The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vries equations over a slowly varying random bottom is rigorously studied. One motivation for studying such a system is better understanding the unidirectional motion of interacting surface and internal waves for a fluid system that is formed of two immiscible layers. It was shown recently by Craig-Guyenne-Sulem [1] that in the regime where the internal wave has a large amplitude and a long wavelength, the dynamics of the surface of the fluid is described by the Schrödinger equation, while that of the internal wave is described by the Korteweg-de Vries equation. The purpose of this letter is to show that in the presence of a slowly varying random bottom, the coupled waves evolve adiabatically over a long time scale. The analysis covers the cases when the surface wave is a stable bound state or a long-lived metastable state.

We study the spectral stability of solitary wave solutions to the nonlinear Dirac equation in one dimension. We focus on the Dirac equation with cubic nonlinearity, known as the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model. Presented numerical computations of the spectrum of linearization at a solitary wave show that the solitary waves are spectrally stable. We corroborate our results by finding explicit expressions for several of the eigenfunctions. Some of the analytic results hold for the nonlinear Dirac equation with generic nonlinearity.

We consider two atomic transitions excited by two variable laser fields in a three-level system. We study the soliton-pair propagation out of resonance and under thermal bath effect. We present general analytical implicit expression of the soliton-pair shape. Furthermore, we show that when the coupling to the environment exceeds a critical value, the soliton-pair propagation through three-level atomic system will be prohibited.

A linear, uniformly stratified ocean model is used to investigate propagation of baroclinic Kelvin waves in a cylindrical basin. It is found that smaller wave amplitudes are inherent to higher mode individual terms of the obtained solutions that are also evanescent away of a costal line toward the center of the circular basin. It is also shown that the individual terms if the obtained solutions can be visualized as spinning patterns in rotating stratified fluid confined in a circular basin. Moreover, the fluid patterns look rotating in an anticlockwise sense looking above the North Pole and that spinning is more intensive for smaller mode numbers. Finally, we observe the existence of the oceanic region where the pressure increases relatively rapidly with the depth.

Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie’s group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.

In this paper, we show finite time blow-up of solutions of the p−wave equation in ℝN, with critical Sobolev exponent. Our work extends a result by Galaktionov and Pohozaev [4]

Two-dimensional inviscid channel flow of an incompressible fluid is considered. It is shown that if the flow is steady and features no horizontal stagnation, then the flow must necessarily be a parallel shear flow.

The influence of a global delayed feedback control which acts on a system governed by a subcritical complex Ginzburg-Landau equation is considered. The method based on a variational principle is applied for the derivation of low-dimensional evolution models. In the framework of those models, one-pulse and two-pulse solutions are found, and their linear stability analysis is carried out. The application of the finite-dimensional model allows to reveal the existence of chaotic oscillatory regimes and regimes with double-period and quadruple-period oscillations. The diagram of regimes resembles those found in the damped-driven nonlinear Schrödinger equation. The obtained results are compared with the results of direct numerical simulations of the original problem.

We review some recent results concerning Gibbs measures for nonlinear Schrödinger equations (NLS), with implications for the theory of the NLS, including stability and typicality of solitary wave structures. In particular, we discuss the Gibbs measures of the discrete NLS in three dimensions, where there is a striking phase transition to soliton-like behavior.

A simple explicit numerical scheme is proposed for the solution of the Gardner–Ostrovsky equation (ut + cux + α uux + α1u2ux + βuxxx)x = γu which is also known as the extended rotation-modified Korteweg–de Vries (KdV) equation. This equation is used for the description of internal oceanic waves affected by Earth’ rotation. Particular versions of this equation with zero some of coefficients, α, α1, β, or γ are also known in numerous applications. The proposed numerical scheme is a further development of the well-known finite-difference scheme earlier used for the solution of the KdV equation. The scheme is of the second order accuracy both on temporal and spatial variables. The stability analysis of the scheme is presented for infinitesimal perturbations. The conditions for the calculations with the appropriate accuracy have been found. Examples of calculations with the periodic boundary conditions are presented to illustrate the robustness of the proposed scheme.

The matrix KdV equation with a negative dispersion term is considered in the right upper quarter–plane. The evolution law is derived for the Weyl function of a corresponding auxiliary linear system. Using the low energy asymptotics of the Weyl functions, the unboundedness of solutions is obtained for some classes of the initial–boundary conditions.

We obtain solvability conditions in H6(ℝ3) for a sixth order partial differential equation which is the linearized Cahn-Hilliard problem using the results derived for a Schrödinger type operator without Fredholm property in our preceding article [18].