Interaction Of Short Waves Research Articles

A numerical study of the semi‐classical limit for three‐coupled long wave–short wave interaction equations

We study numerically the semi‐classical limit for three‐coupled long wave–short wave interaction equations. The Fourier–Galerkin semi‐discretization is proved to be spectrally convergent in an appropriate energy space. We propose a split‐step Fourier method in the semi‐classical regime with the discussion of the meshing strategy, which is necessary to obtain correct numerical solution. Plane wave solution with weak and strong initial phases, solitary wave solution and Gaussian solution are considered to investigate the semi‐classical limit.

WKB analysis for the three coupled long wave–short wave interaction equations

ABSTRACT This paper is devoted to the WKB analysis of the three coupled long wave–short wave interaction (LSI for short) equations. We consider the zero-dispersion limit of the LSI for initial data with Sobolev regularity, before the shocks appear in the limit system. For the smooth solution, the limit system is given by the two fluids equations. The split-step Fourier method is also employed to justify the numerical simulation of the small dispersion limit.

Travelling Wave Solutions for the Long–Short–Wave Interaction System Using the Unified Method

The Long-Short-Wave interaction system plays a significant role between low frequency long waves and high frequency short waves. Besides being an important complex model it is also related with several physical phenomenon such as, gravity and water waves, plasma and bio-physics, nonlinear optic. This system was handled by many researchers and solutions of this system were obtained by using different methods. In this paper, new solitary type solutions for the Long-short-wave interaction system are formaly drived by using different method namely the unified method. The obtained solutions can be considered as improved version of the previous solutions. This study also shows the efficiency of the unified method.

On the classification of conservation laws and soliton solutions of the long short-wave interaction system

In this paper, the classification of conservation laws (Cls) of the long short-wave interaction system (LSWS) which appears in fluid mechanics as well as plasma physics is implemented using two Cls theorems, namely, the multipliers approach and the new conservation theorem. The LSWS describes the interaction between one long longitudinal wave and one short transverse wave propagating in a generalized elastic medium. The zeroth-order multipliers and the nonlinear self-adjoint substitutions of the model are derived. Considering the fact that the new conservation theorem needs Lie point symmetries in constructing Cls, we derive the point symmetries of a system of nonlinear partial differential equations (NPDEs) acquired by transforming the model into real and imaginary components. Moreover, we derive some kink-type, bell-shaped, singular and combined soliton solutions to the model using the powerful sine-Gordon expansion method (SGEM). Some figures are presented to show the physical interpretations of the acquired results.

On the nonintegrability of equations for long- and short-wave interactions

We examine the integrability of two models used for the interaction of long and short waves in dispersive media. One is more classical but arguably cannot be derived from the underlying water wave equations, while the other one was recently derived. We use the method of Zakharov and Schulman to attempt to construct conserved quantities for these systems at different orders in the magnitude of the solutions. The coupled KdV–NLS model is shown to be nonintegrable, due to the presence of fourth-order resonances. A coupled real KdV–complex KdV system is shown to suffer the same fate, except for three special choices of the coefficients, where higher-order calculations or a different approach are necessary to conclude integrability or the absence thereof.

On high-order perturbation expansion for the study of long–short wave interactions

In high-order analysis and simulation of long–short surface wave interaction using mode decomposition, ‘divergent’ terms of the form $k_{S}a_{L}=O(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D716})\gg 1$ appear in the high-order expansions, where $k_{L,S}$, $a_{L,S}$ are respectively the long, short modal wavenumbers and amplitudes, with $\unicode[STIX]{x1D6FE}\equiv k_{S}/k_{L}\gg 1$ and $k_{L}a_{L}\sim k_{S}a_{S}=O(\unicode[STIX]{x1D716})$ finite. We address the effect of these terms on the numerical scheme, showing numerical cancellation at all orders $m$; but increasing ill-conditioning of the numerics with $\unicode[STIX]{x1D6FE}$ and $m$, which we quantify. In the context of mode decomposition, we show theoretical exact cancellation of the divergent terms up to $m=3$, extending the existing result of Brueckner & West (J. Fluid Mech., vol. 196, 1988, pp. 585–592) and supporting the conjecture that this is obtained for all orders $m$. We show the latter by developing a theoretical proof for any $m$ using a Dirichlet–Neumann operator and mathematical induction. The implication of the theoretical proof on the numerical simulation of long–short wave interaction is discussed.

On some new analytical solutions for the nonlinear long–short wave interaction system

This paper deals with exact soliton solutions of the nonlinear long–short wave interaction system, utilizing two analytical methods. The system of coupled long–short wave interaction equations is investigated with the help of two analytical methods, namely, the generalized \(\tan (\phi /2)\)-expansion method and He’s semi-inverse variational method. Moreover, in this paper we generalize two aforementioned methods which give new soliton wave solutions. As a consequence, solutions are including solitons, kink, periodic and rational solutions. Moreover, dark, bright and singular solition solutions of the coupled long–short wave interaction equations have been found. All solutions have been verified back into its corresponding equation with the aid of maple package program. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. Finally, we believe that the executed methods are robust and efficient than other methods and the obtained solutions in this paper can help us to understand the soliton waves in the fields of physics and mechanics.

On the interaction of short linear internal waves with internal solitary waves

Abstract. We study the interaction of small-scale internal wave packets with a large-scale internal solitary wave using high-resolution direct numerical simulations in two dimensions. A key finding is that for wave packets whose constituent waves are short in comparison to the solitary wave width, the interaction leads to an almost complete destruction of the short waves. For mode-1 short waves in the packet, as the wavelength increases, a cutoff is reached, and for larger wavelengths the waves in the packet are able to maintain their structure after the interaction. This cutoff corresponds to the wavelength at which the phase speed of the short waves upstream of the solitary wave exceeds the maximum current induced by the solitary wave. For mode-2 waves in the packet, however, no corresponding cutoff is found. Analysis based on linear theory suggests that the destruction of short waves occurs primarily due to the velocity shear induced by the solitary wave, which alters the vertical structure of the waves so that significant wave activity is found only above (below) the deformed pycnocline for overtaking (head-on) collisions. The deformation of vertical structure is more significant for waves with a smaller wavelength. Consequently, it is more difficult for these waves to adjust to the new solitary-wave-induced background environment. These results suggest that through the interaction with relatively smaller length scale waves, internal solitary waves can provide a means to decrease the power observed in the short-wave band in the coastal ocean.

On the Cauchy problem for the Zakharov-Rubenchik/ Benney-Roskes system

We address various issues concerning the Cauchy problem for the Zakharov-Rubenchik system(known as the Benney-Roskes system in water waves theory), which models the interaction of short and long waves in many physical situations. Motivated by the transverse stability/instability of the one-dimensional solitary wave (line solitary), we study the Cauchy problem in the background of a line solitary wave.

The interaction of long and short waves in dispersive media

The KdV equation models the propagation of long waves in dispersive media, while the NLS equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A system that couples the two equations to model the interaction of long and short waves is mathematically attractive and such a system has been studied over the last decades. We evaluate the validity of this system as a physical model, discussing two main problems. First, only the system coupling the linear Schrödinger equation with KdV has been derived in the literature. Second, the time variables appearing in the equations are of a different order. It appears that in the manuscripts that study the coupled NLS-KdV system, an assumption has been made that the coupled system can be derived, justifying its mathematical study. In fact, this is true even for the papers where the asymptotic derivation with the problems described above is presented. In addition to discussing these inconsistencies, we present an alternative system describing the interaction of long and short waves.

Analytical solutions for nonlinear long–short wave interaction systems with highly complex structure

In this paper, we investigate and use the new modified exp(−Ω(ξ))-expansion method, (MEM). We apply the new MEM to nonlinear long–short-wave interaction systems (NLSWIS). Among our findings are sets of solutions including, but not limited to, new hyperbolic, complex, and dark soliton solutions. Not only is MEM shown to be highly adaptable for partial differential equations with strong nonlinearities, but also, it turns out to be highly efficient, despite its ease.

Comparison between model equations for long waves and blow-up phenomena

Recently, to describe the unidirectional propagation of water waves, Bona et al. [7] introduced a fifth order KdV–BBM type model(0.1)ηt+ηx−16ηxxt+δ1ηxxxxt+δ2ηxxxxx+34(η2)x+γ(η2)xxx−112(ηx2)x−14(η3)x=0, where η=η(x,t) is a real-valued function, and δ1>0, δ2,γ∈R. In this work, we plan to compare solution of the initial value problem (IVP) associated to the fifth-order KDV–BBM type model (0.1) to that of the IVP associated to the fifth-order KdV model(0.2)ut+δ3uxxxxx+c1uxuxx+c2uuxxx+c3u2ux=0, where u=u(x,t) is a real-valued function and δ3, c1, c2 and c3 are real constants with δ3≠0. This later model (0.2) was proposed by Benney in [4] to describe the interaction of long and short waves. Also, we will study the possibility of blow-up phenomenon of the fifth-order KDV–BBM type model under certain restrictions on the coefficients.

Multicomponent long-wave-short-wave resonance interaction system: Bright solitons, energy-sharing collisions, and resonant solitons.

We consider a general multicomponent (2+1)-dimensional long-wave-short-wave resonance interaction (LSRI) system with arbitrary nonlinearity coefficients, which describes the nonlinear resonance interaction of multiple short waves with a long wave in two spatial dimensions. The general multicomponent LSRI system is shown to be integrable by performing the Painlevé analysis. Then we construct the exact bright multisoliton solutions by applying the Hirota's bilinearization method and study the propagation and collision dynamics of bright solitons in detail. Particularly, we investigate the head-on and overtaking collisions of bright solitons and explore two types of energy-sharing collisions as well as standard elastic collision. We have also corroborated the obtained analytical one-soliton solution by direct numerical simulation. Also, we discuss the formation and dynamics of resonant solitons. Interestingly, we demonstrate the formation of resonant solitons admitting breather-like (localized periodic pulse train) structure and also large amplitude localized structures akin to rogue waves coexisting with solitons. For completeness, we have also obtained dark one- and two-soliton solutions and studied their dynamics briefly.

Propagation and interaction of short waves in a homogeneous transversally isotropic elastic medium

The Cauchy problem for the equations of motion of a homogeneous transversally isotropic elastic medium is considered. For its solution, a short-wavelength asymptotic expansion is constructed, which is also applicable near specific directions. The resonance set, i.e., the set of points at which the ray expansion cannot be used is described.

Numerical modelling of long waves amplification at Marina di Carrara Harbour

In this paper we test two different numerical models for the assessment of the long waves climate inside a harbour, induced by nonlinear interaction of short waves. One model is based on the nonlinear Boussinesq Type Equations; given the properties of the incoming short waves it is able to model the growth of the long waves, and their propagation and amplification inside the harbour basin. The other model uses a simplified formula for the calculation of the incoming long waves, and solves the linearized Mild Slope Equation to reproduce the long waves penetration into the harbour. The models are applied to the harbour of Marina di Carrara (Italy), where a field measurement network of both short and long waves is operational since 2005. Short waves records are obtained by a directional buoy deployed at about 1km offshore the harbour entrance, whereas the long waves measurements inside the basin are obtained from 8 pressure transducers installed along the harbour inner perimeter. About 3 years of measurements (December 2005 to October 2008) are considered in this paper. Both the models are shown to provide reasonable estimates of the long waves climate inside the harbour. Advantages and disadvantages of each are discussed.

A Conservative Local Discontinuous Galerkin Method for the Schrödinger-KdV System

AbstractIn this paper we develop a conservative local discontinuous Galerkin (LDG) method for the Schrödinger-Korteweg-de Vries (Sch-KdV) system, which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Conservative quantities in the discrete version of the number of plasmons, energy of the oscillations and the number of particles are proved for the LDG scheme of the Sch-KdV system. Semi-implicit time discretization is adopted to relax the time step constraint from the high order spatial derivatives. Numerical results for accuracy tests of stationary traveling soliton, and the collision of solitons are shown. Numerical experiments illustrate the accuracy and capability of the method.

On the exact solutions to the long—short-wave interaction system

The complete discrimination system for polynomial method is applied to the long—short-wave interaction system to obtain the classifications of single traveling wave solutions. Compared with the solutions given by the (G′/G)-expansion method, we gain some new solutions.

Interaction behaviors between solitons for the (2+1)-dimensional long wave short wave interaction model

With the help of the entangled mapping approach based on the general reduction theory, we at first derive new type of variable separation solution for the (2+1)-dimensional long wave short wave interaction model (LWSWIM). Next, localized coherent structures about multi-valued functions, i.e. special dromion–compacton, special peakon–compacton and semifoldon–compacton, and the interactions between them are discussed. At last, we carry out asymptotic behavior analysis for the localized excitations. Moreover, the phase shifts for all the local excitations are investigated explicitly and graphically.

Transverse linear instability of solitary waves for coupled long-wave–short-wave interaction equations

In this paper, we investigate the transverse linear instability of one-dimensional solitary wave solutions of the coupled system of two-dimensional long-wave–short-wave interaction equations. We show that the one-dimensional solitary waves are linearly unstable to perturbations in the transverse direction if the coefficient of the term associated with transverse effects is negative. This transverse instability condition coincides with the non-existence condition identified in the literature for two-dimensional localized solitary wave solutions of the coupled system.

Periodic and solitary wave solutions of coupled nonlinear wave equations using the first integral method

In this paper, the periodic and solitary wave solutions of two selected systems of nonlinear wave equations are obtained using the first integral method. The long–short-wave interaction system and Bogoyavlenskii equations are considered as examples, and this approach can also be applied to other coupled nonlinear differential equations.