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We consider the cubic-quintic nonlinear Schrödinger equation of up to three space dimensions. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the energy space. The main goal of this paper is to investigate the interplay between dispersion and orbital (in-)stability of solitary waves. In space dimension one, it is already known that all solitons are orbitally stable. In dimension two, we show that if the initial data belong to the conformal space, and have at most the mass of the ground state of the cubic two-dimensional Schrödinger equation, then the solution is asymptotically linear. For larger mass, solitary wave solutions exist, and we review several results on their stability. Finally, in dimension three, relying on previous results from other authors, we show that solitons may or may not be orbitally stable. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for the cubic nonlinear Schrodinger equation, perturbed by (higher order) dissipative nonlinearities. We prove global in-time existence of solutions for general initial data in the energy space. In particular we treat the energy-critical case of a quintic dissipation in three space dimensions. [ABSTRACT FROM AUTHOR]

The aim of this paper is the accurate numerical study of the Kadomtsev–Petviashvili (KP) equation. In particular, we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end, we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey–Stewartson system. In a second step, we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg–de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of the high‐frequency asymptotics of such models is reviewed, in particular we highlight the difficulties in crossing caustics when using (time‐dependent) WKB‐methods. Using Wigner measures we present an alternative approach to such asymptotic problems. We first discuss the connection of the naive WKB solutions to transport equations of Liouville type (with mono‐kinetic solutions) in the pre‐breaking regime. Further we show how the Wigner measure approach can be used to analyze high‐frequency limits in the post‐breaking regime, in comparison with the traditional Fourier integral operator method. Finally we present some illustrating examples. [ABSTRACT FROM AUTHOR]