The nature of transverse instabilities of dark solitons for the (2+1)-dimensional defocusing nonlinear Schrödinger/Gross–Pitaevskĭi (NLS/GP) equation is considered. Special attention is given to the small (shallow) amplitude regime, which limits to the Kadomtsev–Petviashvili (KP) equation. We study analytically and numerically the eigenvalues of the linearized NLS/GP equation. The dispersion re...

We consider the linearization of the three-dimensional water waves equation with surface tension about a flat interface. Using oscillatory integral methods, we prove that solutions of this equation demonstrate dispersive decay at the somewhat surprising rate of t−5/6. This rate is due to competition between surface tension and gravitation at O(1) wave numbers and is connected to the fact that, ...

In this paper, we obtained the 1-soliton solutions of the symmetric regularized long wave (SRLW) equation and the (3+1)-dimensional shallow water wave equations. Solitary wave ansatz method is used to carry out the integration of the equations and obtain topological soliton solutions The physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. Note t...

The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schrödinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U(1). For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the free Schrödinger equation. T...

The fractional derivatives in the sense of Caputo, and the homotopy analysis method (HAM) are used to construct the approximate solutions for nonlinear fractional dispersive long wave equation with reaspect to time fractional derivative. The HAM contains a certain auxiliary parameter which provides us with a simple way to adjust and control the convergence region and rate of convergence of the ...

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...

A Dispersive Wave Equation in 2 + 1 dimensions (2LDW) widely discussed by different authors is shown to be nothing but the modified version of the Generalized Dispersive Wave Equation (GLDW). Using Singularity Analysis and techniques based upon the Painlevé Property leading to the Double Singular Manifold Expansion we shall find the Miura Transformation which converts the 2LDW Equation into the...

Using a traveling wave reduction technique, we have shown that Maccari equation, (2?1)-dimensional nonlinear Schrödinger equation, medium equal width equation, (3?1)-dimensional modified KdV–Zakharov– Kuznetsev equation, (2?1)-dimensional long wave-short wave resonance interaction equation, perturbed nonlinear Schrödinger equation can be reduced to the same family of auxiliary elliptic-like equ...

The propagation of linear water waves over a three-dimensional ocean is modelled using the mildslope equation. Various parabolic wave models are described that approximate the governing elliptic partial differential equation, and so are very convenient for computing wave propagation over large distances. Several aspects are discussed: computation of the reflected wavefield, the construction of ...

The equal width (EW) equation for long waves propagating in the positive x-direction, has the form 0    xxt x t u uu u   (6.1.1) where  and  are positive constants, which require the boundary conditions 0  u as   x .The EW equation is a model nonlinear partial differential equation for the simulation of one-dimensional wave propagation in nonlinear media with dispersion process. Soli...