The Inverse Spectral Transform for the Dunajski hierarchy and some of its reductions, I: Cauchy problem and longtime behavior of solutions

Abstract

In this paper we apply the formal Inverse Spectral Transform for integrable dispersionless PDEs arising from the commutation condition of pairs of one-parameter families of vector fields, recently developed by S. V. Manakov and one of the authors, to one distinguished class of equations, the so-called Dunajski hierarchy. We concentrate, for concreteness, i) on the system of PDEs characterizing a general anti-self-dual conformal structure in neutral signature, ii) on its first commuting flow, and iii) on some of their basic and novel reductions. We formally solve their Cauchy problem and we use it to construct the longtime behavior of solutions, showing, in particular, that unlike the case of soliton PDEs, different dispersionless PDEs belonging to the same hierarchy of commuting flows evolve in time in very different ways, exhibiting either a smooth dynamics or a gradient catastrophe at finite time.

References in corpus (26)

• Integrable Hierarchies and Dispersionless Limit
• Integrable Structure of Interface Dynamics
• On integrability of (2+1)-dimensional quasilinear systems
• Conformal maps and dispersionless integrable hierarchies
• Integrable hydrodynamic chains
• Einstein-Weyl geometry, the dKP equation and twistor theory
• A class of Einstein--Weyl spaces associated to an integrable system of hydrodynamic type
• The Cauchy Problem on the Plane for the Dispersionless Kadomtsev - Petviashvili Equation
• Anti-self-dual four-manifolds with a parallel real spinor
• SDiff(2) Toda equation -- hierarchy, $\tau$ function, and symmetries
• Inverse Scattering Problem for Vector Fields and the Cauchy Problem for the Heavenly Equation
• Hyper-K{ä}hler Hierarchies and their twistor theory
• Twistor theory of hyper-K{ä}hler metrics with hidden symmetries
• Dunajski generalization of the second heavenly equation: dressing method and the hierarchy
• Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains
• Towards a theory of differential constraints of a hydrodynamic hierarchy
• On the solutions of the dKP equation: nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking
• Einstein--Weyl spaces and dispersionless Kadomtsev--Petviashvili equation from Painlevé I and II
• Multi-Hamiltonian structure of Plebanski's second heavenly equation
• Coisotropic deformations of associative algebras and dispersionless integrable hierarchies
• On the dbar-dressing method applicable to heavenly equation
• Bubble break-off in Hele-Shaw flows : Singularities and integrable structures
• Dispersionless integrable equations as coisotropic deformations. Extensions and reductions
• `Interpolating' differential reductions of multidimensional integrable hierarchies
• Grassmannians Gr(N-1,N+1), closed differential N-1 forms and N-dimensional integrable systems
• Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields

Cited by in corpus (4)

• Twistor theory at fifty: from contour integrals to twistor strings
• On the Einstein-Weyl and conformal self-duality equations
• Null Kähler geometry and isomonodromic deformations
• On the dispersionless Davey-Stewartson system: Hamiltonian vector fields Lax pair and relevant nonlinear Riemann-Hilbert problem for dDS-II system