Nonlinear analysis and solitary waves for two superposed streaming electrified fluids of uniform depths with rigid boundaries

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the behavior of small perturbations superimposed on traveling wave trains can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. Wave-like solutions to this equation are examined, and different types of localized excitations (envelope solitary waves) are shown to exist. It is shown that when these perturbations are neutrally stable and sufficiently long, solutions to the nonlinear Schrödinger equation may be approximated by the well-known Korteweg-de Vries equation. The speed of the solitary on the interface is seen to be reduced by the electric field. It is found that there are two critical values of the applied voltage that lead to (i) breaking up of the solitary waves, and (ii) bifurcation of solutions of the governing equations. On the other hand, the complex amplitude of standing wave trains near the marginal state is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation, under a suitable transformation, is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solutions with variable speed. Using the tangent hyperbolic method, it is observed that the wave speed increases as well as decreases, with the increase of electric field values, according to the chosen wavenumbers range. Finally, the nonlinear stability analysis is discussed in view of the coefficients of nonlinear Schrödinger equation to show the effects of various physical parameters, and also to recover the some limiting cases studied earlier in the literature.     

Modulation of an internal gravity wave packet in a stratified shear flow

Modulations of an internal gravity wave packet in a stratified shear flow are discussed in the weakly nonlinear and weakly dispersive context. It is shown that the modulations are described by a variable coefficient nonlinear Schrödinger equation when the modulations are confined to the direction of wave propagation. Transverse modulations couple the nonlinear Schrödinger equation to the mean flow equations. For long waves, it is shown that the modulation equations may be somewhat simplified. An Appendix describes the equations governing long wave resonance.     

The use of Volterra series in the analysis of the nonlinear Schrödinger equation

A Volterra series analysis is used to analyse the dispersive behaviour in the frequency domain for the non-linear Schrödinger equation (NLS). It is shown that the solution of the initial value problem for the nonlinear Schrödinger equation admits a local multi-input Volterra series representation. Higher order spatial frequency responses of the nonlinear Schrödinger equation can therefore be defined in a similar manner as for lumped parameter non-linear systems. A systematic procedure is presented to calculate these higher order spatial frequency response functions analytically. The frequency domain behaviour of the equation, subject to Gaussian initial waves, is then investigated to reveal a variety of non-linear phenomena such as self-phase modulation (SPM), cross-phase modulation (CPM), and Raman effects modelled using the NLS.     

Localized traveling wave solution for a logarithmic nonlinear Schrödinger equation

We investigate localized traveling wave solutions for a Schrödinger equation with two logarithmic nonlinear terms under no external potential. It is shown that it can have solitary wave type solutions whose envelope profile depends on the two types of nonlinearity. Remarkably, the profile has cutoffs in the coordinate of propagation. We argue also some fundamental properties that discriminate it from power law type nonlinear Schrödinger equations.     

General high-order rogue waves to nonlinear Schrödinger–Boussinesq equation with the dynamical analysis

Nonlinear Dynamics - General high-order rogue waves of the nonlinear Schrödinger–Boussinesq equation are obtained by the KP-hierarchy reduction theory, and the N-order rogue waves are...     

Governing equations of envelopes created by nearly bichromatic waves on deep water

In this paper, the author derives the modified Schrödinger equation that governs the envelope created by nearly bichromatic waves, which are defined by the waves whose energy is almost concentrated in two closely approached wavenumbers. The stability of the solution of the modified Schrödinger equation for nearly bichromatic waves on deep water is discussed and the fact that the Benjamin–Feir instability occurs in a condition is shown. Moreover, the solutions of the modified Schrödinger equation for nearly bichromatic waves on deep water are obtained and, in a special case, the solution becomes the standing wave solution is shown.     

Equivalence transformations and differential invariants of a generalized cubic–quintic nonlinear Schrödinger equation with variable coefficients

Nonlinear Dynamics - In this paper, a variable-coefficient cubic–quintic nonlinear Schrödinger equation involving five arbitrary real functions of space and time is analyzed from the...     

Contrast of optical activity and rogue wave propagation in chiral materials

Nonlinear Dynamics - We report the contrast of optical activity and properties of nonparaxial optical rogue waves for the higher-order nonparaxial chiral nonlinear Schrödinger (NLS) equation....     

Nonlinear tunneling for controllable rogue waves in two dimensional graded-index waveguides

With the help of the similarity transformation connected the variable-coefficient nonlinear Schrödinger equation with the standard nonlinear Schrödinger equation, we firstly obtain first-order and second-order rogue wave solutions in two dimensional graded-index waveguides. Then, we investigate the nonlinear tunneling of controllable rogue waves when they pass through nonlinear barrier and nonlinear well. Our results indicate that the propagation behaviors of rogue waves, such as postpone, sustainment and restraint, can be manipulated by choosing the relation between the maximum value of the effective propagation distance Z _m and the effective propagation distance corresponding to maximum amplitude of rogue waves Z ₀. Postponed, sustained and restrained rogue waves can tunnel through the nonlinear barrier or well with increasing, unchanged and decreasing amplitudes by modulating the ratio of the amplitudes of rogue waves to barrier or well height.     

Breathers,solitons and rogue waves of the quintic nonlinear Schrödinger equation on various backgrounds

Nonlinear Dynamics - We investigate the generation of breathers, solitons, and rogue waves of the quintic nonlinear Schrödinger equation (QNLSE) on uniform and elliptical backgrounds. The...     

Well-Posedness for Multicomponent Schrödinger–gKdV Systems and Stability of Solitary Waves with Prescribed Mass

In this paper we prove the well-posedness issues of the associated initial value problem, the existence of nontrivial solutions with prescribed \(L^2\)-norm, and the stability of associated solitary waves for two classes of coupled nonlinear dispersive equations. The first problem here describes the nonlinear interaction between two Schrödinger type short waves and a generalized Korteweg-de Vries type long wave and the second problem describes the nonlinear interaction of two generalized Korteweg-de Vries type long waves with a common Schrödinger type short wave. The results here extend many of the previously obtained results for two-component coupled Schrödinger–Korteweg-de Vries systems.     

Two-dimensional bifurcations of stokes waves

Bifurcation techniques are used to obtain a new class of small amplitude water waves of permanent form. This calculation illustrates an approach which can be applied to nonlinear waves of various types to generate new steady solutions from old.Stokes waves are used as a starting point, and the critical value of steepness at which bifurcation can occur is computed for various choices of modulation wavelength and angular orientation. It is found that, for two-dimensional surfaces, bifurcation can occur at small values of wave steepness.Second-order corrections to the wave amplitude, modulation, frequency, and speed, which apply when one moves off the bifurcation point onto a new branch of solutions, are also computed. Two types of new solutions are found, one symmetric with respect to the carrier wave propagation direction, and one asymmetric.The nonlinear Schrödinger equation is used to model water waves, and thus the calculation can be applied rather directly to other systems governed by the nonlinear Schrödinger equation.     

The Linear Stability of Shock Waves for the Nonlinear Schrödinger–Inviscid Burgers System

We investigate the coupling between the nonlinear Schrödinger equation and the inviscid Burgers equation, a system which models interactions between short and long waves, for instance in fluids. Well-posedness for the associated Cauchy problem remains a difficult open problem, and we tackle it here via a linearization technique. Namely, we establish a linearized stability theorem for the Schrödinger–Burgers system, when the reference solution is an entropy-satisfying shock wave to Burgers equation. Our proof is based on suitable energy estimates and on properties of hyperbolic equations with discontinuous coefficients. Numerical experiments support and expand our theoretical results.     

Solitons and rogue wave solutions of focusing and defocusing space shifted nonlocal nonlinear Schrödinger equation

Nonlinear Dynamics - In this paper, we are concerned with the explicit analytic solutions for the focusing and defocusing space shifted nonlocal nonlinear Schrödinger (NLS) equation introduced...     

Wave jumps in media described by a modified Schrödinger equation

The nonlinear Schrödinger equationA _t ±iA _xx+i‖A‖² A=0 describes an envelope of periodic waves with slowly varying parameters on water, in plasmas, and in nonlinear optics [1]. This equation can also be applied to steady periodic waves (the wave amplitude and wave number do not depend on time, the variablest andx are replaced by the variables of a horizontal coordinate system on the surface of the fluid [2]). In the present paper the properties of a modified Schrödinger equation involving the third and higher derivatives are studied. Solutions describing transition regions between uniform wave states are obtained numerically. If the structure of the transition region whose extent increases with time is not considered, these solutions may be interpreted as jumps.     

Soliton-like excitations in the one-dimensional electrical transmission line

The dynamics of modulated waves are studied in the one-dimensional discrete nonlinear electrical transmission line. The contribution of the linear dispersive capacitance is taken into account, and it is shown via the reductive perturbation method that the evolution of such waves in this system is governed by the higher-order nonlinear Schrödinger equation. Passing through the Stokes analysis, we establish a generalized criterion for the Benjamin-Feir instability in the network and determine the exact solutions of the obtained wave equation by using the Pathria-Morris approach.     

Interaction between envelope solitons as a model for freak wave formations. Part I: Long time interactionInteraction de solitons enveloppes comme modèle pour la formation de « freak waves». Partie I : Interaction à long terme

We are concerned by a special mechanism that can explain the formation of freak waves. We study numerically the long time evolution of a surface gravity wave packet, comparing a fully nonlinear model with Schrödinger-like simplified equations. We observe that the interaction between envelope solitons generates large waves. This is predicted by both models. The fully nonlinear simulations show a qualitative behaviour that differs significantly from the ones preticted by Schrödinger models, however. Indeed, the occurence of freak waves is much more frequent with the fully nonlinear model. This is a consequence of the long-time interaction between envelope solitons, which, in the fully nonlinear model, is totally different from the Schrödinger scenario. The fundamental differences appear for times when the simplified equations cease to be valid. Possible statistical models, based on the latter, should hence under-estimate the probability of freak wave formation. To cite this article: D. Clamond, J. Grue, C. R. Mecanique 330 (2002) 575–580.     

On different aspects of the optical rogue waves nature

Rogue waves are giant nonlinear waves that suddenly appear and disappear in oceans and optics. We discuss the facts and fictions related to their strange nature, dynamic generation, ingrained instability, and potential applications. We present rogue wave solutions to the standard cubic nonlinear Schrödinger equation that models many propagation phenomena in nonlinear optics. We propose the method of mode pruning for suppressing the modulation instability of rogue waves. We demonstrate how to produce stable Talbot carpets—recurrent images of light and plasma waves—by rogue waves, for possible use in nanolithography. We point to instances when rogue waves appear as numerical artefacts, due to an inadequate numerical treatment of modulation instability and homoclinic chaos of rogue waves. Finally, we display how statistical analysis based on different numerical procedures can lead to misleading conclusions on the nature of rogue waves.

     

Stability of Standing Waves for the Nonlinear Fractional Schrödinger Equation

We study the standing waves of the nonlinear fractional Schrödinger equation. We obtain that when \(0<\gamma <2s\), the standing waves are orbitally stable; when \(\gamma =2s\), the ground state solitary waves are strongly unstable to blow-up.