Interaction Of Short Waves Research Articles

Characteristics of stochastic solutions to the long–short-wave interaction model through Brownian process

Characteristics of stochastic solutions to the long–short-wave interaction model through Brownian process

Exact Jacobi elliptic solutions of some models for the interaction of long and short waves

<abstract><p>Some systems were recently put forth by Nguyen et al. as models for studying the interaction of long and short waves in dispersive media. These systems were shown to possess synchronized Jacobi elliptic solutions as well as synchronized solitary wave solutions under certain constraints, i.e., vector solutions, where the two components are proportional to one another. In this paper, the exact periodic traveling wave solutions to these systems in general were found to be given by Jacobi elliptic functions. Moreover, these cnoidal wave solutions are unique. Thus, the explicit synchronized solutions under some conditions obtained by Nguyen et al. are also indeed unique.</p></abstract>

NOVEL INVESTIGATION OF FRACTIONAL LONG- AND SHORT-WAVE INTERACTION SYSTEM

In ocean engineering, the long- and short-wave interaction system represents a crucial nonlinear evolution equation that elucidates the resonant interaction phenomenon between ocean waves. In this study, we describe the fractional long and short-wave interaction (FLSWI) system employing the M-truncated derivative. Subsequently, we employ the extended fractional [Formula: see text] and the fractional [Formula: see text] methods to address the FLSWI system. These two approaches yields novel and intriguing soliton solutions. To further elucidate the derived soliton solutions, three-dimensional visualizations are constructed and analyzed.

Initial value problem of nonlinear KdV-Schrödinger system

The Korteweg-de Vries (KdV) equation is a mathematical model that describes the propagation of long waves in dispersive media. It takes into account both nonlinearity and dispersion, and is particularly useful for modeling phenomena like solitons. The Nonlinear Schrödinger (NLS) equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. It is a useful model for describing many physical systems, including Bose-Einstein condensates, optical fibers, and water waves. A system that couples the KdV and NLS equations can model the interaction of long and short waves. This system combines the strengths of both models. The long waves described by the KdV equation can affect the behavior of the short waves described by the NLS equation, while the short waves can in turn affect the behavior of the long waves. Such a coupled system has been studied extensively over the last few decades, and has led to important insights into many physical systems. This paper considers the existence of local solutions to the Cauchy problem of KdV-Schrödinger nonlinear system on the basis of literature (Bernard D, Nghiem V N, Benjamin L S <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1088/1751-8113/49/41/415501">2016 <i>J. Phys. A: Math. Theor.</i> <b>49</b> 415501</ext-link>), and also gives the existence space of the local solutions.

Some models for the interaction of long and short waves in dispersive media. Part II: Well-posedness

Some models for the interaction of long and short waves in dispersive media. Part II: Well-posedness

Exact solutions of the stochastic fractional long–short wave interaction system with multiplicative noise in generalized elastic medium

The long–short wave interaction system (L-SWIS) is an important model describing the interaction of two waves propagating in a generalized elastic medium. In this study, the bifurcation and the influence of random interaction on the exact solution of the stochastic fractional long–short wave interaction system (SFL-SWIS) with multiplicative Brownian motion are studied, where the derivative refers to the modified Riemann–Liouville definition. After the Hamiltonian system is established by traveling wave transformation and first-order integration, we obtain abundant exact parametric solutions of SFL-SWIS. Also, the three-dimensional graph of some obtained results are drawn by using the symbolic calculation software Maple, which is used to show the influence of fractional derivative and random interaction on the solution. The obtained solitary wave solutions and exact solutions include the elliptic function solution, trigonometric function solution, exponential function solution and singular periodic solution. This study is the first to investigate the bifurcation and exact solution of SFL-SWIS, and the obtained results are theoretically and practically important to the propagation of optical signals, plasma waves and ocean waves.

New behavior of tsunami and tidal oscillations for Long- and short-wave interaction systems

The Jacobi elliptic function expansion method is one of the most powerful tools for exploring exact solutions of nonlinear partial differential models, which is used in this work to characterize the interaction between one long longitudinal wave and one short transverse wave propagation in a generalized elastic medium. First, we applied a wave transform to the proposed system of equations and obtained an ordinary differential equation. We obtain the values of involved free parameters after performing necessary operations, then substitute the obtained values to the ordinary differential equation by considering the constructed solutions to the ordinary and then to the partial differential equation. As a result, different structures of solutions are constructed such as solitary dark–bright, dark, bright, singular, trigonometric function, Jacobi elliptic function, and hyperbolic function solutions. In order to illustrate the tsunami and tidal oscillations, we draw 3D surfaces and 2D graphics for the obtained solutions by giving a specific value for the involved parameters under the given conditions.

On the Stability of the Periodic Waves for the Benney System

We analyze the Benney model for interaction of short and long waves in resonant water wave interactions. Our particular interest is in the periodic traveling waves, which we construct and study in detail. The main results are that, for all natural values of the parameters, the periodic dnoidal waves are spectrally stable with respect to perturbations of the same period. For another natural set of parameters, we construct the snoidal waves, which exhibit instabilities, in the same setup. Our results are the first instability results in this context. On the other hand, the spectral stability established herein improves significantly on the work of Angulo, Corcho, and Hakkaev [Adv. Difference Equ., 16 (2011), pp. 523--550], which established stability of the dnoidal waves, on a subset of parameter space, by relying on the Grillakis--Shatah theory. Our approach, which turns out to give definite answer for the entire domain of parameters, relies on the instability index theory, as developed by Kapitula, Kevrekidis, and Sandstede [Phys. D, 3--4 (2004), pp. 263--282]; Kapitula, Kevrekidis, and Sandstede [Phys. D, 195 (2004), no. 3--4, 263--282], Phys. D, 201 (2005), pp. 199--201]; Lin and Zeng [Instability, Index Theorem, and Exponential Trichotomy for Linear Hamiltonian PDEs, 2021]; and Pelinovsky [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), pp. 783--812]. Interestingly, end even though the linearized operators are explicit, our spectral analysis requires subtle and detailed analysis of matrix Schrödinger operators in the periodic context, which support some interesting features.

A novel approach to study generalized coupled cubic Schrödinger–Korteweg-de Vries equations

The Kortewegde Vries (KdV) equation represents the propagation of long waves in dispersive media, whereas the cubic nonlinear Schrödinger (CNLS) equation depicts the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A model that couples these two equations seems intriguing for simulating the interaction of long and short waves, which is important in many domains of applied sciences and engineering, and such a system has been investigated in recent decades. This work uses a modified Sardar sub-equation procedure to secure the soliton-type solutions of the generalized cubic nonlinear Schrödinger–Korteweg-de Vries system of equations. For various selections of arbitrary parameters in these solutions, the dynamic properties of some acquired solutions are represented graphically and analyzed. In particular, the dynamics of the bright solitons, dark solitons, mixed bright-dark solitons, W-shaped solitons, M-shaped solitons, periodic waves, and other soliton-type solutions. Our results demonstrated that the proposed technique is highly efficient and effective for the aforementioned problems, as well as other nonlinear problems that may arise in the fields of mathematical physics and engineering.

Brownian motion effects on analytical solutions of a fractional-space long–short-wave interaction with conformable derivative

In this article, we consider the stochastic fractional-space long–short-wave interaction system (SFS-LSWIs) forced by multiplicative Brownian motion. To obtain a new exact stochastic fractional-space solutions, we apply two different methods such as sin–cos method and the Riccati–Bernoulli sub-ODE method. These solutions are essential for explaining some difficult and complicated physical phenomena. Because this system has never been investigated using a combination of multiplicative noise and fractional space, we will generalize certain previously obtained results as special cases. Furthermore, we use Matlab to plot 3D surfaces of analytical solutions produced in this study to show the impact of Brownian motion on the analytical solutions of the SFS-LSWIs.

Mathematical Modeling of Soliton-Like Modes at Optical Rectification

We discuss the results of numerical modeling of forming optical-terahertz bullets at the process of optical rectification. Our calculations are based on a generalization of the well-known Yajima - Oikawa system, which describes the nonlinear interaction of short (optical) and long (terahertz) waves. The generalization relates to situations when the optical component is close to a few-cycle pulse. We study the influence of the number of optical pulse oscillations on the formation of an optical-terahertz bullet. We develop original nonlinear conservative pseudo-spectral difference scheme approximating the generalization of the Yajima-Oikawa system. It is realized with the help of FFT algorithm. Mathematical modeling demonstrates scheme efficiency.

The new soliton solutions for long and short-wave interaction system

The goal of this paper is to discover modern soliton solutions to long and short-wave interaction system by procedures called extended rational sine-cosine and rational sinh-cosh methods. We assume that the equation has a hypothetical soliton solutions. By reorganizing the resulting equations, we obtain a system of equations. Using Maple software, we get unknown coefficients in the system and writing them in the original equation, we obtain new solition solutions of the equation. The results show that the soliton solutions generated by the method for the long and short-wave interaction system are bright, kink type, bright periodic and dark solutions. We provided 3-D figures to illustrate the solutions. Computational results indicate that the method employed in this paper is superior than some other methods used in the literature to solve the same system equations.

A new integrable model of long wave-short wave interaction and linear stability spectra.

We consider the propagation of short waves which generate waves of much longer (infinite) wavelength. Model equations of such long wave–short wave (LS) resonant interaction, including integrable ones, are well known and have received much attention because of their appearance in various physical contexts, particularly fluid dynamics and plasma physics. Here we introduce a new LS integrable model which generalizes those first proposed by Yajima and Oikawa and by Newell. By means of its associated Lax pair, we carry out the linear stability analysis of its continuous wave solutions by introducing the stability spectrum as an algebraic curve in the complex plane. This is done starting from the construction of the eigenfunctions of the linearized LS model equations. The geometrical features of this spectrum are related to the stability/instability properties of the solution under scrutiny. Stability spectra for the plane wave solutions are fully classified in the parameter space together with types of modulational instabilities.

Global Smooth Solutions With Large Data for a System Modeling Aurora Type Phenomena in the 2-Torus

We prove the existence and uniqueness of smooth solutions with large initial data for a system of equations modeling the interaction of short waves, governed by a nonlinear Schrodinger equation, an...

Fundamental solutions for the long–short-wave interaction system

Abstract In this article, the system for the long–short-wave interaction (LS) system is considered. In order to construct some new traveling wave solutions, He’s semi-inverse method is implemented. These solutions may be applicable for some physical environments, such as physics and fluid mechanics. These new solutions show that the proposed method is easy to apply and the proposed technique is a very powerful tool to solve many other nonlinear partial differential equations in applied science.

Cauchy matrix structure of the Mel’nikov model of long–short wave interaction

We propose a systematic method to construct the Mel’nikov model of long–short wave interactions, which is a special case of the Kadomtsev–Petviashvili (KP) equation with self-consistent sources (KPSCS). We show details how the Cauchy matrix approach applies to Mel’nikov's model which is derived as a complex reduction of the KPSCS. As a new result we find that in the dispersion relation of a 1-soliton there is an arbitrary time-dependent function that has previously not reported in the literature about the Mel’nikov model. This function brings time variant velocity for the long wave and also governs the short-wave packet. The variety of interactions of waves resulting from the time-freedom in the dispersion relation is illustrated.

Some Models for the Interaction of Long and Short Waves in Dispersive Media: Part I—Derivation

It is universally accepted that the cubic, nonlinear Schrodinger equation (NLS) models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves, while the Korteweg–de Vries equation (KdV) models the propagation of long waves in dispersive media. A system that couples the two equations seems attractive to model the interaction of long and short waves, and such a system has been studied over the last few decades. However, questions about the validity of this system in the study of water waves were raised in a previous work of one of us where the analysis was presented using the fifth-order KdV as the starting point. These questions will now be settled unequivocally in a series of papers. In this first part, we show that the NLS-KdV system (or even the linear Schrodinger–KdV system) cannot be resulted from the full Euler equations formulated in the study of water waves. In the process of so doing, we also propose a few alternative models for describing the interaction of long and short waves.

On complex wave structures related to the nonlinear long–short wave interaction system: Analytical and numerical techniques

This article presents a survey on the exact and numerical solutions of the nonlinear long–short wave interaction system. The system performs an optical domain, which does not alter during multiplication according to a ticklish equipoise between nonlinear and linear influences in elastic surrounding (the medium that can alter the figure due to the existence of a deforming strength and comes back to its original shape in the absence of this force). The wave in this medium is obtained by vibrations that are the outcomes of the acoustic power. The modified auxiliary equation and the quintic B-spline approaches are investigated in our model to obtain a bundle of solutions to discuss new physical behaviors for this model. Moreover, the stability property is discussed for the analytical solutions via the properties related to the Hamiltonian system to show the range of the ability of solutions to be used in the applications of the model. These novel properties are explained by different types of figures. Finally, the convergence and the absolute error between the obtained solutions are discussed in a table.

Disturbance solutions for the long–short-wave interaction system using bi-random Riccati–Bernoulli sub-ODE method

This article applied the Riccati–Bernoulli (RB) sub-ODE method in order to get new exact solutions for the long–short-wave interaction (LS) equations. Namely, we obtain deterministic and random solutions, since we consider the proposed method in deterministic and random cases. The RB sub-ODE technique gives the travelling wave solutions in forms of hyperbolic, trigonometric and rational functions. It is shown that the proposed method gives a robust mathematical tool for solving nonlinear wave equations in applied science. Furthermore, some bi-random variables and some random distributions are used in random case corresponding to the LS system. The stability for the obtained solutions in random case is considered. In addition, there is a display of several numerical simulations, which helps to understand the physical phenomena of these soliton wave solutions.

Efficient and Accurate Numerical Methods for Long-Wave Short-Wave Interaction Equations in the Semiclassical Limit Regime

Efficient and Accurate Numerical Methods for Long-Wave Short-Wave Interaction Equations in the Semiclassical Limit Regime