The long-time behavior of an initial step resulting in a dispersive shock wave (DSW) for the one-dimensional isentropic Euler equations regularized by generic, third-order dispersion is considered by use of Whitham averaging. Under modest assumptions, the jump conditions (DSW locus and speeds) for admissible, weak DSWs are characterized and found to depend only upon the sign of dispersion (conv...

This is a sequel to our paper (Lett. Math. Phys. (2000)), triggered from a question posed by Marcel, Ovsienko, and Roger in their paper (1997). In this paper, we show that the multicomponent (or vector) Ito equation, modified dispersive water wave equation, and modified dispersionless long wave equation are the geodesic flows with respect to an L2 metric on the semidirect product space ̂ Diffs(S...

In this paper, an analytic solution is presented using differential transform method (DTM) for a class of wave equation. The emphasis is on the nonlinear two-dimensional wave equation. The procedures introduced in this paper are in recursive forms which can be used to obtain the closed form of the solutions, if they are required. The method is tested on various examples, and the results reveal ...

This work aims at introducing some energy operators linked to the Teager-Kaiser energy operator and expands it from time to space in the real domain. We then show the decomposition of the functions of space and time differentiable at least twice in R using those energy operators. The second part of the work focuses on using the energy operators to redefine the general wave equation. The method ...

The generalized channel Boussinesq (gcB) two-equation model and the forced channel Kortewegae Vries (cKdV) one-equation model previously derived by the authors are further analysed and discussed in the present study. The gcB model describes the propagation and generation of weakly nonlinear, weakly dispersive and weakly forced long water waves in channels of arbitrary shape that may vary both i...

In this work, we prove a comparison result for general class of nonlinear dispersive unidirectional wave equations. The nature one-dimensional waves occurs because convolution integral in space. For two specific choices the kernel function, Benjamin–Bona–Mahony equation and Rosenau that are particularly suitable to model water elastic waves, respectively, members class. We first an energy estim...

We investigate L(R) → L∞(R4) dispersive estimates for the Schrödinger operator H = −∆ + V when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator Ft satisfying ‖Ft‖L1→L∞ . 1/ log t for t > 2 such that ‖ePac − Ft‖L1→L∞ . t , for t > 2. We als...

We present a method to prove nonlinear instability of solitary waves in dispersive models. Two examples are analyzed: we prove the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow and the transverse nonlinear instability of solitary waves for the cubic nonlinear Schrödinger equation.

The problem of tsunami wave generation by variable meteo-conditions is discussed. The simplified linear and nonlinear shallow water models are derived, and their analytical solutions for a basin of constant depth are discussed. The shallow-water model describes well the properties of the generated tsunami waves for all regimes, except the resonance case. The nonlinear-dispersive model based on ...