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  • Hamiltonian PDEs: Deformations, Integrability, Solutions

    B.Dubrovin∗

    ∗ SISSA, Trieste, and Steklov Mathematical Institute, Moscow

    Abstract

    We review recent classification results on the theory of systems of nonlinear Hamiltonian partial differential equations with one spatial dimension, including a perturbative approach to the integrability theory of such systems, and discuss universality conjectures describing critical behaviour of solutions to such systems.

    1 Introduction

    The main subject of our research is the study of Hamiltonian perturbations of systems of quasilinear hyperbolic1 PDEs

    uit + A i j(u)u

    j x + higher order derivatives = 0, i = 1, . . . , n.

    (Here and below the summation over repeated indices will be assumed.) They can be obtained, in particular, by applying the procedure of weak dispersion expansion: starting from a system of PDEs

    uit + F i(u, ux, uxx, . . . ) = 0, i = 1, . . . , n

    with the analytic right hand side let us introduce slow variables

    x 7→ � x, t 7→ � t.

    Expanding in � one obtains, after dividing by � a system of the above form

    1

    � F i(u, � ux, �

    2uxx, . . . ) = u i t + A

    i j(u)u

    j x + �

    ( Bij(u)u

    j xx +

    1

    2 Cijk(u)u

    j xu

    k x

    ) + . . .

    1Always the strong hyperbolicity will be assumed, i.e., the eigenvalues of the n×n matrix ( Aij(u)

    ) are all real and pairwise distinct for all u = (u1, . . . , un) in the domain under consideration. Certain parts of the formalism developed in this paper will be applicable also to the quasilinear systems with complex distinct eigenvalues of the coefficient matrix.

    1

  • assuming all the dependent variables are slow, i.e., the terms of the order 1/� vanish:

    F i(u, 0, 0, . . . ) ≡ 0, i = 1, . . . , n.

    The celebrated Korteweg - de Vries (KdV) equation

    ut + uux + �2

    12 uxxx = 0 (1.1)

    is one of the most well known examples of such a weakly dispersive Hamiltonian PDE.

    The systems of PDEs under investigation

    uit + A i j(u)u

    j x + �

    ( Bij(u)u

    j xx +

    1

    2 Cijk(u)u

    j xu

    k x

    ) + · · · = 0, i = 1, . . . , n (1.2)

    depending on a small parameter � will be considered as Hamiltonian vector fields on the “infinite dimensional manifold”

    L(Mn)⊗ R[[�]] (1.3)

    where Mn is a n-dimensional manifold (in all our examples it will have the topology of a ball) and

    L(Mn) = { S1 →Mn

    } is the space of loops on Mn. The dependent variables

    u = (u1, . . . , un) ∈Mn

    are local coordinates on Mn. In the expansion (1.2) the terms of order �k are polyno- mials of degree k + 1 in the derivatives ux, uxx, . . . where

    deg u(m) = m, m = 1, 2, . . .

    The coefficients of these polynomials are smooth functions defined in every coordinate chart on Mn.

    The systems of the form (1.2) will be assumed to be Hamiltonian flows

    uit = {ui(x), H} = ∑ k≥0

    �k k+1∑ m=0

    Aijk,m ( u;ux, . . . , u

    (m) ) ∂k−m+1x

    δH

    δuj(x) (1.4)

    with respect to a local Poisson bracket

    {ui(x), uj(y)} = ∑ k≥0

    �k k+1∑ m=0

    Aijk,m ( u(x);ux(x), . . . , u

    (m)(x) ) δ(k−m+1)(x− y)

    (1.5)

    degAijk,m ( u;ux, . . . , u

    (m) )

    = m

    2

  • with local Hamiltonians

    H = ∑ k≥0

    �k ∫ hk ( u;ux, . . . , u

    (k) ) dx = H0 + �H1 + �

    2H2 + . . .

    (1.6)

    Hk =

    ∫ hk ( u;ux, . . . , u

    (k) ) dx, deg hk

    ( u;ux, . . . , u

    (k) )

    = k.

    Here δ(x) is the Dirac delta-function,

    δH

    δuj(x) =

    ∂h

    ∂uj − ∂x

    ∂h

    ∂ujx + ∂2x

    ∂h

    ∂ujxx − . . . for H =

    ∫ h(u;ux, uxx, . . . ) dx.

    the Euler–Lagrange operator of a local Hamiltonian. The coefficients of the Poisson bracket and Hamiltonian densities will always be assumed to be differential polynomi- als in every order in �. The antisymmetry and Jacobi identity for the Poisson bracket (1.5) are understood as identities for formal power series in �. Note that, for a differen- tial polynomial f(u;ux, uxx, . . . , u

    (m)) the local functional ∫ f(u;ux, uxx, . . . , u

    (m)) dx is defined as the class of equivalence of the differential polynomial f modulo Im ∂x where

    ∂x = ∑

    ui,k+1 ∂

    ∂ui,k , ui,k :=

    ∂kui

    ∂xk (1.7)

    is the operator of total x-derivative. The Poisson bracket of local functionals of the form

    F = ∑ k≥0

    �k ∫ fk(u;ux, . . . , u

    (k)) dx, G = ∑ l≥0

    �l ∫ gl(u;ux, . . . , u

    (l)) dx

    deg fk(u;ux, . . . , u (k)) = k, deg gl(u;ux, . . . , u

    (l)) = l. (1.8)

    reads

    {F,G} = ∫

    δF

    δui(x) Aij

    δG

    δuj(x) dx (1.9)

    Aij := ∑ k≥0

    �k k+1∑ m=0

    Aijk,m ( u;ux, . . . , u

    (m) ) ∂k−m+1x

    Therefore a local Poisson bracket (1.5) defines a structure of Lie algebra on the space Gloc of local functionals.

    Example 1.1 The KdV equation (1.1) admits a Hamiltonian representation

    ut + {u(x), H} ≡ ut + ∂x δH

    δu(x) = 0 (1.10)

    with the Poisson bracket {u(x), u(y)} = δ′(x− y) (1.11)

    and the Hamiltonian

    H =

    ∫ ( 1

    6 u3 − �

    2

    24 u2x

    ) dx. (1.12)

    3

  • Let us now introduce the class of “coordinate changes” on the infinite dimensional manifold L(Mn)⊗ R[[�]]. Define a generalized Miura transformation

    ui 7→ ũi = ∑ k≥0

    �kF ik ( u;ux, . . . , u

    (k) )

    (1.13)

    degF ik ( u;ux, . . . , u

    (k) )

    = k

    det

    ( ∂F i0(u)

    ∂uj

    ) 6= 0.

    The coefficients F ik ( u;ux, . . . , u

    (k) )

    are differential polynomials. It is easy to see that the transformations of the form (1.13) form a group. The classes of evolution PDEs (1.2), local Poisson brackets (1.5) and local Hamiltonians (1.6) are invariant with respect to the action of the group of generalized Miura transformations. We say that two objects of our theory (i.e., two evolutionary vector fields of the form (1.2), two local Poisson brackets of the form (1.5), or two local Hamiltonians of the form (1.6)) are equivalent if they are related by a generalized Miura transformation.

    Our main goals are

    • classification of Hamiltonian PDEs • selection of integrable PDEs • study of general properties of solutions to these PDEs

    2 Classification problems

    Setting � = 0 in (1.4)–(1.6) one obtains the so-called dispersionless limit of the PDE

    uit = A i j(u)u

    j x. (2.1)

    The dispersionless limit is itself a Hamiltonian PDE

    uit = {ui(x), H0}0 (2.2)

    with respect to the Poisson bracket

    {ui(x), uj(y)}0 = Aij0,0(u)δ′(x− y) + A ij 0,1(u;ux)δ(x− y) (2.3)

    with a very simple local Hamiltonian

    H0 =

    ∫ h0(u) dx. (2.4)

    The original system (1.4) with its Hamiltonian structure (1.5), (1.6) can be considered as a deformation of (2.1)–(2.4).

    4

  • The problem of classification of Hamiltonian systems of our class can therefore be decomposed in two steps:

    • classification of dispersionless Hamiltonian systems; • classification of their perturbations. The dispersionless systems and their Hamiltonian structures will be classified with

    respect to the action of the group of (local) diffeomorphisms, while the perturbations have to be classified up to equivalencies established by the action of the group of generalized Miura transformations.

    The answer to the first question was obtained by S.P.Novikov and the author in [16] under the so-called non-degeneracy assumption2

    det ( Aij0,0(u)

    ) 6= 0 (2.5)

    The non-degeneracy condition is invariant with respect to the action of the group of (local) diffeomorphisms.

    Theorem 2.1 Poisson brackets of the form (2.3) satisfying the non-degeneracy con- dition (2.5) by a change of local coordinates ũi = ũi(u1, . . . , un), det (∂ũi/∂uj) 6= 0 can be reduced to the following standard form

    {ũi(x), ũj(y)}0 = ηijδ′(x− y) (2.6)

    where (ηij) is a constant symmetric non-degenerate matrix.

    The crucial step in the proof of this theorem is the observation that the Riemannian or pseudo-Riemannian metric

    ds2 = gij(u)du iduj, (gij(u)) =

    ( Aij0,0(u)

    ) (2.7)

    defined by the inverse to the matrix ( Aij0,0(u)

    ) has vanishing curvature. Moreover, the

    second coefficient of the bracket (2.3) can be expressed via the Christoffel coefficients of the Levi-Civita connection for the metric

    Aij0,1(u;ux) = −Ais0,0(u)Γ j sk(u)u

    k x. (2.8)

    Choosing locally a system of flat coordinates one reduces the metric to a constant form

    ds2 = ηijdv idvj. (2.9)

    The Christoffel coefficients in these coordinates vanish.

    The matrix (ηij) in (2.6) is the inverse to (ηij).

    We will omit tildes in the notation for the flat coordinates. From the previous theorem one obtains

    2To avoid confusion with non-degeneracy of the Poisson bracket. The bracket (2.3) satisfying (2.5) is always degenerate. The symplectic leaves of this bracket have codimension n.

    5

  • Corollary 2.2 Any Hamiltonian system of dispersionles