Solitary wave families of a generalized microstructure PDE

Wave propagation in a generalized microstructure PDE, under the Mindlin relations, is considered. Limited analytic results exist for the occurrence of one family of solitary wave solutions of these equations. Since solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions here (via normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves, we find a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. The new family of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work are also mentioned.     

Soliton and Periodic Wave Solutions of the Nonlinear Loaded (3+1)-Dimensional Version of the Benjamin-Ono Equation by Functional Variable Method

In this article, we establish new travelling wave solutions for the nonlinear loaded (3+1)-dimensional version of the Benjamin-Ono equation by the functional variable method. The performance of this method is reliable and effective and the method provides the exact solitary wave solutions and periodic wave solutions. The solution procedure is very simple and the traveling wave solutions are expressed by hyperbolic functions and trigonometric functions. After visualizing the graphs of the soliton solutions and the periodic wave solutions, the use of distinct values of random parameters is demonstrated to better understand their physical features. It has been shown that the method provides a very effective and powerful mathematical tool for solving nonlinear equations in mathematical physics.     

Construction of traveling wave solutions of the (2 + 1)-dimensional modified KdV-KP equation

Traveling wave solutions have played a vital role in demonstrating the wave character of nonlinear problems emerging in the field of mathematical sciences and engineering. To depict the nature of propagation of the nonlinear waves in nature, a range of nonlinear evolution equations has been proposed and investigated in the existing literature. In this article, solitary and traveling periodic wave solutions for the (2 + 1)-dimensional modified KdV-KP equation are derived by employing an ansatz method, named the enhanced (G′/G)-expansion method. For this continued equation, abundant solitary wave solutions and nonlinear periodic wave solutions, along with some free parameters, are obtained. We have derived the exact expressions for the solitary waves that arise in the continuum-modified KdV-KP model. We study the significance of parameters numerically that arise in the obtained solutions. These parameters play an important role in the physical structure and propagation directions of the wave that characterizes the wave pattern. We discuss the relation between velocity and parameters and illustrate them graphically. Our numerical analysis suggests that the taller solitons are narrower than shorter waves and can travel faster. In addition, graphical representations of some obtained solutions along with their contour plot and wave train profiles are presented. The speed, as well as the profile of these solitary waves, is highly sensitive to the free parameters. Our results establish that the continuum-modified KdV-KP system supports solitary waves having different shapes and speeds for different values of the parameters.     

On the Stability of Solitary Water Waves with a Point Vortex

This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small-amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite-dimensional Hamiltonian systems in the presence of symmetry. This is in the spirit of the seminal work of Grillakis, Shatah, and Strauss (GSS) , but with hypotheses that are relaxed in a number of ways necessary for the point vortex system, and for other hydrodynamical applications more broadly. In particular, we are able to allow the Poisson map to have merely dense range, as opposed to being surjective, and to be state-dependent. As a second application of the general theory, we consider a family of nonlinear dispersive PDEs that includes the generalized Korteweg–de Vries (KdV) and Benjamin-Ono equations. The stability or instability of solitary waves for these systems has been studied extensively, notably by Bona, Souganidis, and Strauss , who used a modification of the GSS method. We provide a new, more direct proof of these results, as a straightforward consequence of our abstract theory. At the same time, we allow fractional dispersion and obtain a new instability result for fractional KdV. © 2020 Wiley Periodicals, Inc.     

Stability and long time evolution of the periodic solutions to the two coupled nonlinear Schrödinger equations

The system of two coupled nonlinear Schrödinger equations has wide applications in physics. In the past, the main attention has been their solitary waves. Here we turn our attention to their periodic wave solutions. In this paper, the stability of the periodic solutions is studied analytically and the criteria for the stability are obtained. The long time evolution of the solutions to the coupled system is studied numerically for the unstable case emphasizing wave–wave interactions in nonlinear optics. Different kinds of evolution are observed depending on the coefficients of the system and the parameters of the unperturbed waves and perturbation. For certain ranges of parameters, the evolution appears to be periodic, while for some other ranges of parameters, solitary wave or solitary wave pairs can be excited among the irregular background although often the evolution is completely chaotic.     

Solitary wave solutions to the Isobe-Kakinuma model for water waves

We consider the Isobe-Kakinuma model for two-dimensional water waves in the case of a flat bottom. The Isobe-Kakinuma model is a system of Euler-Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. We show theoretically the existence of a family of small amplitude solitary wave solutions to the Isobe-Kakinuma model in the long wave regime. Numerical analysis for large amplitude solitary wave solutions is also provided and suggests the existence of a solitary wave of extreme form with a sharp crest.     

A Numerical Study of the Exact Evolution Equations for Surface Waves in Water of Finite Depth

We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate the exact evolution of nonlinear free surface waves governed by the two-dimensional Euler equations. These equations are written in the transformed plane where the free surface is mapped onto a flat surface and do not require the common assumption that the waves have small amplitude used in deriving the weakly nonlinear Korteweg–de Vries and Boussinesq long-wave equations. We compare the solution of the exact reduced equations with these weakly nonlinear long-wave models and with the nonlinear long-wave equations of Su and Gardner that do not assume the waves have small amplitude. The Su and Gardner solutions are in remarkably close agreement with the exact Euler solutions for large amplitude solitary wave interactions while the interactions of low-amplitude solitary waves of all four models agree. The simulations demonstrate that our method is an efficient and accurate approach to integrate all of these equations and conserves the mass, momentum, and energy of the Euler equations over very long simulations.     

Quadratic trigonometric b-spline galerkin methods for the regularized long wave equation

In this study, a numerical solution of the Regularized Long Wave (RLW) equation is obtained using Galerkin finite element method, based on two and three steps Adams Moulton method for the time integration and quadratic trigonometric B-spline functions for the space integration. After two different linearization techniques are applied, the proposed algorithms are tested on the problems of propagation of a solitary wave and interaction of two solitary waves. For the first test problem, the rate of convergence and the running time of the proposed algorithms are computed and the error norm $L_{\infty }$ is used to measure the differences between exact and numerical solutions. The three conservation quantities of the motion are calculated to determine the conservation properties of the proposed algorithms for both of the test problems.     

Exact and approximate nonlinear waves generated by the periodic superposition of solitons

Toda [1], Boyd [2], Zaitsev [3], Korpel & Banerjee [4], and Whitham [5] have proved that many species of solitons may be cloned and superposed with even spacing to generateexact nonlinear, spatially periodic solutions (“cnoidal waves”). The equations solved by such “imbricate” series of solitary waves include the Korteweg-deVries, Cubic Schroedinger, Benjamin-Ono, and resonant triad equations. However, all existing theorems apply only when the solitons arerational ormeromorphic functions and the cnoidal waves areelliptic functions. In this note, we ask: does the exact soliton-superposition apply to non-elliptic solitons and cnoidal waves? Although a complete answer to this (very broad!) question eludes us, it is possible to offer a revealing counterexample. The quartic Korteweg-deVries equation has solutions which arehyperelliptic, and thus very special. Nevertheless, its periodic solutions are not the exact superposition of the infinite number of copies of a soliton. This is highly suggestive that non-elliptic extensions of the Toda theorem are rare or non-existent. It is intriguing, however, that the soliton-superposition generates a very goodapproximation to the hypercnoidal wave even when the solitons strongly overlap.     

组合Zakharov-Kuznetsov方程的显式孤波解

借助于Ｍａｔｈｅｍａｔｉｃａ是吴消元法,本文通过用一个新的假设,获得了组合Ｚａ－ｋｈａｒｏｖ－Ｋｕｚｎｅｔｓｏｖ方程的１２种孤波解,其中包括钟状与扭状组合型孤波解和周期型孤波解。这种假设也能用于其他的非线性演化方程（组）。     

Solitary wave solutions of the high-order KdV models for bi-directional water waves

An approach, which allows us to construct specific closed-form solitary wave solutions for the KdV-like water-wave models obtained through the Boussinesq perturbation expansion for the two-dimensional water wave problem in the limit of long wavelength/small amplitude waves, is developed. The models are relevant to the case of the bi-directional waves with the amplitude of the left-moving wave of O(ϵ) (ϵ is the amplitude parameter) as compared with that of the right-moving wave. We show that, in such a case, the Boussinesq system can be decomposed into a system of coupled equations for the right- and left-moving waves in which, to any order of the expansion, one of the equations is dependent only on the (main) right-wave elevation and takes the form of the high-order KdV equation with arbitrary coefficients whereas the second equation includes both elevations. Then the explicit solitary wave solutions constructed via our approach may be treated as the exact solutions of the infinite-order perturbed KdV equations for the right-moving wave with the properly specified high-order coefficients. Such solutions include, in a sense, contributions of all orders of the asymptotic expansion and therefore may be considered to a certain degree as modelling the solutions of the original water wave problem under proper initial conditions. Those solitary waves, although stemming from the KdV solitary waves, possess features found neither in the KdV solitons nor in the solutions of the first order perturbed KdV equations.     

Solitary waves and their linear stability in weakly coupled KdV equations

We consider a system of weakly coupled KdV equations developed initially by Gear & Grimshaw to model interactions between long waves. We prove the existence of a variety of solitary wave solutions, some of which are not constrained minimizers. We show that such solutions are always linearly unstable. Moreover, the nature of the instability may be oscillatory and as such provides a rigorous justification for the numerically observed phenomenon of “leapfrogging.”     

Solitary waves and their linear stability in weakly coupled KdV equations

We consider a system of weakly coupled KdV equations developed initially by Gear & Grimshaw to model interactions between long waves. We prove the existence of a variety of solitary wave solutions, some of which are not constrained minimizers. We show that such solutions are always linearly unstable. Moreover, the nature of the instability may be oscillatory and as such provides a rigorous justification for the numerically observed phenomenon of “leapfrogging.”     

Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions

We derive the Whitham modulation equations for the Zakharov–Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all periodic solutions traveling along the first spatial coordinate are linearly unstable with respect to purely transversal perturbations, and we obtain an explicit expression for the growth rate of perturbations in the long wave limit. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. We also calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches. Finally, we generalize the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and we determine the resulting domains of stability and instability.     

一类二次KdV类型水波方程的行波解

应用平面动力系统理论研究了一类非线性KdV方程的行波解的动力学行为．在参数空间的不同区域内,给出了系统存在孤立波解,周期波解,扭子和反扭子波解的充分条件,并计算出所有可能的精确行波解的参数表示．     

General exact solutions for linear and nonlinear waves in a Thirring model

In the present paper, we construct exact solutions to a system of partial differential equations iu_x + v + u | v | ² = 0, iv_t + u + v | u | ² = 0 related to the Thirring model. First, we introduce a transform of variables, which puts the governing equations into a more useful form. Because of symmetries inherent in the governing equations, we are able to successively obtain solutions for the phase of each nonlinear wave in terms of the amplitudes of both waves. The exact solutions can be described as belonging to two classes, namely, those that are essentially linear waves and those which are nonlinear waves. The linear wave solutions correspond to waves propagating with constant amplitude, whereas the nonlinear waves evolve in space and time with variable amplitudes. In the traveling wave case, these nonlinear waves can take the form of solitons, or solitary waves, given appropriate initial conditions. Once the general solution method is outlined, we focus on a number of more specific examples in order to show the variety of physical solutions possible. We find that radiation naturally emerges in the solution method: if we assume one of u or v with zero background, the second wave will naturally include both a solitary wave and radiation terms. The solution method is rather elegant and can be applied to related partial differential systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Smooth and non-smooth traveling wave solutions of some generalized Camassa–Holm equations

In this paper we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a recently-derived integrable family of generalized Camassa–Holm (GCH) equations. A recent, novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of three of the GCH NLPDEs, i.e. the possible non-smooth peakon and cuspon solutions. One of the considered GCH equations supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. The second equation does not support singular traveling waves and the last one supports four-segmented, non-smooth M-wave solutions.Moreover, smooth traveling waves of the three GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of their traveling-wave equations, corresponding to pulse (kink or shock) solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding GCH equations, as well as ensure simultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. We also show the traveling wave nature of these pulse and front solutions to the GCH NLPDEs.     

Coupled-mode envelope solitary waves in a pair of cubic Schrödinger equations with cross modulation: Analytical solution and collisions with application to Rossby waves

Coupled-Nonlinear Schrödinger equations, linked by cross modulation terms, arise in both nonlinear optics and in Rossby waves in the atmosphere and ocean. In this paper, we derive exact, analytic solutions for the “bright” coupled-mode soliton, for which the envelope in each mode asymptotes to zero at spatial infinity, and for its spatially periodic generalization. We then numerically study the collisions of the coupled-mode solitary wave both with a conventional envelope soliton, confined to a single mode, and also with a second coupled-mode solitary wave. The collisions are sensitive to both the relative speed and phase of the solitons. In some parameter ranges, the collisions are nearly elastic, but in others, one or both solitons fission.     

Convective Linear Stability of Solitary Waves for Boussinesq Equations

Boussinesq was the first to explain the existence of Scott Russell's solitary wave mathematically. He employed a variety of asymptotically equivalent equations to describe water waves in the small-amplitude, long-wave regime. We study the linearized stability of solitary waves for three linearly well-posed Boussinesq models. These are problems for which well-developed Lyapunov methods of stability analysis appear to fail. However, we are able to analyze the eigenvalue problem for small-amplitude solitary waves, by comparison to the equation that Boussinesq himself used to describe the solitary wave, which is now called the Korteweg–de Vries equation. With respect to a weighted norm designed to diminish as perturbations convect away from the wave profile, we prove that nonzero eigenvalues are absent in a half-plane of the form R λ>− b for some b >0, for all three Boussinesq models. This result is used to prove the decay of solutions of the evolution equations linearized about the solitary wave, in two of the models. This "convective linear stability" property has played a central role in the proof of nonlinear asymptotic stability of solitary-wave-like solutions in other systems.     

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.