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16
On universality of critical behaviour in Hamiltonian PDEs, in Geometry
 Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser
"... on the occasion of his 70th birthday. Abstract. Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the subclass of Hamiltonian PDEs with one spatial dimensi ..."
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on the occasion of his 70th birthday. Abstract. Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the subclass of Hamiltonian PDEs with one spatial dimension. For the systems of order one or two we describe the local structure of singularities of a generic solution to the unperturbed system near the point of “gradient catastrophe ” in terms of standard objects of the classical singularity theory; we argue that their perturbed companions must be given by certain special solutions of Painlevé equations and their generalizations. Contents
Dispersive deformations of hydrodynamic reductions of 2D dispersionless integrable systems
 J. Phys. A: Math. Theor
, 1989
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On Properties of Hamiltonian Structures for a Class of Evolutionary PDEs
, 711
"... In [17] it is proved that for certain class of perturbations of the hyperbolic equation ut = f(u)ux, there exist changes of coordinate, called quasiMiura transformations, that reduce the perturbed equations to the unperturbed one. We prove in the present paper that if in addition the perturbed equa ..."
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In [17] it is proved that for certain class of perturbations of the hyperbolic equation ut = f(u)ux, there exist changes of coordinate, called quasiMiura transformations, that reduce the perturbed equations to the unperturbed one. We prove in the present paper that if in addition the perturbed equations possess Hamiltonian structures of certain type, the same quasiMiura transformations also reduce the Hamiltonian structures to their leading terms. By applying this result, we obtain a criterion of the existence of Hamiltonian structures for a class of scalar evolutionary PDEs and an algorithm to find out the Hamiltonian structures. Key words: Hamiltonian structure, quasiMiura transformation, quasitriviality 1
Zhang Y.: Jacobi structures of evolutionary partial differential equations
 Adv. Math
, 2011
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Hamiltonian PDEs: deformations, integrability, solutions
 J. Phys. A: Math. Theor
"... 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 behav ..."
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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.
Integrable equations in 2 + 1 dimensions: deformations of dispersionless limits
, 903
"... We classify integrable third order equations in 2 + 1 dimensions which generalize the examples of KadomtsevPetviashvili, VeselovNovikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of integrable systems in 2 + 1 dimensions possess infinitely many mul ..."
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We classify integrable third order equations in 2 + 1 dimensions which generalize the examples of KadomtsevPetviashvili, VeselovNovikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of integrable systems in 2 + 1 dimensions possess infinitely many multiphase solutions coming from the socalled hydrodynamic reductions. In this paper we adopt a novel perturbative approach to the classification problem. Based on the method of hydrodynamic reductions, we first classify integrable quasilinear systems which may (potentially) occur as dispersionless limits of soliton equations in 2 + 1 dimensions. To reconstruct dispersive deformations, we require that all hydrodynamic reductions of the dispersionless limit are inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third order equations, some of which are apparently new.
Zhang Y.: Bihamiltonian Cohomologies and Integrable Hierarchies II: The General Case
 In preparation
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A deformation of the method of characteristics and the Cauchy problem for Hamiltonian PDEs in the small dispersion limit
"... We introduce a deformation of the method of characteristics valid for Hamiltonian perturbations of a scalar conservation law in the small dispersion limit. Our method of analysis is based on the “variational string equation”, a functionaldifferential relation originally introduced by Dubrovin in a ..."
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We introduce a deformation of the method of characteristics valid for Hamiltonian perturbations of a scalar conservation law in the small dispersion limit. Our method of analysis is based on the “variational string equation”, a functionaldifferential relation originally introduced by Dubrovin in a particular case, of which we lay the mathematical foundation. Starting from first principles, we construct the string equation explicitly up to the fourth order in perturbation theory, and we show that the solution to the Cauchy problem of the Hamiltonian partial differential equation (PDE) satisfies the appropriate string equation in the small dispersion limit. We apply our construction to explicitly compute the first two perturbative corrections of the solution to the general Hamiltonian PDE. In the Korteweg–de Vries (KdV) case, we prove the existence of a quasitriviality transformation at any order and for arbitrary initial data.
dispersionless limits
"... We classify integrable third order equations in 2 + 1 dimensions which generalize the examples of KadomtsevPetviashvili, VeselovNovikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of integrable systems in 2 + 1 dimensions possess infinitely many mul ..."
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We classify integrable third order equations in 2 + 1 dimensions which generalize the examples of KadomtsevPetviashvili, VeselovNovikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of integrable systems in 2 + 1 dimensions possess infinitely many multiphase solutions coming from the socalled hydrodynamic reductions. In this paper we adopt a novel perturbative approach to the classification problem. Based on the method of hydrodynamic reductions, we first classify integrable quasilinear systems which may (potentially) occur as dispersionless limits of soliton equations in 2 + 1 dimensions. To reconstruct dispersive deformations, we require that all hydrodynamic reductions of the dispersionless limit are inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third order equations, some of which are apparently new.
Frobenius Manifolds and . . . The DrinfeldSokolov Bihamiltonian Structures
, 2007
"... The DrinfeldSokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete set of invariants of the related bihamiltonian structures with respect to the group of Miura type transformations. ..."
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The DrinfeldSokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete set of invariants of the related bihamiltonian structures with respect to the group of Miura type transformations.