On critical behaviour in generalized Kadomtsev--Petviashvili equations

Abstract

An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev--Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painleve' I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture.

References in corpus (4)

• On Hamiltonian perturbations of hyperbolic systems of conservation laws
• Dispersive shock waves in the Kadomtsev-Petviashvili and Two Dimensional Benjamin-Ono equations
• Remarks on the mass constraint for KP type equations
• Shock formation in the dispersionless Kadomtsev-Petviashvili equation

Cited by in corpus (7)

• Symmetry multi-reduction method for partial differential equations with conservation laws
• The quadric ansatz for the $mn$-dispersionless KP equation, and supersymmetric Einstein-Weyl spaces
• Whitham modulation theory for (2+1)-dimensional equations of Kadomtsev-Petviashvili type
• Oblique spatial dispersive shock waves in nonlinear Schrödinger flows
• Spatial structure of shock formation
• Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation
• On the dispersionless Kadomtsev-Petviashvili equation with arbitrary nonlinearity and dimensionality: exact solutions, longtime asymptotics of the Cauchy problem, wave breaking and discontinuous shocks