On the Whitham hierarchy: dressing scheme, string equations and additional symmetrie

Abstract

A new description of the universal Whitham hierarchy in terms of a factorization problem in the Lie group of canonical transformations is provided. This scheme allows us to give a natural description of dressing transformations, string equations and additional symmetries for the Whitham hierarchy. We show how to dress any given solution and prove that any solution of the hierarchy may be undressed, and therefore comes from a factorization of a canonical transformation. A particulary important function, related to the $\tau$-function, appears as a potential of the hierarchy. We introduce a class of string equations which extends and contains previous classes of string equations considered by Krichever and by Takasaki and Takebe. The scheme is also applied for an convenient derivation of additional symmetries. Moreover, new functional symmetries of the Zakharov extension of the Benney gas equations are given and the action of additional symmetries over the potential in terms of linear PDEs is characterized.

References in corpus (18)

• Integrable Hierarchies and Dispersionless Limit
• Integrable Structure of Interface Dynamics
• Conformal maps and dispersionless integrable hierarchies
• SDiff(2) KP hierarchy
• Normal random matrix ensemble as a growth problem
• Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP hierarchy
• SDiff(2) Toda equation -- hierarchy, $\tau$ function, and symmetries
• Laplacian Growth and Whitham Equations of Soliton Theory
• Dispersionless Toda hierarchy and two-dimensional string theory
• Complex Curve of the Two Matrix Model and its Tau-function
• Viscous fingering and a shape of an electronic droplet in the Quantum Hall regime
• Quasi-classical limit of Toda hierarchy and W-infinity symmetries
• On Associativity Equations in Dispersionless Integrable Hierarchies
• On the Whitham Hierarchies: Reductions and Hodograph Solutions
• Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy
• On the r-th dispersionless Toda hierarchy I: Factorization problem, symmetries and some solutions
• Exact Solutions of Integrable 2D Contour Dynamics
• Solutions of the Dispersionless Toda Hierarchy Constrained by String Equations

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• The multicomponent 2D Toda hierarchy: dispersionless limit
• Integrable (2+1)-dimensional systems of hydrodynamic type
• A Two-Component Generalization of the Integrable rdDym Equation
• Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions
• Dispersionful analogue of the Whitham hierarchy
• Semiclassical expansions in the Toda hierarchy and the hermitian matrix model
• The integrable heavenly type equations and their Lie-algebraic structure
• Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics
• Loewner equations, Hirota equations and reductions of universal Whitham hierarchy
• Integrability of the Egorov hydrodynamic type systems
• Non-degenerate solutions of universal Whitham hierarchy
• Kernel Formula Approach to the Universal Whitham Hierarchy