905) (119888
1119906119909+ 1198882119906119905)
119862119909
3= 119891 (119905) (119888
1119906119909+ 1198882119906119905) (1 + 119886119906
119898)
+ 119887119891 (119905) (1198881119906119909119909119905+ 1198882119906119909119905119905)
minus 1198911015840(119905) (1198881119906 + 1198882int119906119905119889119909 + 119892 (119905))
(36)
Here we have used (1)
Remark 5 In fact (2) admits a Lagrangian
1198712= minus
1
2
V119909V119905minus
1
2
V2119909minus
119886
(119898 + 1) (119898 + 2)
V119898+2119909
+
119887
2
V119909119905V119909119909
(37)
and has the following Noether symmetry generators
1198835=
120597
120597119909
1198836=
120597
120597119905
1198837= 119891 (119905)
120597
120597Vwith gangue function 119860119905 = 0
119860119909= minus
1
2
1198911015840(119905) V
(38)
where119891(119905) is a differentiable function in 119905 Noetherrsquos theoremon conservation laws [18] can yield the conserved quantitiesof (2) which are the same as the ones in [11] Those wereconstructed by the so-called ldquopartial Noether approachrdquo
4 Conclusions
In this paper we investigate the Lagrangians and self-adjointnesses of the GRLW equation (1) and its transformedequation (2) We show that (1) has a nonlocal Lagrangianwith an auxiliary function and is strictly self-adjoint itstransformed equation is nonlinearly self-adjoint and theminimal order of the differential substitution is equal to oneThen by Ibragimovrsquos theorem on conservation laws we obtainsome conserved qualities of the generalized regularized long-wave equation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 5
Acknowledgments
The first author is partly supported by Zhejiang Provin-cial Natural Science Foundation of China under Grantno LY12A01003 and Subjects Research and DevelopmentFoundation of Hangzhou Dianzi University under Grantno ZX100204004-6 The second author is partly supportedby NSFC under Grant no 11101111 and Zhejiang Provin-cial Natural Science Foundation of China under Grant noLY14A010029
References
[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966
[2] D H Peregrine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967
[3] M Dehghan and R Salehi ldquoThe solitary wave solution of thetwo-dimensional regularized long-wave equation in fluids andplasmasrdquoComputer Physics Communications vol 182 no 12 pp2540ndash2549 2011
[4] J C Eilbeck and G R McGuire ldquoNumerical study of the reg-ularized long-wave equation II interaction of solitary wavesrdquoJournal of Computational Physics vol 23 no 1 pp 63ndash73 1977
[5] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979
[6] Q Chang ldquoConservative scheme for a model of nonlineardispersive waves and its solitary waves induced by boundarymotionrdquo Journal of Computational Physics vol 93 no 2 pp360ndash375 1991
[7] D Bhardwaj and R Shankar ldquoComputational method forregularized long wave equationrdquo Computers and Mathematicswith Applications vol 40 no 12 pp 1397ndash1404 2000
[8] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004
[9] A-M Wazwaz ldquoA sine-cosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004
[10] A Mohebbi ldquoSolitary wave solutions of the nonlinear general-ized Pochhammer-Chree and regularized long wave equationsrdquoNonlinear Dynamics vol 70 no 4 pp 2463ndash2474 2012
[11] R Naz M D Khan and I Naeem ldquoConservation laws andexact solutions of a class of non linear regularized long waveequations via double reduction theory and Lie symmetriesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 4 pp 826ndash834 2013
[12] A Biswas ldquoSolitary waves for power-law regularized long-waveequation and 119877(119898 119899) equationrdquo Nonlinear Dynamics vol 59no 3 pp 423ndash426 2010
[13] W Sarlet ldquoComment on lsquoconservation laws of higher ordernonlinear PDEs and the variational conservation laws in theclass withmixed derivativesrsquordquo Journal of Physics AMathematicaland Theoretical vol 43 no 45 Article ID 458001 2010
[14] N H Ibragimov ldquoA new conservation theoremrdquo Journal ofMathematical Analysis and Applications vol 333 no 1 pp 311ndash328 2007
[15] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsand Fractals vol 19 no 4 pp 847ndash851 2004
[16] N H Ibragimov ldquoNonlinear self-adjointness and conservationlawsrdquo Journal of Physics A Mathematical and Theoretical vol44 no 43 Article ID 432002 2011
[17] P J Olver Application of Lie Groups To Differential EquationsSpringer Berlin Germany 1986
[18] E Noether ldquoInvariante variationsproblemerdquo Nachrichten vonder Gesellschaft derWissenschaften zuGottingenMathematisch-Physikalische Klasse vol 2 pp 235ndash257 1918
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
3 Self-Adjointness and Conservation Laws
In this section we will investigate the self-adjointness andconservation laws of (1) by Ibragimovrsquos theorem
31 Self-Adjointness for (1) and (2) Let
119867 = 119906119905+ 119906119909+ 119886119906119898119906119909+ 119887119906119909119909119905 (20)
then we have the following formal Lagrangian for (1)
L1= 119908 (119906
119905+ 119906119909+ 119886119906119898119906119909+ 119887119906119909119909119905) (21)
where 119908 = 119908(119909 119905) is a new function Computing the vari-ational derivative of this formal Lagrangian we obtain thefollowing adjoint equation of (1)
119867lowast= minus119908119905minus 119908119909minus 119886119908119909119906119898minus 119887119908119909119909119905= 0 (22)
Assume that119867lowast|119908=120601(119905119909119906)
= 120582119867 for a certain function 120582Then it is easy to obtain that 120601 = 119888
1119906 + 1198882and 120582 = minus120601
119906= minus1198881
where 1198881 1198882are two arbitrary constantsThis implies that (1) is
quasiself-adjoint especially if 1198881= 1 and 119888
2= 0 that is 120601 = 119906
then (1) is strictly self-adjoint (see [16] for the definitions)Thus we have demonstrated the following statement
Theorem3 Equation (1) is quasiself-adjoint with the substitu-tion 119908 = 119888
1119906 + 1198882 where 119888
1= 0 1198882are two arbitrary constants
Especially (1) is strictly self-adjoint with the substitution119908 = 119906
Next we discuss the self-adjointness of (2) and then bythe nonlinear self-adjointness and Ibragimovrsquos theorem onconservation laws we construct some conserved quantities of(1)
Similar to the above we set
119864 = V119909119905+ V119909119909+ 119886V119898119909V119909119909+ 119887V119909119909119909119905
(23)
then the formal Lagrangian for (2) is
L2= 119908 (V
119909119905+ V119909119909+ 119886V119898119909V119909119909+ 119887V119909119909119909119905
) (24)
and the adjoint equation of (1) is
119864lowast= 119908119909119905+ 119908119909119909+ 119886119908119909119909V119898119909+ 119886119898119908
119909V119898minus1119909
V119909119909+ 119887119908119909119909119909119905
= 0
(25)
According to [16] (2) will be nonlinearly self-adjoint ifthere exists a differential substitution
119908 = ℎ (119909 119905 V V119909 V119905 V119909119909 ) ℎ = 0 (26)
involving a finite number of partial derivatives of V withrespect to 119909 and 119905 such that the equation
119864lowast1003816100381610038161003816119908=ℎ
= 1205830119864 + 1205831119863119909119864 + 1205832119863119905119864 + 12058331198632
119909119864 + sdot sdot sdot (27)
holds identically in the variables 119905 119909 119906 119906119909 119906119905 where
1205830 1205831 are undetermined variable coefficients different
from infin on the solutions of (2) The highest order ofderivatives involved in 119906 is called the order of the differentialsubstitution (26) The calculation provides the followingresult
Theorem 4 Equation (2) is nonlinearly self-adjoint Theminimal order of the differential substitution (26) satisfying(27) is equal to one and is given by the function ℎ = 119888
1V119909+
1198882V119905+ 119892(119905) where 119888
1and 1198882are two constants and 119892(119905) is a
differentiable function of 119905
32 Conservation Laws of (1) Now we study the conserva-tion laws of (1) We will first construct the conservation lawsof (2) and then reduce them to the ones of (1) From theclassical Lie group theory [17] we assume that a Lie pointsymmetry of (2) is a vector field
119883 = 120585119909(119905 119909 V)
120597
120597119909
+ 120585119905(119905 119909 V)
120597
120597119905
+ 120588 (119905 119909 V)120597
120597V(28)
on R+ times R times R such that 119883(4)119864 = 0 when 119864 = 0 where 119864is given by (23) Taking into account (2) the operator 119883(4) isgiven as follows
119883(4)= 119883 + 120588
(1)
119909
120597
120597V119909
+ 120588(2)
119909119905
120597
120597V119909119905
+ 120588(2)
119909119909
120597
120597V119909119909
+ 120588(4)
119909119909119909119905
120597
120597V119909119909119909119905
(29)where
120588(1)
119909= 119863119909120588 minus (119863
119909120585119909) V119909minus (119863119909120585119905) V119905
120588(2)
119909119905= 119863119905120588(1)
119909minus (119863119905120585119909) V119909119909minus (119863119905120585119905) V119909119905
120588(2)
119909119909= 119863119909(120588(1)
119909) minus (119863
119909120585119909) V119909119909minus (119863119909120585119905) V119909119905
120588(4)
119909119909119909119905= 119863119909(120588(3)
119909119909119909) minus (119863
119909120585119909) V119909119909119909119909
minus (119863119909120585119905) V119909119909119909119905
(30)
The condition 119883(4)119864|119864=0
= 0 yields determining equationsSolving these determining equations we can obtain thesymmetries of (5) as follows
1198831=
120597
120597119909
1198832=
120597
120597119905
1198833= 119891 (119905)
120597
120597V(31)
with 119891(119905) as an arbitrary function Ibragimovrsquos theorem onconservation laws yields the following conserved quantitiesof (2) for a general Lie symmetry 119883 = 120585
119909(120597120597119909) + 120585
119905(120597120597119905) +
120588(120597120597V) Consider the following
119862119905= 120585119905L2minus119882(
119908119909
2
+
119887119908119909119909119909
4
) +119882119909(
119908
2
+
119887119908119909119909
4
)
minus119882119909119909(
119887119908119909
4
) +119882119909119909119909(
119887119908
4
)
119862119909= 120585119909L2+119882(119886119898V119898minus1
119909V119909119909119908 minus 119863
119909[119908 (1 + 119886V119898
119909)]
minus
119908119905
2
minus
3119887119908119909119909119905
4
)
+119882119909((1 + 119886V119898
119909) 119908 +
119887119908119909119905
2
) +119882119905(
119908
2
+
119887119908119909119909
4
)
minus119882119909119909(
119887119908119905
4
)
minus119882119909119905
119887119908119909
2
+119882119909119909119905(
3119887119908
4
)
(32)
4 Abstract and Applied Analysis
where119882 = 120588 minus 120585119909V119909minus 120585119905V119905is the Lie characteristic function
and119908 is given by the differential substitution119908 = 1198881V119909+1198882V119905+
119892(119905) Let us construct the conserved vector corresponding tothe space translation group with the generator
1198831=
120597
120597119909
(33)
In this case 119882 = minusV119909 Taking account of the differential
substitution 119908 = 1198881V119909+ 1198882V119905+ 119892(119905) and V
119909= 119906 from
(32) we obtain the conserved quantities of (2) which can betransformed to the ones of (1) as follows
119862119905
1=
1
4
1198882(2119906119906119905+ 119887119906119906
119909119909119905minus 2119906119909int119906119905119889119909
minus 119887119906119909119906119909119905+ 119887119906119905119906119909119909minus 119887119906119909119909119909int119906119905119889119909)
minus
1
4
119887119892 (119905) 119906119909119909119909
minus
1
2
119892 (119905) 119906119909
=
1
2
1198882(2119906119906119905+ 119887119906119906
119909119909119905+ 119887119906119905119906119909119909)
minus
1
4
1198882119863119909(119887119906119906119909119905+ 2119906int119906
119905119889119909 + 119887119906
119909119909int119906119905119889119909)
minus
1
4
119863119909[119887119892 (119905) 119906
119909119909+ 2119892 (119905) 119906]
119862119909
1=
1
4
1198882(3119887119906119906
119909119905119905minus 2119887119906
119909119906119905119905+ 119887119906119905119906119909119905+ 119887119906119909119909int119906119905119905119889119909
+ 2119906int119906119905119905119889119909 + 4119886119906
119898+1119906119905
+4119906119906119905+ 2119906119905int119906119905119889119909 + 119887119906
119909119909119905int119906119905119889119909)
+
119887
4
[119892 (119905) 119906119909119909119905+ 1198921015840(119905) 119906119909119909]
+ 1198881119906 (119906119909+ 119906119905+ 119886119906119898119906119909+ 119887119906119909119909119905) +
1
2
[1199061198921015840(119905) + 119892 (119905) 119906
119905]
= 1198881119906 (119906119909+ 119906119905+ 119886119906119898119906119909+ 119887119906119909119909119905)
+
1
2
1198882(119887119906119906119909119905119905minus 119887119906119909119906119905119905+ 2119886119906
119898+1119906119905+ 2119906119906
119905)
+
1
4
1198882119863119905(119887119906119906119909119905+ 2119906int119906
119905119889119909 + 119887119906
119909119909int119906119905119889119909)
+
1
4
119863119905[119887119892 (119905) 119906
119909119909+ 2119892 (119905) 119906]
(34)
which can be reduced to the form
119862119905
1=
1
2
1198882(2119906119906119905+ 119887119906119906
119909119909119905+ 119887119906119905119906119909119909)
119862119909
1=
1198882
2
(119887119906119906119909119905119905minus 119887119906119909119906119905119905+ 2119886119906
119898+1119906119905+ 2119906119906
119905)
(35)
Here we have used (1)
By the procedure analogous to that used above weobtain the conserved quantities of (2) corresponding tothe generators 119883
2and 119883
3 which can be reduced by the
differential substitutions119908 = 1198881V119909+ 1198882V119905+ 119892(119905) and V
119909= 119906 to
the conserved quantities of (1) as follows
119862119905
2=
1198881119887
2
(119906119906119909119909119905minus 119906119905119906119909119909) + 119886119892 (119905) 119906
119898119906119909
119862119909
2=
1198881119887
2
(119906119909119906119905119905minus 119906119906119909119905119905) minus 119886119892 (119905) 119906
119898119906119905
+ 1198921015840(119905) (int 119906
119905119889119909 + 119906 + 119887119906
119909119905)
119862119905
3= 119891 (119905) (119888
1119906119909+ 1198882119906119905)
119862119909
3= 119891 (119905) (119888
1119906119909+ 1198882119906119905) (1 + 119886119906
119898)
+ 119887119891 (119905) (1198881119906119909119909119905+ 1198882119906119909119905119905)
minus 1198911015840(119905) (1198881119906 + 1198882int119906119905119889119909 + 119892 (119905))
(36)
Here we have used (1)
Remark 5 In fact (2) admits a Lagrangian
1198712= minus
1
2
V119909V119905minus
1
2
V2119909minus
119886
(119898 + 1) (119898 + 2)
V119898+2119909
+
119887
2
V119909119905V119909119909
(37)
and has the following Noether symmetry generators
1198835=
120597
120597119909
1198836=
120597
120597119905
1198837= 119891 (119905)
120597
120597Vwith gangue function 119860119905 = 0
119860119909= minus
1
2
1198911015840(119905) V
(38)
where119891(119905) is a differentiable function in 119905 Noetherrsquos theoremon conservation laws [18] can yield the conserved quantitiesof (2) which are the same as the ones in [11] Those wereconstructed by the so-called ldquopartial Noether approachrdquo
4 Conclusions
In this paper we investigate the Lagrangians and self-adjointnesses of the GRLW equation (1) and its transformedequation (2) We show that (1) has a nonlocal Lagrangianwith an auxiliary function and is strictly self-adjoint itstransformed equation is nonlinearly self-adjoint and theminimal order of the differential substitution is equal to oneThen by Ibragimovrsquos theorem on conservation laws we obtainsome conserved qualities of the generalized regularized long-wave equation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 5
Acknowledgments
The first author is partly supported by Zhejiang Provin-cial Natural Science Foundation of China under Grantno LY12A01003 and Subjects Research and DevelopmentFoundation of Hangzhou Dianzi University under Grantno ZX100204004-6 The second author is partly supportedby NSFC under Grant no 11101111 and Zhejiang Provin-cial Natural Science Foundation of China under Grant noLY14A010029
References
[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966
[2] D H Peregrine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967
[3] M Dehghan and R Salehi ldquoThe solitary wave solution of thetwo-dimensional regularized long-wave equation in fluids andplasmasrdquoComputer Physics Communications vol 182 no 12 pp2540ndash2549 2011
[4] J C Eilbeck and G R McGuire ldquoNumerical study of the reg-ularized long-wave equation II interaction of solitary wavesrdquoJournal of Computational Physics vol 23 no 1 pp 63ndash73 1977
[5] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979
[6] Q Chang ldquoConservative scheme for a model of nonlineardispersive waves and its solitary waves induced by boundarymotionrdquo Journal of Computational Physics vol 93 no 2 pp360ndash375 1991
[7] D Bhardwaj and R Shankar ldquoComputational method forregularized long wave equationrdquo Computers and Mathematicswith Applications vol 40 no 12 pp 1397ndash1404 2000
[8] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004
[9] A-M Wazwaz ldquoA sine-cosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004
[10] A Mohebbi ldquoSolitary wave solutions of the nonlinear general-ized Pochhammer-Chree and regularized long wave equationsrdquoNonlinear Dynamics vol 70 no 4 pp 2463ndash2474 2012
[11] R Naz M D Khan and I Naeem ldquoConservation laws andexact solutions of a class of non linear regularized long waveequations via double reduction theory and Lie symmetriesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 4 pp 826ndash834 2013
[12] A Biswas ldquoSolitary waves for power-law regularized long-waveequation and 119877(119898 119899) equationrdquo Nonlinear Dynamics vol 59no 3 pp 423ndash426 2010
[13] W Sarlet ldquoComment on lsquoconservation laws of higher ordernonlinear PDEs and the variational conservation laws in theclass withmixed derivativesrsquordquo Journal of Physics AMathematicaland Theoretical vol 43 no 45 Article ID 458001 2010
[14] N H Ibragimov ldquoA new conservation theoremrdquo Journal ofMathematical Analysis and Applications vol 333 no 1 pp 311ndash328 2007
[15] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsand Fractals vol 19 no 4 pp 847ndash851 2004
[16] N H Ibragimov ldquoNonlinear self-adjointness and conservationlawsrdquo Journal of Physics A Mathematical and Theoretical vol44 no 43 Article ID 432002 2011
[17] P J Olver Application of Lie Groups To Differential EquationsSpringer Berlin Germany 1986
[18] E Noether ldquoInvariante variationsproblemerdquo Nachrichten vonder Gesellschaft derWissenschaften zuGottingenMathematisch-Physikalische Klasse vol 2 pp 235ndash257 1918
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
where119882 = 120588 minus 120585119909V119909minus 120585119905V119905is the Lie characteristic function
and119908 is given by the differential substitution119908 = 1198881V119909+1198882V119905+
119892(119905) Let us construct the conserved vector corresponding tothe space translation group with the generator
1198831=
120597
120597119909
(33)
In this case 119882 = minusV119909 Taking account of the differential
substitution 119908 = 1198881V119909+ 1198882V119905+ 119892(119905) and V
119909= 119906 from
(32) we obtain the conserved quantities of (2) which can betransformed to the ones of (1) as follows
119862119905
1=
1
4
1198882(2119906119906119905+ 119887119906119906
119909119909119905minus 2119906119909int119906119905119889119909
minus 119887119906119909119906119909119905+ 119887119906119905119906119909119909minus 119887119906119909119909119909int119906119905119889119909)
minus
1
4
119887119892 (119905) 119906119909119909119909
minus
1
2
119892 (119905) 119906119909
=
1
2
1198882(2119906119906119905+ 119887119906119906
119909119909119905+ 119887119906119905119906119909119909)
minus
1
4
1198882119863119909(119887119906119906119909119905+ 2119906int119906
119905119889119909 + 119887119906
119909119909int119906119905119889119909)
minus
1
4
119863119909[119887119892 (119905) 119906
119909119909+ 2119892 (119905) 119906]
119862119909
1=
1
4
1198882(3119887119906119906
119909119905119905minus 2119887119906
119909119906119905119905+ 119887119906119905119906119909119905+ 119887119906119909119909int119906119905119905119889119909
+ 2119906int119906119905119905119889119909 + 4119886119906
119898+1119906119905
+4119906119906119905+ 2119906119905int119906119905119889119909 + 119887119906
119909119909119905int119906119905119889119909)
+
119887
4
[119892 (119905) 119906119909119909119905+ 1198921015840(119905) 119906119909119909]
+ 1198881119906 (119906119909+ 119906119905+ 119886119906119898119906119909+ 119887119906119909119909119905) +
1
2
[1199061198921015840(119905) + 119892 (119905) 119906
119905]
= 1198881119906 (119906119909+ 119906119905+ 119886119906119898119906119909+ 119887119906119909119909119905)
+
1
2
1198882(119887119906119906119909119905119905minus 119887119906119909119906119905119905+ 2119886119906
119898+1119906119905+ 2119906119906
119905)
+
1
4
1198882119863119905(119887119906119906119909119905+ 2119906int119906
119905119889119909 + 119887119906
119909119909int119906119905119889119909)
+
1
4
119863119905[119887119892 (119905) 119906
119909119909+ 2119892 (119905) 119906]
(34)
which can be reduced to the form
119862119905
1=
1
2
1198882(2119906119906119905+ 119887119906119906
119909119909119905+ 119887119906119905119906119909119909)
119862119909
1=
1198882
2
(119887119906119906119909119905119905minus 119887119906119909119906119905119905+ 2119886119906
119898+1119906119905+ 2119906119906
119905)
(35)
Here we have used (1)
By the procedure analogous to that used above weobtain the conserved quantities of (2) corresponding tothe generators 119883
2and 119883
3 which can be reduced by the
differential substitutions119908 = 1198881V119909+ 1198882V119905+ 119892(119905) and V
119909= 119906 to
the conserved quantities of (1) as follows
119862119905
2=
1198881119887
2
(119906119906119909119909119905minus 119906119905119906119909119909) + 119886119892 (119905) 119906
119898119906119909
119862119909
2=
1198881119887
2
(119906119909119906119905119905minus 119906119906119909119905119905) minus 119886119892 (119905) 119906
119898119906119905
+ 1198921015840(119905) (int 119906
119905119889119909 + 119906 + 119887119906
119909119905)
119862119905
3= 119891 (119905) (119888
1119906119909+ 1198882119906119905)
119862119909
3= 119891 (119905) (119888
1119906119909+ 1198882119906119905) (1 + 119886119906
119898)
+ 119887119891 (119905) (1198881119906119909119909119905+ 1198882119906119909119905119905)
minus 1198911015840(119905) (1198881119906 + 1198882int119906119905119889119909 + 119892 (119905))
(36)
Here we have used (1)
Remark 5 In fact (2) admits a Lagrangian
1198712= minus
1
2
V119909V119905minus
1
2
V2119909minus
119886
(119898 + 1) (119898 + 2)
V119898+2119909
+
119887
2
V119909119905V119909119909
(37)
and has the following Noether symmetry generators
1198835=
120597
120597119909
1198836=
120597
120597119905
1198837= 119891 (119905)
120597
120597Vwith gangue function 119860119905 = 0
119860119909= minus
1
2
1198911015840(119905) V
(38)
where119891(119905) is a differentiable function in 119905 Noetherrsquos theoremon conservation laws [18] can yield the conserved quantitiesof (2) which are the same as the ones in [11] Those wereconstructed by the so-called ldquopartial Noether approachrdquo
4 Conclusions
In this paper we investigate the Lagrangians and self-adjointnesses of the GRLW equation (1) and its transformedequation (2) We show that (1) has a nonlocal Lagrangianwith an auxiliary function and is strictly self-adjoint itstransformed equation is nonlinearly self-adjoint and theminimal order of the differential substitution is equal to oneThen by Ibragimovrsquos theorem on conservation laws we obtainsome conserved qualities of the generalized regularized long-wave equation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 5
Acknowledgments
The first author is partly supported by Zhejiang Provin-cial Natural Science Foundation of China under Grantno LY12A01003 and Subjects Research and DevelopmentFoundation of Hangzhou Dianzi University under Grantno ZX100204004-6 The second author is partly supportedby NSFC under Grant no 11101111 and Zhejiang Provin-cial Natural Science Foundation of China under Grant noLY14A010029
References
[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966
[2] D H Peregrine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967
[3] M Dehghan and R Salehi ldquoThe solitary wave solution of thetwo-dimensional regularized long-wave equation in fluids andplasmasrdquoComputer Physics Communications vol 182 no 12 pp2540ndash2549 2011
[4] J C Eilbeck and G R McGuire ldquoNumerical study of the reg-ularized long-wave equation II interaction of solitary wavesrdquoJournal of Computational Physics vol 23 no 1 pp 63ndash73 1977
[5] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979
[6] Q Chang ldquoConservative scheme for a model of nonlineardispersive waves and its solitary waves induced by boundarymotionrdquo Journal of Computational Physics vol 93 no 2 pp360ndash375 1991
[7] D Bhardwaj and R Shankar ldquoComputational method forregularized long wave equationrdquo Computers and Mathematicswith Applications vol 40 no 12 pp 1397ndash1404 2000
[8] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004
[9] A-M Wazwaz ldquoA sine-cosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004
[10] A Mohebbi ldquoSolitary wave solutions of the nonlinear general-ized Pochhammer-Chree and regularized long wave equationsrdquoNonlinear Dynamics vol 70 no 4 pp 2463ndash2474 2012
[11] R Naz M D Khan and I Naeem ldquoConservation laws andexact solutions of a class of non linear regularized long waveequations via double reduction theory and Lie symmetriesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 4 pp 826ndash834 2013
[12] A Biswas ldquoSolitary waves for power-law regularized long-waveequation and 119877(119898 119899) equationrdquo Nonlinear Dynamics vol 59no 3 pp 423ndash426 2010
[13] W Sarlet ldquoComment on lsquoconservation laws of higher ordernonlinear PDEs and the variational conservation laws in theclass withmixed derivativesrsquordquo Journal of Physics AMathematicaland Theoretical vol 43 no 45 Article ID 458001 2010
[14] N H Ibragimov ldquoA new conservation theoremrdquo Journal ofMathematical Analysis and Applications vol 333 no 1 pp 311ndash328 2007
[15] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsand Fractals vol 19 no 4 pp 847ndash851 2004
[16] N H Ibragimov ldquoNonlinear self-adjointness and conservationlawsrdquo Journal of Physics A Mathematical and Theoretical vol44 no 43 Article ID 432002 2011
[17] P J Olver Application of Lie Groups To Differential EquationsSpringer Berlin Germany 1986
[18] E Noether ldquoInvariante variationsproblemerdquo Nachrichten vonder Gesellschaft derWissenschaften zuGottingenMathematisch-Physikalische Klasse vol 2 pp 235ndash257 1918
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Acknowledgments
The first author is partly supported by Zhejiang Provin-cial Natural Science Foundation of China under Grantno LY12A01003 and Subjects Research and DevelopmentFoundation of Hangzhou Dianzi University under Grantno ZX100204004-6 The second author is partly supportedby NSFC under Grant no 11101111 and Zhejiang Provin-cial Natural Science Foundation of China under Grant noLY14A010029
References
[1] DH Peregrine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 no 2 pp 321ndash3301966
[2] D H Peregrine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967
[3] M Dehghan and R Salehi ldquoThe solitary wave solution of thetwo-dimensional regularized long-wave equation in fluids andplasmasrdquoComputer Physics Communications vol 182 no 12 pp2540ndash2549 2011
[4] J C Eilbeck and G R McGuire ldquoNumerical study of the reg-ularized long-wave equation II interaction of solitary wavesrdquoJournal of Computational Physics vol 23 no 1 pp 63ndash73 1977
[5] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979
[6] Q Chang ldquoConservative scheme for a model of nonlineardispersive waves and its solitary waves induced by boundarymotionrdquo Journal of Computational Physics vol 93 no 2 pp360ndash375 1991
[7] D Bhardwaj and R Shankar ldquoComputational method forregularized long wave equationrdquo Computers and Mathematicswith Applications vol 40 no 12 pp 1397ndash1404 2000
[8] D Kaya ldquoA numerical simulation of solitary-wave solutionsof the generalized regularized long-wave equationrdquo AppliedMathematics and Computation vol 149 no 3 pp 833ndash8412004
[9] A-M Wazwaz ldquoA sine-cosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004
[10] A Mohebbi ldquoSolitary wave solutions of the nonlinear general-ized Pochhammer-Chree and regularized long wave equationsrdquoNonlinear Dynamics vol 70 no 4 pp 2463ndash2474 2012
[11] R Naz M D Khan and I Naeem ldquoConservation laws andexact solutions of a class of non linear regularized long waveequations via double reduction theory and Lie symmetriesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 4 pp 826ndash834 2013
[12] A Biswas ldquoSolitary waves for power-law regularized long-waveequation and 119877(119898 119899) equationrdquo Nonlinear Dynamics vol 59no 3 pp 423ndash426 2010
[13] W Sarlet ldquoComment on lsquoconservation laws of higher ordernonlinear PDEs and the variational conservation laws in theclass withmixed derivativesrsquordquo Journal of Physics AMathematicaland Theoretical vol 43 no 45 Article ID 458001 2010
[14] N H Ibragimov ldquoA new conservation theoremrdquo Journal ofMathematical Analysis and Applications vol 333 no 1 pp 311ndash328 2007
[15] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsand Fractals vol 19 no 4 pp 847ndash851 2004
[16] N H Ibragimov ldquoNonlinear self-adjointness and conservationlawsrdquo Journal of Physics A Mathematical and Theoretical vol44 no 43 Article ID 432002 2011
[17] P J Olver Application of Lie Groups To Differential EquationsSpringer Berlin Germany 1986
[18] E Noether ldquoInvariante variationsproblemerdquo Nachrichten vonder Gesellschaft derWissenschaften zuGottingenMathematisch-Physikalische Klasse vol 2 pp 235ndash257 1918
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![MJEN - Manas Universityjournals.manas.edu.kg/mjen/archives/Y2017_V5_I1/d301cea55e284a… · Adomian Decomposition method [9-19], the Modified Decomposition method [10, 13, 20-27]](https://static.fdocuments.net/doc/165x107/6063d5cfc05d7c1ca23044c2/mjen-manas-adomian-decomposition-method-9-19-the-modified-decomposition-method.jpg)
