Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy

Abstract

We consider two-component integrable generalizations of the dispersionless 2DTL hierarchy connected with non-Hamiltonian vector fields, similar to the Manakov-Santini hierarchy generalizing the dKP hierarchy. They form a one-parametric family connected by hodograph type transformations. Generating equations and Lax-Sato equations are introduced, a dressing scheme based on the vector nonlinear Riemann problem is formulated. The simplest two-component generalization of the dispersionless 2DTL equation is derived, its differential reduction analogous to the Dunajski interpolating system is presented. A symmetric two-component generalization of the dispersionless elliptic 2DTL equation is also constructed.

10 pages, the text of the talk at NEEDS 09. Notations clarified, references added

References in corpus (4)

• Interpolating Dispersionless Integrable System
• The dispersionless 2D Toda equation: dressing, Cauchy problem, longtime behavior, implicit solutions and wave breaking
• A hierarchy of integrable PDEs in 2+1 dimensions associated with 2 - dimensional vector fields
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• Second-order PDEs in 3D with Einstein-Weyl conformal structure
• Dunajski-Tod equation and reductions of the generalized dispersionless 2DTL hierarchy
• Matrix extension of the Manakov-Santini system and integrable chiral model on Einstein-Weyl background
• Projective differential geometry of multidimensional dispersionless integrable hierarchies
• Dispersionless integrable systems and the Bogomolny equations on an Einstein-Weyl geometry background
• Lax representations with non-removable parameter and exotic cohomology of symmetry algebras of PDEs
• From the conformal self-duality equations to the Manakov-Santini system