Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations

In this article we study global existence and blow-up of solutions for a general class of nonlocal nonlinear wave equations with power-type nonlinearities, $u_{tt}-Lu_{xx}=B(- |u|^{p-1}u)_{xx}, ~(p>1)$, where the nonlocality enters through two pseudo-differential operators $L$ and $B$. We establish thresholds for global existence versus blow-up using the potential well method which relies essentially on the ideas suggested by Payne and Sattinger. Our results improve the global existence and blow-up results given in the literature for the present class of nonlocal nonlinear wave equations and cover those given for many well-known nonlinear dispersive wave equations such as the so-called double-dispersion equation and the traditional Boussinesq-type equations, as special cases.Comment: 17 pages. Accepted for publication in Nonlinear Analysis:Theory, Methods & Application

Derivation of the Camassa-Holm equations for elastic waves

In this paper we provide a formal derivation of both the Camassa-Holm equation and the fractional Camassa-Holm equation for the propagation of small-but-finite amplitude long waves in a nonlocally and nonlinearly elastic medium. We first show that the equation of motion for the nonlocally and nonlinearly elastic medium reduces to the improved Boussinesq equation for a particular choice of the kernel function appearing in the integral-type constitutive relation. We then derive the Camassa-Holm equation from the improved Boussinesq equation using an asymptotic expansion valid as nonlinearity and dispersion parameters tend to zero independently. Our approach follows mainly the standard techniques used widely in the literature to derive the Camassa-Holm equation for shallow water waves. The case where the Fourier transform of the kernel function has fractional powers is also considered and the fractional Camassa-Holm equation is derived using the asymptotic expansion technique.Comment: 15 page

The Camassa-Holm equation as the long-wave limit of the improved Boussinsq equation and of a class of nonlocal wave equations

In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters ϵ and δ measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximatio

Unidirectional wave motion in a nonlocally and nonlinearly elastic medium: the KdV, BBM, and CH equations

We consider unidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral with a suitable kernel function. We first give a brief review of asymptotic wave models describing the unidirectional propagation of small-but-finite amplitude long waves. When the kernel function is the well-known exponential kernel, the asymptotic description is provided by the Korteweg–de Vries (KdV) equation, the Benjamin–Bona–Mahony (BBM) equation, or the Camassa–Holm (CH) equation. When the Fourier transform of the kernel function has fractional powers, it turns out that fractional forms of these equations describe unidirectional propagation of the waves. We then compare the exact solutions of the KdV equation and the BBM equation with the numerical solutions of the nonlocal model. We observe that the solution of the nonlocal model is well approximated by associated solutions of the KdV equation and the BBM equation over the time interval considered

Existence and stability of traveling waves for a class of nonlocal nonlinear equations

In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: u_tt−Lu_xx=B(±|u|^(p−1)u)_xx, p>1. The main characteristic of this class of equations is the existence of two sources of dispersion, characterized by two coercive pseudo-differential operators L and B. Members of the class arise as mathematical models for the propagation of dispersive waves in a wide variety of situations. For instance, all Boussinesq-type equations and the so-called double-dispersion equation are members of the class. We first establish the existence of traveling wave solutions to the nonlocal wave equations considered. We then obtain results on the orbital stability or instability of traveling waves. For the case L=I, corresponding to a class of Klein-Gordon-type equations, we give an almost complete characterization of the values of the wave velocity for which the traveling waves are orbitally stable or unstable by blow-up

Coupled quintic nonlinear Schrodinger equations in a generalized elastic solid

In the present study, the nonlinear modulation of transverse waves propagating in a cubically nonlinear dispersive elastic medium is studied using a multiscale expansion of wave solutions. It is found that the propagation of quasimonochromatic transverse waves is described by a pair of coupled nonlinear Schrodinger (CNLS) equations. In the process of deriving the amplitude equations, it is observed that for a specific choice of material constants and wavenumber, the coefficient of nonlinear terms becomes zero, and the CNLS equations are no longer valid for describing the behaviour of transverse waves. In order to balance the nonlinear effects with the dispersive effects, by intensifying the nonlinearity, a new perturbation expansion is used near the critical wavenumber. It is found that the long time behaviour of the transverse waves about the critical wavenumber is given by a pair of coupled quintic nonlinear Schrodinger (CQNLS) equations. In the absence of one of the transverse waves, the CQNLS equations reduce to the single quintic nonlinear Schrodinger (QNLS) equation which has already been obtained in the context of water waves. By using a modified form of the so-called tanh method, some travelling wave solutions of the CQNLS equations are presented.Publisher's Versio

A semi-discrete numerical method for convolution-type unidirectional wave equations

Numerical approximation of a general class of nonlinear unidirectional wave equations with a convolution-type nonlocality in space is considered. A semi-discrete numerical method based on both a uniform space discretization and the discrete convolution operator is introduced to solve the Cauchy problem. The method is proved to be uniformly convergent as the mesh size goes to zero. The order of convergence for the discretization error is linear or quadratic depending on the smoothness of the convolution kernel. The discrete problem defined on the whole spatial domain is then truncated to a finite domain. Restricting the problem to a finite domain introduces a localization error and it is proved that this localization error stays below a given threshold if the finite domain is large enough. For two particular kernel functions, the numerical examples concerning solitary wave solutions illustrate the expected accuracy of the method. Our class of nonlocal wave equations includes the Benjamin-Bona-Mahony equation as a special case and the present work is inspired by the previous work of Bona, Pritchard and Scott on numerical solution of the Benjamin-Bona-Mahony equation

Some remarks on the stability and instability properties of solitary waves for the double dispersion equation

In this article we give a review of our recent results on the instability and stability properties of travelling wave solutions of the double dispersion equation utt − uxx + auxxxx − buxxtt = −(|u|p−1u)xx for p > 1, a ≥ b > 0. After a brief reminder of the general class of nonlocal wave equations to which the double dispersion equation belongs, we summarize our findings for both the existence and orbital stability/instability of travelling wave solutions to the general class of nonlocal wave equations. We then state (i) the conditions under which travelling wave solutions of the double dispersion equation are unstable by blow-up and (ii) the conditions under which the travelling waves are orbitally stable. We plot the instability/stability regions in the plane defined by wave velocity and the quotient b/a for various values of p.publisher versio

Comparison of nonlocal nonlinear wave equations in the long-wave limit

We consider a general class of convolution-type nonlocal wave equations modeling bidirectional nonlinear wave propagation. The model involves two small positive parameters measuring the relative strengths of the nonlinear and dispersive effects. We take two different kernel functions that have similar dispersive characteristics in the long-wave limit and compare the corresponding solutions of the Cauchy problems with the same initial data. We prove rigorously that the difference between the two solutions remains small over a long time interval in a suitable Sobolev norm. In particular our results show that, in the long-wave limit, solutions of such nonlocal equations can be well approximated by those of improved Boussinesq-type equations

Existence of solitary wave solutions for long wave-short wave interaction equations

Sürekli bir ortamda yayılan iki kısa ve bir uzun dalganın rezonans etkileşimi bir boyutlu üç kuple uzun dalga-kısa dalga (LSI) etkileşim denklemleri ile temsil edilmektedir. Kısa dalgaların aynı grup hızına sahip olması ve bu grup hızının uzun dalganın faz hızına eşit olması rezonans durumunu oluşturmaktadır. LSI denklemleri yüzey su dalgalarının etkileşimi, elastik bir ortamda yayılan iç elastik dalgaların rezonans etkileşimi gibi fiziksel olayları tanımlayan denklemler olarak türetilmiştir. Bu çalışmada LSI denklem sisteminin yalnız dalga çözümlerinin varlığı ispat edilmiştir. İspat esas olarak kübik nonlineerliğe sahip tek bileşenli nonlineer Schrödinger (NLS) denklemi için önerilmiş yaklaşımın genişletilmesi üzerine inşa edilmiştir ve kısıtlamasız bir varyasyonel problemin çözümüne dayanmaktadır. Bunun için öncelikle LSI denklemlerinin yalnız dalga çözümlerinin sağlamış olduğu iki-kuple denklem sistemi elde edilmiştir. Daha sonra Euler-Lagrange denklemleri bu iki-kuple denklem sistemini veren bir  fonksiyoneli, Gagliardo-Nirenberg eşitsizliği yardımıyla tanımlanmıştır. Böylece, yalnız dalgaların varlığı problemi  fonksiyonelinin bir minimumunun varlığının gösterilmesi problemine indirgenmiştir. Fonksiyonelin minimumunun varlığını göstermek için Lieb’in kompaktlık lemması kullanılmıştır. NLS denkleminin yalnız dalga çözümlerinin varlığını göstermek amacıyla literatürde tanımlanmış olan  fonksiyoneli  uzay boyutu için geçerli iken, şimdiki çalışmada tanımlanmış olan fonksiyonel   uzay boyutunda da geçerlidir. Ayrıca, çözümlerin pozitif tanımlılığı gösterilmiş ve fonksiyoneli ile enerji fonksiyoneli arasındaki ilişki de verilmiştir. Anahtar Kelimeler: Uzun dalga-kısa dalga etkileşim denklemleri, yalnız dalgalar.Nonlinear dispersive wave equations arise in many areas of physics, such as solid mechanics, nonlinear optics and plasma. Solitary wave solutions occur as a result of the balance of dispersive and nonlinear effects. This balance makes the localized waves travel without change of form. So there is a wide interest for the behaviour of solitary wave solutions of nonlinear wave equations: existence and uniqueness, continuous dependence to initial data, stability. In the present study we are interested in establishing the existence of solitary wave solutions for a particular nonlinear dispersive wave equation. Short waves propagating in various continuous media are governed by the nonlinear Schrödinger (NLS) equation and its generalizations. In one dimensional case there exists a unique localized solitary wave solution of the NLS equation and can easily be evaluated. The existence problem in the multidimensional case has generally been investigated by using constrained or unconstrained variational methods. The constrained variational methods are based on minimization of energy functionals, which take the infimum value at the ground state solutions, under some constraints. Weinstein proved the existence problem of solitary wave solutions of the NLS equation in the multidimensional case by the help of an unconstrained variational problem. In his study a functional  associated with the Gagliardo-Nirenberg inequality is defined. The solution of the Euler-Lagrange equation of is the solitary wave solution of the NLS equation. So the existence of solitary wave solutions was shown by proving the existence of a minimizer of the functional . The resonant interaction among two short wave modes with equal group speeds and one long wave mode whose phase speed is equal to the group speed of short waves is represented by the three coupled long wave-short wave interaction (LSI) equations of the form; where,  is the spatial coordinate, is the time; and  and  are  real constants. Here represents the long wave mode;  and  denote short wave modes. The above three-component system of long wave-short wave interaction equations describes the resonant wave propagation in various continuous media, for instance, the surface of water and a bulk elastic medium. The aim of the present study is to prove the existence of solitary wave solutions for the LSI system by using variational methods. Substitution of solitary wave solutions of the form with  into the LSI system yields the coupled system  In spite of a wide interest in this system and its generalizations, a few rigorous results on the existence of solutions exist up to present. In recent years there has been a large number of studies devoted to the problem of existence of solutions for various settings of this system. In these studies, the problem of existence of nontrivial solutions has been investigated extensively using variational methods based on minimization of certain energy functionals under some constraints. Here, to prove the existence of positive solutions to the above two-component system, we use a different approach based on adapting the method used by Weinstein for the NLS equation. As the one-variable functional defined by Weinstein is valid for , a new functional associated with the Gagliardo-Nirenberg  inequality  has been defined for . The new two-variable functional assumes the solutions of two coupled system as the critical points. And the existence of a minimizer for is proved using Lieb's compactness Lemma. The positivity of solutions and the relation between and the energy functional are also investigated. Keywords: Long wave-short wave interaction equations, existence of solitary waves