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  • This content has been downloaded from IOPscience. Please scroll down to see the full text.

    Download details:

    IP Address: 222.66.117.23

    This content was downloaded on 17/11/2015 at 06:09

    Please note that terms and conditions apply.

    Integrable discretizations and self-adaptive moving mesh method for a coupled short pulse

    equation

    View the table of contents for this issue, or go to the journal homepage for more

    2015 J. Phys. A: Math. Theor. 48 385202

    (http://iopscience.iop.org/1751-8121/48/38/385202)

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  • Integrable discretizations and self-adaptivemoving mesh method for a coupled shortpulse equation

    Bao-Feng Feng1, Junchao Chen2, Yong Chen2,Ken-ichi Maruno3 and Yasuhiro Ohta4

    1Department of Mathematics, The University of Texas-Rio Grade Valley, Edinburg,TX 78541, USA2 Shanghai Key Laboratory of Trustworthy Computing, East China Normal University,Shanghai 200062, Peopleʼs Republic of China3Department of Applied Mathematics, School of Fundamental Science andEngineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan4Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

    E-mail: [email protected]

    Received 24 November 2014, revised 23 July 2015Accepted for publication 29 July 2015Published 25 August 2015

    AbstractIn the present paper, integrable semi-discrete and fully discrete analogues of acoupled short pulse (CSP) equation are constructed. The key to the con-struction are the bilinear forms and determinant structure of the solutions ofthe CSP equation. We also construct N-soliton solutions for the semi-discreteand fully discrete analogues of the CSP equations in the form of Casoratideterminants. In the continuous limit, we show that the fully discrete CSPequation converges to the semi-discrete CSP equation, then further to thecontinuous CSP equation. Moreover, the integrable semi-discretization of theCSP equation is used as a self-adaptive moving mesh method for numericalsimulations. The numerical results agree with the analytical results very well.

    Keywords: coupled short pulse equation, integrable discretization, self-adaptive moving mesh method

    1. Introduction

    The short pulse (SP) equation

    u u u1

    61.1xt xx

    3( ) ( )= +

    Journal of Physics A: Mathematical and Theoretical

    J. Phys. A: Math. Theor. 48 (2015) 385202 (21pp) doi:10.1088/1751-8113/48/38/385202

    1751-8113/15/385202+21$33.00 © 2015 IOP Publishing Ltd Printed in the UK 1

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  • was derived by Schäfer and Wayne to describe the propagation of ultra-short opticalpulses in nonlinear media [1, 2]. Here, u u x t,( )= represents the magnitude of theelectric field, while the subscripts t and x denote partial differentiations. The SP equationrepresents an alternative approach in contrast with the slowly varying envelopeapproximation which leads to the nonlinear Schödinger (NLS) equation. As the pulseduration shortens, the NLS equation becomes less accurate, while the SP equationprovides an increasingly better approximation to the corresponding solution of theMaxwell equations [2]. With the rapid progress of ultra-short optical pulse techniques, itis expected that the SP equation and its multi-component generalization will play moreand more important roles in applications.

    The SP equation has been shown to be completely integrable[3–5]. The loop-solitonsolutions as well as smooth-soliton solutions of the SP solution were found in [6–8]. Multi-soliton solutions including multi-loop and multi-breather ones were given in [9]. Periodicsolutions to the SP equation were discussed in [10].

    Similar to the case of the NLS equation [11], it is necessary to consider its two-com-ponent or multi-component generalizations of the SP equation for describing the effects ofpolarization or anisotropy. As a matter of fact, several integrable coupled SP equations havebeen proposed in the literature [12–17]. A complex version of the integrable coupled SPequation in [14, 15] was studied in [18]. The bi-Hamiltonian structures for the above two-component SP equations were obtained in [19].

    Integrable discretizations of soliton equations have received considerable attentionrecently [20–23]. Integrable semi- and full discretizations of the SP equation were constructedvia Hirotaʼs bilinear method [24]. The same discretizations were reconstructed from the pointof view of geometry in [25]. Most recently, an integrable discretization for a coupled SPequation proposed in [14, 15] was constructed [26].

    In the present paper, we consider integrable discretizations of another coupled short pulse(CSP) equation proposed by one of the authors [16]

    u u u v u1

    6

    1

    2, 1.2xt xx xx

    3 2( ) ( )= + +

    v v v u v1

    6

    1

    2. 1.3xt xx xx

    3 2( ) ( )= + +It was shown in [16] that equations (1.2) and (1.3) can be derived from bilinear equations

    D D f f f f1

    2, 1.4s y 2

    2( )· ¯ ( )= -D D f f f f

    1

    2, 1.5s y

    2 2( )¯ · ¯ ¯ ( )= -D D g g g g

    1

    2, 1.6s y 2 2( )· ¯ ( )= -

    D D g g g g1

    2, 1.7s y 2 2( )¯ · ¯ ¯ ( )= -

    through a hodograph transformation

    x y FF t sln , , 1.8s( )( )¯ ( )= - =

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    2

  • and dependent variable transformations

    uF

    Fv

    G

    Gi ln , i ln , 1.9

    s s

    ¯ ¯( )

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟= =

    where F fg G fg, ¯= = , F̄ and Ḡ stand for the complex conjugates of F and G, respectively.Meanwhile, N-soliton solutions of the CSP equation in parametric form are given, and theproperties of one-soliton, soliton-breather solutions are investigated in detail in [16]. The bi-Hamiltonian structure of the CSP equations (1.2) and (1.3) was derived by Brunelli andSakovich [19].

    The rest of the present paper is organized as follows. In section 2, we propose anintegrable semi-discrete analogue of the CSP equation by constructing a Bäcklund transformof the bilinear equations of the CSP equation. Meanwhile, an N-soliton solution is provided interms of the Casorati determinant form. In section 3, starting from two sets of Bäcklundtransforms to the bilinear equations of the CSP equation, the fully discrete analogue of theCSP equation is proposed by introducing two auxiliary variables. Moreover, the N-solitonsolution is presented to confirm the integrability. Section 4 contributes to the self-adaptivemoving mesh method by applying the semi-implicit Euler scheme to the semi-discrete CSP.The paper is concluded by section 5. Appendices A, B and C present the proofs of proposition1, theorems 1 and 2, respectively.

    2. Integrable semi-discretization of the CSP equation

    We start with two sets of bilinear equations for the semi-discrete two-dimensional Toda-lattice (2DTL) equations with the same discrete parameter a

    aD k k k k

    11 1 1 0, 2.1x n n n n1 11 ( ) · ( ) ( ) ( ) ( )⎜ ⎟

    ⎛⎝

    ⎞⎠t t t t- + + + =+ --

    aD k k k k

    11 1 1 0, 2.2x n n n n1 11 ( ) · ( ) ( ) ( ) ( )⎜ ⎟

    ⎛⎝

    ⎞⎠t t t t- ¢ + ¢ + ¢ + ¢ =+ --

    which is linked by a Bäcklund transformation [27]

    D k k k k1 0. 2.3x n n n n1 11( ) ( ) · ( ) ( ) ( ) ( )t t t t- ¢ + ¢ =+ --

    Proposition 1. The bilinear equations (2.1) and (2.3) admit the following determinantsolutions

    x k

    k k k

    k k k

    k k k

    , , 2.4n

    n n n N

    n n n N

    nN

    nN

    n NN

    1

    11

    11

    1

    21

    21

    2

    1 1

    ( )( ) ( ) ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    t

    f f f

    f f f

    f f f

    =-

    + + -

    + + -

    + + -

    x k

    k k k

    k k k

    k k k

    , , 2.5n

    n n n N

    n n n N

    nN

    nN

    n NN

    1

    11

    11

    1

    21

    21

    2

    1 1

    ( )( ) ( ) ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    t

    y y y

    y y y

    y y y

    ¢ =-

    + + -

    + + -

    + + -

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    3

  • where

    k p ap e q aq e1 1 , 2.6ni

    in

    ik

    in

    ik

    i i( ) ( )( ) ( )( )f = - + -x h- -

    k p p ap e q q aq e1 1 1 1 , 2.7ni

    in

    i ik

    in

    i ik

    i i( )( ) ( )( )( ) ( )( )y = - - + - -x h- -

    with

    p x q x, .i i i i i i1

    1 01

    1 0x x h h= + = +-

    --

    -

    Here pi, qi, i0x and i0h are arbitrary parameters which can take either real or complex values.

    The proof is presented in appendix A.Applying a 2-reduction condition q pi i= - , then we could have each of the τ sequences

    become a sequence of period 2, i.e., n n 2t t + , n n 2t t¢ ¢ + . Here means two τ functions areequivalent up to a constant multiple. Furthermore, by choosing particular values of phaseconstants, we can make nt and n 1t + complex conjugate to each other. Based on the bilinearequations with 2-reduction, we construct the semi-discrete analogue of the CSP equationusing the following theorem:

    Theorem 1. The following equations constitute an integrable semi-discretization of the CSPequations (1.2) and (1.3)

    su u u u u u v v

    d

    d

    1

    2

    1

    2, 2.8k k k k k

    kk k k k1 1 1 1

    2 2( )( ) ( ) ( ) ( )d d- = + - - -+ + + +

    sv v v v v v u u

    d

    d

    1

    2

    1

    2, 2.9k k k k k

    kk k k k1 1 1 1

    2 2( )( ) ( ) ( ) ( )d d- = + - - -+ + + +

    su u v v

    d

    d

    1

    2. 2.10k k k k k1

    2 21

    2 2( ) ( )d = - - + -+ +Furthermore, the N-soliton solution to the above semi-discrete CSP equation is of thefollowing form

    uf g

    f gv

    f g

    f giln , iln , 2.11k

    k k

    k k s

    kk k

    k k s

    ¯ ¯ ¯

    ¯( )

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟= =

    x ka f f g g2 ln , 2.12k k k k k s( )( )¯ ¯ ( )= -

    x xa f f

    f f

    f f

    f f

    g g

    g g

    g g

    g g2, 2.13k k k

    k k

    k k

    k k

    k k

    k k

    k k

    k k

    k k1

    1

    1

    1

    1

    1

    1

    1

    1¯ ¯

    ¯ ¯

    ¯ ¯¯ ¯

    ( )⎛⎝⎜

    ⎞⎠⎟d = - = + + ++

    +

    +

    +

    +

    +

    +

    +

    +

    where fk, gk, fk̄ and gk̄ are tau-functions defined by

    fs

    k fs

    k gs

    k gs

    k2

    , ,2

    , ,2

    , ,2

    , ,

    2.14

    k k k k0 1 0 1¯ ¯

    ( )

    ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎛⎝

    ⎞⎠

    ⎛⎝

    ⎞⎠

    ⎛⎝

    ⎞⎠

    ⎛⎝

    ⎞⎠t t t t= = = ¢ = ¢

    with

    sk

    sk

    2, ,

    2, , 2.15n n j

    i

    i j Nn n j

    i

    i j N1 1 , 1 1 ,( )( )

    ( )( )( )⎜ ⎟ ⎜ ⎟

    ⎛⎝

    ⎞⎠

    ⎛⎝

    ⎞⎠ t f t y= ¢ =+ - + -

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    4

  • k p ap p ap

    k p p ap p p ap

    1 e 1 e ,

    1 1 e 1 1 e .

    ni

    in

    ik p

    si

    ni

    k ps

    ni

    in

    i ik p

    si

    ni i

    k ps

    12

    12

    12

    12

    ii

    ii

    ii

    ii

    0 0

    0 0

    ( ) ( )( )( ) ( )( )

    ( ) ( )

    ( ) ( )

    ( )

    ( )

    f

    y

    = - + - +

    = - - + - + +

    x h

    x h

    - + - - +

    - + - - +

    The proof is presented in appendix B. In the process of the proof, the multi-soliton solutionexpressed in determinant form is obvious. Next, we show that the semi-discrete CSP equationconverges to the CSP equation in the continuous limit.

    In the continuous limit a 0 ( 0kd ), we have

    u u u

    x

    u uu,

    2, 2.16k k

    k

    k k1 1 ( )d-

    ¶¶

    ++ +

    x

    s

    x

    s su v u v u v

    d

    d

    1

    2

    1

    2. 2.17

    j

    kj

    j

    k

    j j j j0

    0

    1

    0

    1

    12

    12 2 2 2 2( ) ( ) ( )å åd¶¶ =

    ¶¶

    + = - + - - - +=

    -

    =

    -

    + +

    Thus

    x

    su v

    1

    2. 2.18s t x t x2 2( ) ( )¶ = ¶ + ¶

    ¶¶ ¶- + ¶

    Consequently, equation (B.25) converges to

    u v u u u1

    21 ,t x x x

    2 2 2( )( )⎜ ⎟⎛⎝⎞⎠¶ - + ¶ = +

    which is nothing but the first equation of the CSP equation (1.2).It can be shown in the same way that equation (B.26) converges to equation (1.3), the

    second equation of the CSP equation.

    3. Fully discretization of the CSP equation

    To construct a fully discrete analogue of the CSP equation, we introduce one more discretevariable l which corresponds to the discrete time variable.

    It is known in [29] that the τ-functions

    k l k l k l k l, , , , , ,

    3.1

    n n ji

    i j Nn n j

    i

    i j N1 1 , 1 1 ,( ) ( ) ( ) ( )

    ( )

    ( )( )

    ( )( )

    t f t y= ¢ =+ - + -

    with

    k l p ap bp

    q aq bq

    k l p p ap bp

    q q

    aq bq

    , 1 11

    e 1 11

    e ,

    , 1 1 11

    e 1

    1 11

    e

    ni

    in

    ik

    i

    l

    ps

    in

    ik

    i

    l

    qs

    ni

    in

    i ik

    i

    l

    ps

    in

    i

    ik

    i

    ls

    12

    12

    12

    ii

    ii

    ii

    qi i

    0 0

    0

    12 0

    ( ) ( )

    ( )( ) ( )

    ( )

    ( )

    ( )

    ( )

    ( )

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟

    f

    y

    = - - + - -

    = - - - + -

    ´ - -

    x h

    x

    h

    --

    + --

    +

    --

    +

    --

    +

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    5

  • satisfy bilinear equations

    aD k l k l k l k l

    21 1, , 1, , 0, 3.2s n n n n1 1( ) · ( ) ( ) ( ) ( )⎜ ⎟

    ⎛⎝

    ⎞⎠t t t t- + + + =+ -

    aD k l k l k l k l

    21 1, , 1, , 0, 3.3s n n n n1 1( ) · ( ) ( ) ( ) ( )⎜ ⎟

    ⎛⎝

    ⎞⎠t t t t- ¢ + ¢ + ¢ + ¢ =+ -

    and

    bD k l k l k l k l2 1 , 1 , , , 1 0. 3.4s n n n n1 1( ) ( ) · ( ) ( ) ( ) ( )t t t t- + + + =+ +bD k l k l k l k l2 1 , 1 , , , 1 0. 3.5s n n n n1 1( ) ( ) · ( ) ( ) ( ) ( )t t t t- ¢ + ¢ + ¢ ¢ + =+ +

    Here n k l, , are integers, a b, are real numbers, p q, , ,i i i i0 0x h are arbitrary complex numbers.By applying a 2-reduction condition: q pi i= - , we have n n 2t t + , n n 2t t¢ ¢ + . We can

    further get

    f k l f k l g k l g k l, , , , , , , ,k l k l k l k l, 0 , 1 , 0 , 1( ) ¯ ( ) ( ) ¯ ( )t t t t= = = ¢ = ¢

    by adjusting the phases in k l,in ( )( )f and k l,i

    n ( )( )y . Here f̄ and ḡ represent complex conjugatefunctions of f and g, respectively. A fully discrete CSP equation can be constructed asfollows:

    Theorem 2. The fully discrete analogue of the CSP equations (1.2) and (1.3) is of the form

    bu u u u v v v v

    y y u u v v

    y y u u v v

    1

    , 3.6

    k l k l k l k l k l k l k l k l

    k l k l k l k l k l k l

    k l k l k l k l k l k l

    1, 1 1, , 1 , 1, 1 1, , 1 ,

    1, , 1 1, 1 , 1, 1 ,

    1, 1 , , 1 1, , 1 1,

    ( )

    ( )( )( )( ) ( )

    - - + + - - +

    = - + + +

    + - + + +

    + + + + + + + +

    + + + + + +

    + + + + + +

    bu u u u v v v v

    z z u u v v

    z z u u v v

    1

    , 3.7

    k l k l k l k l k l k l k l k l

    k l k l k l k l k l k l

    k l k l k l k l k l k l

    1, 1 1, , 1 , 1, 1 1, , 1 ,

    1, , 1 1, 1 , 1, 1 ,

    1, 1 , , 1 1, , 1 1,

    ( )

    ( )( )( )( ) ( )

    - - + - + + -

    = - + - -+ - + - -

    + + + + + + + +

    + + + + + +

    + + + + + +

    y y y yb

    y y

    u u v v

    u u u u v v v v

    1

    1

    4, 3.8

    k l k l k l k l k l k l

    k l k l k l k l

    k l k l k l k l k l k l k l k l

    1, 1 1, , 1 , , 1 1,

    , 1 1, , 1 1,

    1, 1 1, , 1 , 1, 1 1, , 1 ,

    ( )( )

    ( ) ( )

    ⎜ ⎟⎛⎝⎞⎠- - + + -

    = - + + +

    ´ + - - + + - -

    + + + + + +

    + + + +

    + + + + + + + +

    z z z zb

    z z

    u u v v

    u u u u v v v v

    1

    1

    4, 3.9

    k l k l k l k l k l k l

    k l k l k l k l

    k l k l k l k l k l k l k l k l

    1, 1 1, , 1 , , 1 1,

    , 1 1, , 1 1,

    1, 1 1, , 1 , 1, 1 1, , 1 ,

    ( )

    ( )

    ( ) ( )

    ⎜ ⎟⎛⎝⎞⎠- - + + -

    = - + - -

    ´ + - - - - + +

    + + + + + +

    + + + +

    + + + + + + + +

    where

    uf g

    f gv

    f g

    f giln , iln , 3.10k l

    k l k l

    k l k l s

    k lk l k l

    k l k l s

    ,, ,

    , ,,

    , ,

    , ,

    ¯ ¯ ¯

    ¯( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟= =

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    6

  • y ka f f z ka g gln , ln , 3.11k l k l k l s k l k l k l s, , , , , ,( ) ( )( ) ( )¯ ¯ ( )= - = -and

    x y z ka f f g g2 ln . 3.12k l k l k l k l k l k l k l s, , , , , , ,( )( )¯ ¯ ( )= + = -The proof is presented in appendix C.

    Finally we show that equations (3.6)–(3.9) converge to the semi-discrete CSPequations (2.8)–(2.10) by taking a continuous limit in time (b 0 ). Under this limit,equations (3.6)–(3.9) become

    su u

    sv v y y u u v v

    d

    d

    d

    d, 3.13k k k k k k k k k k1 1 1 1 1( ) ( ) ( )( ) ( )- + - = - + + ++ + + + +

    su u

    sv v z z u u v v

    d

    d

    d

    d, 3.14k k k k k k k k k k1 1 1 1 1( ) ( ) ( )( ) ( )- - - = - + - -+ + + + +

    sy y u u v v u u v v

    d

    d

    1

    4, 3.15k k k k k k k k k k1 1 1 1 1( ) ( )( ) ( )- = - + + + - + -+ + + + +

    sz z u u v v u u v v

    d

    d

    1

    4, 3.16k k k k k k k k k k1 1 1 1 1( ) ( )( ) ( )- = - + - - - - ++ + + + +

    where FF Fb s2

    l l1 ¶-+ (b 0 ) is used. Obviously, we have

    su u u u v v z z y y

    d

    d

    1

    2

    1

    2, 3.17k k k k k k k k k k k1 1 1 1 1( )( ) ( ) ( ) ( )d- = + - + - - ++ + + + +

    sv v v v u u z z y y

    d

    d

    1

    2

    1

    2, 3.18k k k k k k k k k k k1 1 1 1 1( )( ) ( ) ( ) ( )d- = + - + - - ++ + + + +

    from (3.13) and (3.14), and

    sx x u u v v

    d

    d

    1

    2, 3.19k k k k k k1 1

    2 21

    2 2( )( ) ( )- = - - + -+ + +by adding (3.15) and (3.16). Equation (3.19) coincides with equation (2.10).

    Finally, in view of the relations (C.17)–(C.22), we have

    z z y y

    z z y y z z y y

    a f f

    f f

    f f

    f f

    g g

    g g

    g g

    g g

    u u v v

    4

    . 3.20

    k k k k k

    k k k k k k k k

    k k

    k k

    k k

    k k

    k k

    k k

    k k

    k k

    k k k k

    1 1

    1 1 1 1

    21

    1

    1

    1

    21

    1

    1

    1

    2

    1 1

    ( )( )( )

    ¯ ¯¯ ¯

    ¯ ¯¯ ¯

    ( )( ) ( )

    ⎡⎣⎢⎢

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟

    ⎤⎦⎥⎥

    d- - +

    = - - + - + -

    = - + - +

    = - -

    + +

    + + + +

    +

    +

    +

    +

    +

    +

    +

    +

    + +

    A substitution of (3.20) into (3.17) and (3.18) yields (2.8) and (2.9).From the construction of the fully discrete analogue of the CSP equation, the multi-

    soliton solution can be expressed in the following determinant form

    uf g

    f g

    f

    f

    g

    g

    f

    f

    g

    giln

    i

    2, 3.21k l

    k l k l

    k l k l s

    k l

    k l

    k l

    k l

    k l

    k l

    k l

    k l,

    , ,

    , ,

    ,

    ,

    ,

    ,

    ,

    ,

    ,

    ,

    ¯ ¯ ¯¯

    ¯

    ¯( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟= =

    ¢+

    ¢-

    ¢-

    ¢

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    7

  • vf g

    f g

    f

    f

    g

    g

    f

    f

    g

    giln

    i

    2, 3.22k l

    k l k l

    k l k l s

    k l

    k l

    k l

    k l

    k l

    k l

    k l

    k l,

    , ,

    , ,

    ,

    ,

    ,

    ,

    ,

    ,

    ,

    ,

    ¯

    ¯

    ¯¯

    ¯

    ¯( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟= =

    ¢+

    ¢-

    ¢-

    ¢

    y ka f f kaf

    f

    f

    fln

    1

    2, 3.23k l k l k l s

    k l

    k l

    k l

    k l, , ,

    ,

    ,

    ,

    ,( )( )¯

    ¯¯ ( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟= - = -

    ¢+

    ¢

    z ka g g kag

    g

    g

    gln

    1

    2, 3.24k l k l k l s

    k l

    k l

    k l

    k l, , ,

    ,

    ,

    ,

    ,( )( )¯

    ¯

    ¯( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟= - = -

    ¢+

    ¢

    thus

    x ka f f g g kaf

    f

    g

    g

    f

    f

    g

    g2 ln 2

    1

    2. 3.25k l k l k l k l k l s

    k l

    k l

    k l

    k l

    k l

    k l

    k l

    k l, , , , ,

    ,

    ,

    ,

    ,

    ,

    ,

    ,

    ,( )( )¯ ¯

    ¯¯

    ¯

    ¯( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟= - = -

    ¢+

    ¢+

    ¢+

    ¢

    Here

    f k l f k l g k l g k l, , , , , , , , 3.26k l k l k l k l, 0 , 1 , 0 , 1( ) ¯ ( ) ( ) ¯ ( ) ( )t t t t= = = ¢ = ¢

    f k l f k l g k l g k l, , , , , , , , 3.27k l k l k l k l, 0 , 1 , 0 , 1( ) ¯ ( ) ( ) ¯ ( ) ( )r r r r¢ = ¢ = ¢ = ¢ ¢ = ¢

    with k l,n ( )t and k l,n ( )t¢ defined the same as (3.1), and k l,n ( )r and k l,n ( )r¢ defined as

    k l k l k l k l, , , , , ,

    3.28

    n n ji

    i j N n n ji

    i j N2 1 , 3 1 ,( ) ( ) ( ) ( )

    ( )

    ( )( )

    ( )( )

    r f r y= ¢ =+ - + -

    under the 2-reduction condition q pi i= - (i N1, ,= ).

    Remark 3.1. Two intermediate variables yk and zk are introduced in constructing the fullydiscrete CSP equation. This often happens when we construct the full discretizations of acoupled system such as the coupled modified KdV equation [28].

    4. Integrable self-adaptive moving mesh method

    In this section, we propose a self-adaptive moving mesh method for the CSP equation (1.2)–(1.3) and demonstrate the advantage of this integrable scheme by performing severalnumerical experiments.

    4.1. Numerical scheme

    One of the self-adaptive moving mesh methods for the CSP equation can be constructed byapplying a semi-implicit Euler scheme to its integrable semi-discrete CSP equations (2.8)–(2.10). The resulting numerical scheme reads

    p pt

    u ut

    p q v v2 2

    , 4.1kn

    kn

    kn

    kn

    kn

    kn k

    nkn

    kn

    kn1

    1 1( ) ( ) ( )d d= +D

    + -D

    ++ + +

    q qt

    v vt

    p q u u2 2

    , 4.2kn

    kn

    kn

    kn

    kn

    kn k

    nkn

    kn

    kn1

    1 1( ) ( ) ( )d d= +D

    + -D

    ++ + +

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    8

  • tu v u v

    2. 4.3k

    nkn

    kn

    kn

    kn1

    11 2

    11 2 1 2 1 2( )( ) ( ) ( ) ( ) ( )d = -D + - -+ ++ ++ + +

    Here p u uk k k1= -+ , q v vk k k1= -+ , x xk k k1d = -+ . The superscript n represents thenumerical value at t n t= D . Periodic boundary conditions are applied. For convenience, wereserve the time t t - so that the left-moving wave becomes the right-moving one. In whatfollows, we report the numerical results for one- and two-soliton solutions.

    4.2. Numerical experiments

    For the sake of numerical experiments, we list exact one- and two-soliton solutions for thecontinuous, semi- and fully-discrete CSP equation.

    (1). One-soliton solution: the τ-functions for the one-soliton solution of the CSPequation (1.2)–(1.3) are

    f g s1 ie 1 i e , 4.41 ( )µ + µ +q q

    where s p p1 11 1 1( ) ( )= - + , p y s p y1 1 0q = + + . This leads to the one-soliton solution inparametric form

    u y sp

    ,1

    sech sgn s sech , 4.51

    1( )( ) ( ) ( ) ( )q q= + - D

    v y sp

    ,1

    sech sgn s sech , 4.61

    1( )( ) ( ) ( ) ( )q q= - - D

    x yp

    1tanh tanh . 4.7

    1

    ( ( ) ( )) ( )q q= - + - D

    where sexp 1( ) ∣ ∣-D = .For the semi-discrete CSP equation, the τ-functions are

    fap

    apg s

    ap

    ap1 i

    1

    1e , 1 i

    1

    1e . 4.8k

    ks p y

    k

    ks p y1

    11

    1

    1

    1 0 1 0 ( )⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟µ +

    +-

    µ ++-

    + +

    Finally for the fully discrete CSP equation, the τ-functions are

    fap

    ap

    bp

    bp1 i

    1

    1

    1

    1, 4.9k l

    k l

    ,1

    1

    11

    11

    ( )⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟µ +

    +-

    +

    -

    -

    -

    g sap

    ap

    bp

    bp1 i

    1

    1

    1

    1. 4.10k l

    k l

    , 11

    1

    11

    11

    ( )⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟µ +

    +-

    +

    -

    -

    -

    The initial conditions for one-soliton propagation are taken from (4.4) with parametersy 00 = , and p 0.91 = , p 2.01 = . The initial profiles are shown in figures 1(a) and (b),respectively. The simulations are run on a domain 40, 40[ ]- with 800 grid points, thus theaverage mesh size is 0.1.

    When p 0.91 = , u is symmetric with two spikes, and v is antisymmetric. The com-parison between the numerical and analytical results is shown in figure 2, together withthe non-uniform mesh. It can be seen that the non-uniform mesh is dense around the peakpoints of the solitons. Moreover, the most dense part of the non-uniform mesh movesalong with the peak point. When p 2.01 = , u is antisymmetric, and v is symmetric with a

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    9

  • Figure 1. Initial conditions for CSP equation. (a) p 0.91 = ; (b) p 2.01 = .

    Figure 2. Comparison between numerical and analytical solutions for one-solitonsolution to the CSP equation with p 0.91 = at t 4.0= ; solid line: analytical solution,blue dot: numerical solution, red dot: self-adaptive mesh; (a) profile of u, (b) profileof v.

    Figure 3. Comparison between numerical and analytical results for one-soliton solutionto the CSP equation with p 2.01 = at t 12.0= ; solid line: analytical solution, blue dot:numerical solution, red dot: self-adaptive mesh; (a) profile of u, (b) profile of v.

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    10

  • loop structure. The comparison between the numerical and analytical results is shown infigure 3. The error between the numerical solution and the analytical one is displayed infigure 4.

    (2). Two-soliton solution: the τ-functions for two-soliton solution of the CSPequations (1.2) and (1.3) are

    f b1 ie ie e , 4.11121 2 1 2 ( )µ + + -q q q q+

    g s s b1 i e i e e , 4.121 2 121 2 1 2 ( )µ + + - ¢q q q q+

    with s p p1 1i i i( ) ( )= - + , p y s p yi i i1 0q = + + i 1, 2( )= , and b p p12 1 2 2( )= -p p1 2

    2( )+ , and b b s s12 12 1 2* *¢ = .For the semi-discrete CSP equation, the τ-functions are

    fap

    ap

    ap

    ap

    ap ap

    ap apb

    1 i1

    1e i

    1

    1e

    1 1

    1 1e , 4.13

    k

    k k

    k

    1

    1

    2

    2

    1 2

    1 212

    1 2

    1 2( )( )( )( ) ( )

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟

    µ ++-

    ++-

    -+ +

    - -

    x x

    x x+

    g sap

    aps

    ap

    ap

    bap ap

    ap ap

    1 i1

    1e i

    1

    1e

    1 1

    1 1e , 4.14

    k

    k k

    k

    11

    12

    2

    2

    121 2

    1 2

    1 2

    1 2( )( )( )( ) ( )

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟

    µ ++-

    ++-

    - ¢+ +

    - -

    x x

    x x+

    with p s yi i1

    0+- i 1, 2( )= .

    Figure 4. The error between numerical and analytical results for the one-solitonsolution to the CSP equation with p 2.01 = at t 12.0= ; (a) error in u, (b) error in v.

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    11

  • For the fully discrete CSP equation, the τ-functions are

    fap

    ap

    bp

    bp

    ap

    ap

    bp

    bp

    bap ap

    ap ap

    bp bp

    bp bp

    1 i1

    1

    1

    1i

    1

    1

    1

    1

    1 1

    1 1

    1 1

    1 1,

    4.15

    k l

    k l k l

    k l

    ,1

    1

    11

    11

    2

    2

    21

    21

    121 2

    1 2

    11

    21

    11

    21

    ( )( )( )( )

    ( )( )( )( ) ( )

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟

    ⎝⎜⎜

    ⎠⎟⎟

    µ ++-

    +

    -+

    +-

    +

    -

    -+ +

    - -

    + +

    - -

    -

    -

    -

    -

    - -

    - -

    g sap

    ap

    bp

    bps

    ap

    ap

    bp

    bp

    bap ap

    ap ap

    bp bp

    bp bp

    1 i1

    1

    1

    1i

    1

    1

    1

    1

    1 1

    1 1

    1 1

    1 1. 4.16

    k l

    k l k l

    k l

    , 11

    1

    11

    11 2

    2

    2

    21

    21

    121 2

    1 2

    11

    21

    11

    21

    ( )( )( )( )

    ( )( )( )( ) ( )

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟

    ⎝⎜⎜

    ⎠⎟⎟

    µ ++-

    +

    -+

    +-

    +

    -

    - ¢+ +

    - -

    + +

    - -

    -

    -

    -

    -

    - -

    - -

    As pointed out in [16], when p1 and p2 are complex conjugate to each other, the two-soliton solution becomes a breather solution. Equations (4.11) and (4.12) are used as initialconditions with parameters chosen as p 0.4 i1 = + , p 0.4 i2 = - , y y 010 20= = . Thenumerical results at t = 10 are displayed in figure 5 in comparison with the analyticalsolution. Here a grid point of 800 is used on the domain 40, 40[ ]- , the time step size is takenas t 0.005D = . The error between the numerical solution and the analytical one is displayedin figure 6. It can be seen that the numerical results are in good agreement with the analy-tical ones.

    5. Conclusions

    In this paper, we proposed integrable semi-discrete and fully discrete analogues of a coupledshort pulse equation. The determinant formulae of N-soliton solutions for the semi-discreteand fully discrete analogues of the CSP equations are also presented. In the continuous limit,the fully discrete CSP equation converges to the semi-discrete CSP equation, then furtherconverges to the continuous CSP equation.

    Figure 5. Comparison between numerical and analytical results for breather solution tothe CSP equation for p 0.4 i1 = + , p 0.4 i2 = - at t = 10; solid line: analyticalsolution, dashed line: numerical solution; (a) profile of u, (b) profile of v.

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    12

  • In a series of papers by one of the authors, we have constructed integrable discretizationsfor a class of soliton equations with hodograph transformation, and successfully used them asself-adaptive moving mesh methods for the Camassa–Holm equation [30, 31] and the SPequation [16]. Based on the semi-discrete CSP equations (1.2) and (1.3), a self-adaptivemoving mesh method is constructed and used for the numerical simulation of the CSPequation. It should be pointed out that the feature of self-adaptivity of the mesh is due to thehodograph transformation. In other words, the hodograph transformation converts the uni-form and time-independent mesh into a non-uniform and time-dependent mesh. It is a furthertopic to seek this kind of self-adaptive moving method when the hodograph transformation isnot present. The numerical results confirm that it is an excellent scheme due to the nature ofintegrability and self-adaptivity of the mesh. This is the first time this superior numericalmethod has been extended to a coupled system.

    Acknowledgments

    This work is partially supported by the National Natural Science Foundation of China (Nos.11428102, 11075055, 11275072).

    Appendix A. Proof of proposition 1

    Proof. For simplicity, we introduce a convenient notation

    N

    k k k

    k k k

    k k k

    0 , 1 , , 1 , A.1k k k

    n n n N

    n n n N

    nN

    nN

    n NN

    11

    11

    1

    21

    21

    2

    1 1

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    f f f

    f f f

    f f f

    - =

    + + -

    + + -

    + + -

    N

    k k k

    k k k

    k k k

    0 , 1 , , 1 . A.2k k k

    n n n N

    n n n N

    nN

    nN

    n NN

    11

    11

    1

    21

    21

    2

    1 1

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( ) ( ) ( )

    y y y

    y y y

    y y y

    ¢ ¢ - ¢ =

    + + -

    + + -

    + + -

    Figure 6. The error between numerical and analytical results for the one-breathersolution to the CSP equation at t 10.0= ; (a) error in u, (b) error in v.

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    13

  • Since

    k k k k a k, 1 1 , A.3x ni

    ni

    ni

    ni

    ni

    1 11 ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )f f f f f¶ = + - = +- +-

    k k k k a k, 1 1 , A.4x ni

    ni

    ni

    ni

    ni

    1 11 ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )y y y y y¶ = + - = +- +-

    k k k , A.5ni

    ni

    ni

    1( ) ( ) ( ) ( )( ) ( ) ( )f y f- = +

    we can verify the following relations

    k N N1 , 1 , , 2 , 1 , A.6x n k k k k1 ( ) ( )t¶ = - - --

    k N N N

    aN N N

    1 1 , 2 , , 2 , 1 ,

    11 , 2 , , 2 , 1 , 1 , A.7

    n k k k k k

    k k k k k

    1 1

    1

    ( )

    ( )

    t + = - -

    = - - -

    + +

    +

    k N N

    N N

    1 0 , 1 , , 2 , 1

    0 , 1 , , 2 , 1 , A.8

    n k k k k

    k k k k

    1 1 1 1

    1

    ( )( )

    t + = - -

    = - -+ + + +

    +

    k N N N N

    N N

    a N N

    1 1 , 1 , , 2 , 1 0 , 1 , , 2 , 2

    1 , 1 , , 2 , 1

    0 , 1 , , 2 , 1 .

    A.9

    x n k k k k k k k k

    k k k k

    k k k k

    1 1

    1

    1

    1 ( )

    ( )

    t¶ + = - - - + - -

    = - - -

    + - -

    + +

    +

    +

    -

    Combining (A.8) with (A.9), we have

    ak

    aN N

    11 1

    11 , 1 , , 2 , 1 . A.10x n k k k k 11 ( ) ( )⎜ ⎟

    ⎛⎝

    ⎞⎠t¶ - + = - - - +-

    Therefore, the Plücker relation for determinants

    N N N N

    N N N N

    N N N N

    0 , 1 , , 2 , 1 1 , 1 , , 2 , 1

    0 , 1 , , 2 , 1 1 , 1 , , 2 , 1

    1 , , 2 , 1 , 1 1 , 0 , 1 , , 2 0 A.11

    k k k k k k k k

    k k k k k k k k

    k k k k k k k k

    1

    1

    1 ( )

    - - - - -

    - - - - - -

    + - - - - - =

    +

    +

    +

    gives

    ak k

    ak

    k k k

    11 1

    11

    1 0, A.12

    x n n n

    x n n n1 1

    1

    1

    ( ) ( ) ( )

    ( ) ( ) ( ) ( )

    ⎜ ⎟⎛⎝⎞⎠t t t

    t t t

    ¶ - + ´ - +

    ´ ¶ + + =+ -

    -

    -

    which is nothing but the bilinear equation (2.1). Equation (2.2) can be proved in the same way.Now we proceed to the proof of equation (2.3). Similarly we can verify the following

    relations

    k N N

    N N

    0 , 1 , , 2 , 1

    0 , 1 , , 2 , 1 , A.13

    n k k k k

    k k k k

    ( )

    ( )

    t = - -

    = ¢ ¢ - ¢ -

    k N N

    N N

    1 , 2 , , 1 ,

    1 , 2 , , 1 , 1 , A.14

    n k k k k

    k k k k

    1( )

    ( )

    t = ¢ ¢ - ¢

    = ¢ ¢ - ¢ -

    +

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    14

  • k N N1 1 , 1 , , 2 , 1 . A.15x n k k k k1( ) ( ) ( )t¶ - = - ¢ ¢ - ¢ -- Therefore the Plücker relation for determinants:

    N N N N

    N N N N

    N N N

    1 , 1 , , 2 , 1 0 , 1 , , 2 , 1

    0 , 1 , , 2 , 1 1 , 1 , , 2 , 1

    1 , 0 , 1 , , 2 1 , , 1 , 1 0,

    A.16

    k k k k k k k k

    k k k k k k k k

    k k k k k k k

    ( )

    - ¢ ¢ - ¢ - ¢ ¢ - ¢ - ¢

    - ¢ ¢ - ¢ - - ¢ ¢ - ¢ - ¢

    + - ¢ ¢ ¢ - ¢ ¢ - ¢ - =

    gives

    k k k k k k1 0. A.17x n n n x n n n1 11 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )t t t t t t¶ - ´ ¢ - ´ ¶ ¢ + ¢ =+ -- -

    Therefore, equation (2.3) is proved. ,

    Appendix B. Proof of theorem 1

    Proof. By putting s x2 1= - , k fk0 ( )t = , k fk1( ) ¯t = , (2.1) can be converted into

    aD f f f f

    21 , B.1s k k k k1 1· ¯ ¯ ( )⎜ ⎟

    ⎛⎝

    ⎞⎠- = -+ +

    aD f f f f

    21 , B.2s k k k k1 1¯ · ¯ ( )⎜ ⎟

    ⎛⎝

    ⎞⎠- = -+ +

    while by putting k gk0 ( )t¢ = , k gk1( ) ¯t¢ = , (2.2) can be converted into

    aD g g g g

    21 , B.3s k k k k1 1· ¯ ¯ ( )⎜ ⎟

    ⎛⎝

    ⎞⎠- = -+ +

    aD g g g g

    21 . B.4s k k k k1 1¯ · ¯ ( )⎜ ⎟

    ⎛⎝

    ⎞⎠- = -+ +

    Furthermore, the above bilinear equations can be rewritten as the following logarithmicderivatives

    a

    f

    f

    f f

    f f

    2ln 1 , B.5k

    k s

    k k

    k k

    1 1

    1

    ¯ ¯( )

    ⎛⎝⎜

    ⎞⎠⎟ - = -

    + +

    +

    a

    f

    f

    f f

    f f

    2ln 1 , B.6k

    k s

    k k

    k k

    1 1

    1

    ¯¯ ¯ ¯ ( )

    ⎛⎝⎜

    ⎞⎠⎟ - = -

    + +

    +

    a

    g

    g

    g g

    g g

    2ln 1 , B.7k

    k s

    k k

    k k

    1 1

    1

    ¯ ¯( )

    ⎛⎝⎜

    ⎞⎠⎟ - = -

    + +

    +

    a

    g

    g

    g g

    g g

    2ln 1 . B.8k

    k s

    k k

    k k

    1 1

    1

    ¯¯ ¯ ¯

    ( )⎛⎝⎜

    ⎞⎠⎟ - = -

    + +

    +

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    15

  • Introducing two intermediate variable transformations

    sf s

    f ss

    g s

    g s2i ln , 2i ln ,k

    k

    kk

    k

    k

    ( )¯ ( )

    ( )( )

    ¯ ( )( )

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟s s= ¢ =

    one arrives at a pair of semi-discrete sine-Gordon equations

    a

    1

    2sin

    2, B.9k k s

    k k1

    1( ) ( )⎜ ⎟⎛⎝⎞⎠s s

    s s- =

    ++

    +

    a

    1

    2sin

    2. B.10k k s

    k k1

    1( ) ( )⎛⎝⎜

    ⎞⎠⎟s s

    s s¢ - ¢ =

    ¢ + ¢+

    +

    It then follows that

    acos

    2

    1

    4, B.11k k

    sk s k s

    11,

    2,

    2( )( ) ( ) ( )⎜ ⎟⎛⎝⎜⎛⎝

    ⎞⎠

    ⎞⎠⎟

    s ss s

    += - -+ +

    acos

    2

    1

    4, B.12k k

    s

    k s k s1

    1,2

    ,2( )( ) ( ) ( )⎛⎝⎜⎜

    ⎛⎝⎜

    ⎞⎠⎟

    ⎞⎠⎟⎟

    s ss s

    ¢ + ¢= - ¢ - ¢+ +

    where k s,s denotes the derivative of ks with respective to s.Next, we introduce dependent variable transformations

    uf g

    f g

    1

    2i ln , B.13k k s k s

    k k

    k k s

    , ,( )¯ ¯

    ( )⎛⎝⎜

    ⎞⎠⎟s s= + ¢ =

    vf g

    f g

    1

    2i ln , B.14k k s k s

    k k

    k k s

    , ,( )¯

    ¯( )

    ⎛⎝⎜

    ⎞⎠⎟s s= - ¢ =

    and discrete hodograph transformation

    x ka f f g g2 ln . B.15k k k k k s( )( )¯ ¯ ( )= -Then the non-uniform mesh can be derived as

    B.16

    x x

    af f g g

    f f g g

    a f f

    f f

    f f

    f f

    g g

    g g

    g g

    g g

    a

    2 ln

    2

    cos2

    cos2

    .

    k k k

    k k k k

    k k k k s

    k k

    k k

    k k

    k k

    k k

    k k

    k k

    k k

    k k k k

    1

    1 1 1 1

    1

    1

    1

    1

    1

    1

    1

    1

    1 1 ( )

    ¯ ¯¯ ¯

    ¯ ¯¯ ¯

    ¯ ¯¯ ¯

    ⎜ ⎟

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜⎜ ⎛⎝

    ⎞⎠

    ⎛⎝⎜

    ⎞⎠⎟

    ⎞⎠⎟⎟

    d

    s s s s

    = -

    = -

    = + + +

    =+

    +¢ + ¢

    +

    + + + +

    +

    +

    +

    +

    +

    +

    +

    +

    + +

    Taking the derivative with respect to s results in

    B.17

    sa

    u u v v

    d

    dcos

    2cos

    2

    1

    2.

    k k k

    s

    k k

    s

    k k k k

    1 1

    12 2

    12 2( ) ( )

    ⎜ ⎟⎛⎝⎜⎜ ⎛⎝

    ⎞⎠

    ⎛⎝⎜

    ⎞⎠⎟

    ⎞⎠⎟⎟d s s

    s s=

    ++

    ¢ + ¢

    =- - + -

    + +

    + +

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    16

  • Furthermore, assuming

    p qsec4

    , sec4

    ,kk k k k

    kk k k k1 1 1 1

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟

    s s s s s s s s=

    + + ¢ + ¢=

    + - ¢ - ¢+ + + +

    we have

    a

    p q

    2, B.18k

    k k

    ( )d =

    dp

    d s s

    u u

    q u u

    q

    a

    u u

    q

    au u

    d

    dcos

    4

    sin4 2

    2sin

    2sin

    2 2

    2

    1

    2 2

    4, B.19

    k k k k k

    k k k k k k

    k k k k k k k

    kk k k k s

    k k

    kk k

    11 1

    1 1 1

    1 1 1

    1 11

    12 2( )( )

    ( )

    ⎜ ⎟

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜⎜ ⎛⎝

    ⎞⎠

    ⎛⎝⎜

    ⎞⎠⎟

    ⎞⎠⎟⎟

    s s s s

    s s s s

    s s s s

    s s s s

    =+ + ¢ + ¢

    =-+ + ¢ + ¢ +

    =-+

    -¢ + ¢ +

    =- - + ¢ - ¢+

    =- -

    -+ +

    + + +

    + + +

    + ++

    +

    and similarly

    q

    s

    p

    av v

    d

    d 4. B.20k k k k

    1

    12 2( ) ( )= - -

    -

    +

    Thus, in turn, equations (B.19) and (B.20) become

    p

    sp

    u ud

    d, B.21k k

    k k

    k

    22 1

    2 2

    ( )d

    =-+

    and

    q

    sq

    v vd

    d, B.22k k

    k k

    k

    22 1

    2 2

    ( )d

    =-+

    respectively.On the other hand, with the help of trigonometric identity, pk

    2 can be expressed as

    p

    p

    p q

    au u

    u u

    1 tan4

    1 sin4

    14

    1 , B.23

    kk k k k

    kk k k k

    k kk k

    k k

    k

    2 2 1 1

    2 2 1 1

    2 2

    2 12 2

    12

    ( )( )

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜

    ⎞⎠⎟

    s s s s

    s s s s

    d

    = ++ + ¢ + ¢

    = ++ + ¢ + ¢

    = + -

    = +-

    + +

    + +

    +

    +

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    17

  • and similarly

    qv v

    1 . B.24kk k

    k

    2 12

    ( )⎛⎝⎜

    ⎞⎠⎟d= +

    -+

    Therefore, we finally have

    s

    u u u u u ud

    d1

    2, B.25k k

    k

    k k

    k

    k k1 12

    1 ( )⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜⎜

    ⎛⎝⎜

    ⎞⎠⎟

    ⎞⎠⎟⎟d d

    -= +

    - ++ + +

    s

    v v v v v vd

    d1

    2. B.26k k

    k

    k k

    k

    k k1 12

    1 ( )⎛⎝⎜

    ⎞⎠⎟

    ⎛⎝⎜⎜

    ⎛⎝⎜

    ⎞⎠⎟

    ⎞⎠⎟⎟d d

    -= +

    - ++ + +

    Substituting (B.17) into (B.25) and (B.26), one arrives at (2.8) and (2.9), the first twoequations of the semi-discrete CSP equation. ,

    Appendix C. Proof of theorem 2

    Proof. The bilinear equations (3.2)–(3.5) imply the following bilinear equations

    aD f f f f

    21 0, C.1s k l k l k l k l1, , 1, ,· ¯ ¯ ( )⎜ ⎟

    ⎛⎝

    ⎞⎠- + =+ +

    aD f f f f

    21 0, C.2s k l k l k l k l1, , 1, ,¯ · ¯ ( )⎜ ⎟

    ⎛⎝

    ⎞⎠- + =+ +

    aD g g g g

    21 0, C.3s k l k l k l k l1, , 1, ,· ¯ ¯ ( )⎜ ⎟

    ⎛⎝

    ⎞⎠- + =+ +

    aD g g g g

    21 0, C.4s k l k l k l k l1, , 1, ,¯ · ¯ ( )⎜ ⎟

    ⎛⎝

    ⎞⎠- + =+ +

    bD f f f f2 1 0, C.5s k l k l k l k l, 1 , , , 1( ) · ¯ ¯ ( )- + =+ +bD f f f f2 1 0, C.6s k l k l k l k l, 1 , , , 1( ) ¯ · ¯ ( )- + =+ +bD g g g g2 1 0, C.7s k l k l k l k l, 1 , , , 1( ) · ¯ ¯ ( )- + =+ +bD g g g g2 1 0, C.8s k l k l k l k l, 1 , , , 1( ) ¯ · ¯ ( )- + =+ +

    which can be rewritten by logarithmic derivatives as

    a

    f

    f

    f f

    f f

    2ln 1 , C.9k l

    k l s

    k l k l

    k l k l

    1,

    ,

    1, ,

    1, ,

    ¯ ¯( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟ = -+ +

    +

    a

    f

    f

    f f

    f f

    2ln 1 , C.10k l

    k l s

    k l k l

    k l k l

    1,

    ,

    1, ,

    1, ,

    ¯¯ ¯ ¯ ( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟ = -+ +

    +

    a

    g

    g

    g g

    g g

    2ln 1 , C.11k l

    k l s

    k l k l

    k l k l

    1,

    ,

    1, ,

    1, ,

    ¯ ¯( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟ = -+ +

    +

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    18

  • a

    g

    g

    g g

    g g

    2ln 1 , C.12k l

    k l s

    k l k l

    k l k l

    1,

    ,

    1, ,

    1, ,

    ¯¯ ¯ ¯

    ( )⎛⎝⎜⎜

    ⎞⎠⎟⎟ = -+ +

    +

    bf

    f

    f f

    f f2 ln 1 , C.13k l

    k l s

    k l k l

    k l k l

    , 1

    ,

    , , 1

    , 1 ,¯

    ¯¯ ( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟ = -+ +

    +

    bf

    f

    f f

    f f2 ln 1 C.14k l

    k l s

    k l k l

    k l k l

    , 1

    ,

    , , 1

    , 1 ,

    ¯ ¯¯ ( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟ = -+ +

    +

    bg

    g

    g g

    g g2 ln 1 , C.15k l

    k l s

    k l k l

    k l k l

    , 1

    ,

    , , 1

    , 1 ,¯¯

    ¯( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟ = -+ +

    +

    bg

    g

    g g

    g g2 ln 1 . C.16k l

    k l s

    k l k l

    k l k l

    , 1

    ,

    , , 1

    , 1 ,

    ¯ ¯¯

    ( )⎛⎝⎜⎜

    ⎞⎠⎟⎟ = -+ +

    +

    Based on the dependent variable transformation (3.10) and discretehodograph transformation (3.11), we can verify the following relations

    u ua f f

    f f

    g g

    g g

    f f

    f f

    g g

    g g

    i

    2, C.17k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l1, ,

    1, ,

    1, ,

    1, ,

    1, ,

    1, ,

    1, ,

    1, ,

    1, ,

    ¯ ¯ ¯ ¯¯ ¯ ¯ ¯

    ( )⎛⎝⎜⎜

    ⎞⎠⎟⎟- = + - -+ +

    +

    +

    +

    +

    +

    +

    +

    u ub

    f f

    f f

    g g

    g g

    f f

    f f

    g g

    g g

    i

    2, C.18k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l, 1 ,

    , , 1

    , 1 ,

    , , 1

    , 1 ,

    , , 1

    , 1 ,

    , , 1

    , 1 ,

    ¯¯

    ¯¯

    ¯¯

    ¯¯

    ( )⎛⎝⎜⎜

    ⎞⎠⎟⎟+ = + - -+ +

    +

    +

    +

    +

    +

    +

    +

    v va f f

    f f

    g g

    g g

    f f

    f f

    g g

    g g

    i

    2, C.19k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l1, ,

    1, ,

    1, ,

    1, ,

    1, ,

    1, ,

    1, ,

    1, ,

    1, ,

    ¯ ¯ ¯ ¯¯ ¯ ¯ ¯

    ( )⎛⎝⎜⎜

    ⎞⎠⎟⎟- = - - ++ +

    +

    +

    +

    +

    +

    +

    +

    v vb

    f f

    f f

    g g

    g g

    f f

    f f

    g g

    g g

    i

    2, C.20k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l

    k l k l, 1 ,

    , , 1

    , 1 ,

    , , 1

    , 1 ,

    , , 1

    , 1 ,

    , , 1

    , 1 ,

    ¯¯

    ¯¯

    ¯¯

    ¯¯

    ( )⎛⎝⎜⎜

    ⎞⎠⎟⎟+ = - - ++ +

    +

    +

    +

    +

    +

    +

    +

    y ya f f

    f f

    f f

    f f2, C.21k l k l

    k l k l

    k l k l

    k l k l

    k l k l1, ,

    1, ,

    1, ,

    1, ,

    1, ,

    ¯ ¯¯ ¯ ( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟- = ++ +

    +

    +

    +

    z za g g

    g g

    g g

    g g2, C.22k l k l

    k l k l

    k l k l

    k l k l

    k l k l1, ,

    1, ,

    1, ,

    1, ,

    1, ,

    ¯ ¯¯ ¯

    ( )⎛⎝⎜⎜

    ⎞⎠⎟⎟- = ++ +

    +

    +

    +

    y yb b

    f f

    f f

    f f

    f f

    1 1

    2, C.23k l k l

    k l k l

    k l k l

    k l k l

    k l k l, 1 ,

    , , 1

    , 1 ,

    , , 1

    , 1 ,

    ¯¯

    ¯¯ ( )

    ⎛⎝⎜⎜

    ⎞⎠⎟⎟- = - + ++ +

    +

    +

    +

    z zb b

    g g

    g g

    g g

    g g

    1 1

    2. C.24k l k l

    k l k l

    k l k l

    k l k l

    k l k l, 1 ,

    , , 1

    , 1 ,

    , , 1

    , 1 ,

    ¯¯

    ¯¯

    ( )⎛⎝⎜⎜

    ⎞⎠⎟⎟- = - + ++ +

    +

    +

    +

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    19

  • Then, the ratios on the rhs of equations (C.17)–(C.24) can be solved as

    f f

    f f ay y u u v v

    1 i

    2, C.25k l k l

    k l k lk l k l k l k l k l k l

    1, ,

    1, ,1, , 1, , 1, ,

    ¯ ¯( ) ( )

    ⎡⎣⎢

    ⎤⎦⎥= - - - + -

    +

    ++ + +

    f f

    f f ay y u u v v

    1 i

    2, C.26k l k l

    k l k lk l k l k l k l k l k l

    1, ,

    1, ,1, , 1, , 1, ,¯ ¯ ( ) ( )

    ⎡⎣⎢

    ⎤⎦⎥= - + - + -

    +

    ++ + +

    f f

    f fb y y u u v v1

    i

    2,

    C.27

    k l k l

    k l k lk l k l k l k l k l k l

    , , 1

    , 1 ,, 1 , , 1 , , 1 ,

    ¯¯ ( )

    ( )

    ⎡⎣⎢

    ⎤⎦⎥= + - - + + +

    +

    ++ + +

    f f

    f fb y y u u v v1

    i

    2,

    C.28

    k l k l

    k l k lk l k l k l k l k l k l

    , , 1

    , 1 ,, 1 , , 1 , , 1 ,

    ¯¯ ( )

    ( )

    ⎡⎣⎢

    ⎤⎦⎥= + - + + + +

    +

    ++ + +

    g g

    g g az z u u v v

    1 i

    2, C.29k l k l

    k l k lk l k l k l k l k l k l

    1, ,

    1, ,1, , 1, , 1, ,

    ¯ ¯( ) ( )

    ⎡⎣⎢

    ⎤⎦⎥= - - - - +

    +

    ++ + +

    g g

    g g az z u u v v

    1 i

    2, C.30k l k l

    k l k lk l k l k l k l k l k l

    1, ,

    1, ,1, , 1, , 1, ,¯ ¯

    ( ) ( )⎡⎣⎢

    ⎤⎦⎥= - + - - +

    +

    ++ + +

    g g

    g gb z z u u v v1

    i

    2, C.31k l k l

    k l k lk l k l k l k l k l k l

    , , 1

    , 1 ,, 1 , , 1 , , 1 ,

    ¯¯

    ( ) ( )⎡⎣⎢

    ⎤⎦⎥= + - - + - -

    +

    ++ + +

    g g

    g gb z z u u v v1

    i

    2. C.32k l k l

    k l k lk l k l k l k l k l k l

    , , 1

    , 1 ,, 1 , , 1 , , 1 ,

    ¯¯

    ( ) ( )⎡⎣⎢

    ⎤⎦⎥= + - + + - -

    +

    ++ + +

    By making a shift of l l 1 + in (C.25), then dividing it by (C.26), meanwhile, dividing(C.27) by (C.28) after a shift of k k 1 + , one obtains

    y y u u v v

    y y u u v v

    b y y u u v v

    b y y u u v v

    1

    1. C.33

    k l k l k l k l k l k l

    k l k l k l k l k l k l

    k l k l k l k l k l k l

    k l k l k l k l k l k l

    1, 1 , 1i2 1, 1 , 1 1, 1 , 1

    1, ,i2 1, , 1, ,

    1, 1 1,i2 1, 1 1, 1, 1 1,

    , 1 ,i2 , 1 , , 1 ,

    ( )

    ( )

    ( )

    ( )( )

    ⎡⎣ ⎤⎦⎡⎣ ⎤⎦

    - - - + -

    - + - + -

    =+ - - + + +

    + - + + + +

    + + + + + + + + +

    + + +

    + + + + + + + + +

    + + +

    Similarly, one can obtain

    z z u u v v

    z z u u v v

    b z z u u v v

    b z z u u v v

    1

    1, C.34

    k l k l k l k l k l k l

    k l k l k l k l k l k l

    k l k l k l k l k l k l

    k l k l k l k l k l k l

    1, 1 , 1i2 1, 1 , 1, 1 , 1

    1, ,i2 1, , 1, ,

    1, 1 1,i2 1, 1 1, 1, 1 1,

    , 1 ,i2 , 1 , , 1 ,

    ( )

    ( )

    ( )

    ( )( )

    ⎡⎣ ⎤⎦⎡⎣ ⎤⎦

    - - - - +

    - + - - +

    =+ - - + - -

    + - + + - -

    + + + + + + + +

    + + +

    + + + + + + + + +

    + + +

    from relations (C.29)–(C.31). Equating the real parts and imaginary parts of (C.33) and(C.34), we have the fully discrete CSP equations (3.6)–(3.9). ,

    J. Phys. A: Math. Theor. 48 (2015) 385202 B-F Feng et al

    20

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    1. Introduction2. Integrable semi-discretization of the CSP equation3. Fully discretization of the CSP equation4. Integrable self-adaptive moving mesh method4.1. Numerical scheme4.2. Numerical experiments

    5. ConclusionsAcknowledgmentsAppendix A.Appendix B.Appendix C.References