A hierarchy of integrable PDEs in 2+1 dimensions associated with 2 - dimensional vector fields

Abstract

We introduce a hierarchy of integrable PDEs in 2+1 dimensions arising from the commutation of 2-dimensional vector fields. We also solve the associated Cauchy problems, using the recently developed Inverse Scattering Transform for 1-parameter families of multidimensional vector fields.

arXiv comment12 pages. To be published in Theoretical Mathematical Physics

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• On a class of multidimensional integrable hierarchies and their reductions
• Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
• Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy
• Nonlocality and the inverse scattering transform for the Pavlov equation
• Dunajski-Tod equation and reductions of the generalized dispersionless 2DTL hierarchy
• On the dressing method for Dunajski anti-self-duality equation
• Dispersionless integrable systems and the Bogomolny equations on an Einstein-Weyl geometry background
• Generalized dKP: Manakov-Santini hierarchy and its waterbag reduction
• Hodograph solutions for the generalized dKP equation
• Dispersionless BKP equation, the Manakov-Santini system and Einstein-Weyl structures
• On an integrable multi-dimensionally consistent 2n+2n-dimensional heavenly-type equation
• Frobenius manifold for the dispersionless Kadomtsev-Petviashvili equation
• The Cauchy problem for the Pavlov equation
• The Cauchy problem for the Pavlov equation with large data