Three Dimensional Reductions of Four-Dimensional Quasilinear Systems

In this paper we show that integrable four dimensional linearly degenerate equations of second order possess infinitely many three dimensional hydrodynamic reductions. Furthermore, they are equipped infinitely many conservation laws and higher commuting flows. We show that the dispersionless limits of nonlocal KdV and nonlocal NLS equations (the so-called Breaking Soliton equations introduced by O.I. Bogoyavlenski) are one and two component reductions (respectively) of one of these four dimensional linearly degenerate equations

Numerical Approach to Painlevé Transcendents on Unbounded Domains

International audienceA multidomain spectral approach for Painlevé transcendents on unbounded domains is presented. This method is designed to study solutions determined uniquely by a, possibly divergent, asymptotic series valid near infinity in a sector and approximates the solution on straight lines lying entirely within said sector without the need of evaluating truncations of the series at any finite point. The accuracy of the method is illustrated for the example of the tritronquée solution to the Painlevé I equation

Numerical Study of Blow-Up Mechanisms for Davey-Stewartson II Systems

International audienceWe present a detailed numerical study of various blow‐up issues in the context of the focusing Davey–Stewartson II equation. To this end, we study Gaussian initial data and perturbations of the lump and the explicit blow‐up solution due to Ozawa. Based on the numerical results it is conjectured that the blow‐up in all cases is self‐similar, and that the time‐dependent scaling behaves as in the Ozawa solution and not as in the stable blow‐up of standard $L^2$ critical nonlinear Schrödinger equation. The blow‐up profile is given by a dynamically rescaled lump

The WDVV Associativity Equations as a High-Frequency Limit

International audienceIn this paper, we present a new “Hamiltonian” approach for construction of integrable systems. We found an intermediate dispersive system of a Camassa–Holm type. This three-component system has simultaneously a high-frequency (short wave) limit equivalent to the remarkable WDVV associativity equations and a dispersionless (long wave) limit coinciding with a dispersionless limit of the Yajima–Oikawa system