Existence Of Solitary Waves Research Articles

On solitary-wave solutions of Rosenau-type equations

The present paper is concerned with the existence of solitary wave solutions of Rosenau-type equations. By using two standard theories, Normal Form Theory and Concentration-Compactness Theory, some results of existence of solitary waves of three different forms are derived. The results depend on some conditions on the speed of the waves with respect to the parameters of the equations. They are discussed for several families of Rosenau equations present in the literature. The analysis is illustrated with a numerical study of generation of approximate solitary-wave profiles from a numerical procedure based on the Petviashvili iteration.

Existence and stability of solitary waves to the rotation–Camassa–Holm equation

In this paper, we investigate the existence and stability of solitary waves to the rotation–Camassa–Holm equation which can be considered as a model in the shallow water for the long‐crested waves propagating near the equator with effect of the Coriolis force due to the Earth's rotation. We prove the existence of solitary waves by performing a phase plane analysis. Moreover, utilizing the approach proposed by Grillakis–Shatah–Strauss, we prove the stability of solitary waves.

On the Kadomtsev–Petviashvili equation with double-power nonlinearities

In this paper, we study the generalized KP equation with double-power nonlinearities. Our investigation covers various aspects, including the existence of solitary waves, their nonlinear stability, and instability. Notably, we address a broader class of nonlinearities represented by μ1|u|p1−1u+μ2|u|p2−1u, with p1>p2, encompassing cases where μ1>0 and μ1<0<μ2. One of the distinct features of our work is the absence of scaling, which introduces several challenges in establishing the existence of ground states. To overcome these challenges, we employ two different minimization problems, offering novel approaches to address this issue. Furthermore, our study includes a nuanced analysis to ascertain the stability of these ground states. Intriguingly, we extend our stability analysis to encompass cases where the convexity of the Lyapunov function is not guaranteed. This expansion of stability criteria represents a significant contribution to the field. Moving beyond the analysis of solitary waves, we shift our focus to the associated Cauchy problem. Here, we derive criteria that determine whether solutions exhibit finite-time blow-up or remain uniformly bounded within the energy space. Remarkably, our study unveils a notable gap in the existing literature, characterized by the absence of both theoretical evidence of blow-up and uniform boundedness. To explore this intriguing scenario, we employ the integrating factor method, providing a numerical investigation of solution behavior. This method distinguishes itself by offering spectral-order accuracy in space and fourth-order accuracy in time. Lastly, we rigorously establish the strong instability of the ground states, adding another layer of understanding to the complex dynamics inherent in the generalized KP equation.

Persistence of Solitary Waves and Periodic Waves of a Singularly Perturbed Generalized Drinfel’d–Sokolov System

This paper focuses on the persistence of solitary waves and periodic waves of a singularly perturbed generalized Drinfel’d–Sokolov system. Geometric singular perturbation theory is first employed to reduce the higher-dimensional system to the perturbed planar system. By perturbation analysis and Abelian integrals theory, we are then able to find some sufficient conditions about the wave speed to guarantee the existence of homoclinic orbits and periodic orbits, which indicates the existence of solitary waves and periodic waves. Furthermore, we find the lower and upper bounds of the limit wave speed.

Korteweg–de Vries waves in peridynamical media

AbstractWe consider a one‐dimensional peridynamical medium and show the existence of solitary waves with small amplitudes and long wavelength. Our proof uses nonlinear Bochner integral operators and characterizes their asymptotic properties in a singular scaling limit.

Solitary wave solutions for the fourth-order nonlinear Schrödinger equation with variables coefficients

We study the existence and propagation properties of solitary waves within the framework of a fourth-order nonlinear Schrödinger equation with variables coefficients. The model applies to the description of the transport of biological energy inside alpha helical proteins. Both gray, dipole and tripole types are found by adopting the complex multipole ansatz solution. Parametric conditions for the existence of the solitary wave solutions are presented and their dynamic behavior is analyzed for different choice of system parameters. In addition, the stability of the tripole-managed solitons under some initial perturbations is studied by employing the numerical simulation methods. Finally, the collision dynamics of two similar tripole solitons in an inhomogeneous periodic medium is explored.

Periodic travelling waves for a fourth order nonlinear evolution equation

In this article we provide a travelling wave analysis for a fourth order non linear evolution equation. In particular we prove the existence of periodic travelling waves while we exclude the existence of solitary waves for proper values of the parameters. Moreover, we analyse the set of stationary solutions and provide a new proof of the existence of limit cycle for a related equation, that is, the Van der Pol equation.

A local Palais-Smale condition and existence of solitary waves for a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations

&lt;abstract&gt;&lt;p&gt;This paper is concerned with a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \bigg\{ \begin{array}{rl} &amp;amp; u_t + (|u|^{p-2}u)_x + u_{xxx} +h_x(x-\tau t, y) +\beta \nabla_y v = 0, \\ &amp;amp; v_x = \nabla_y u.\end{array} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;By proving a local Palais-Smale condition, we manage to prove the existence of solitary waves with the help of a variational characterization on the smallest positive constant of an anisotropic Sobolev inequality (Huang and Rocha, J. Inequal. Appl., 2018,163). The novelty is to give an &lt;bold&gt;explicit estimate&lt;/bold&gt; on the sufficient condition of $ h $ to get the existence of solitary waves.&lt;/p&gt;&lt;/abstract&gt;

Chirped localized pulses in a highly nonlinear optical fiber with quintic non-Kerr nonlinearities

We study the existence and propagation properties of chirped localized pulses in a highly nonlinear fiber medium exhibiting self-steepening, self-frequency shift, and quintic non-Kerr nonlinearities. Pulse evolution in such fiber system is governed by a higher-order nonlinear Schrödinger equation incorporating the derivative Kerr and non-Kerr nonlinear terms. We show that bright, dark and kink type solitary waves exist in the presence of all physical processes. A special ansatz is introduced to analyze the existence of solitary waves on a continuous-wave background in the optical fiber medium. It is shown that the obtained localized pulses exhibit a nonlinear chirp which has a quadratic dependence on light intensity. We also find that the magnitude of the associated frequency chirp can be controlled effectively by varying the parameters of self-steepening, self-frequency shift, and derivative non-Kerr nonlinearity effects. The restrictions on the optical fiber parameters are also extracted for the existence of these nonlinearly chirped solitary waves. Results in this study may be useful for experimental realization of shape-preserved pulses in optical fibers and further understanding of their optical transmission properties.

Explicit Exact Solutions to the Wave Equation for Nonlinear Elastic Rods

The sine-cosine method was applied to the wave equation for nonlinear elastic rods, and some new periodic and solitary solutions to the equation were obtained (with material constant <i>n</i> other than 1). The graphs of some solutions were given through the math software. The results are helpful to further research on existence of solitary waves in the nonlinear elastic rods.

Dynamics of solitary waves and periodic waves for a generalized KP-MEW-Burgers equation with damping

&lt;p style='text-indent:20px;'&gt;This paper discusses the existence of solitary waves and periodic waves for a generalized (2+1)-dimensional Kadomtsev-Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation with small damping and a weak local delay convolution kernel by using the dynamical systems approach, specifically based on geometric singular perturbation theory and invariant manifold theory. Moreover, the monotonicity of the wave speed is proved by analyzing the ratio of Abelian integrals. The upper and lower bounds of the limit wave speed are given. In addition, the upper and lower bounds and monotonicity of the period &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ T $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; of traveling wave when the small positive parameter &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \tau\rightarrow 0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; are also obtained. Perhaps this paper is the first discussion on the solitary waves and periodic waves for the delayed KP-MEW-Burgers equations and the Abelian integral theory may be the first application to the study of the (2+1)-dimensional equation.&lt;/p&gt;

Global bifurcation of solitary waves to the Boussinesq abcd system

The Boussinesq abcd system arises in the modeling of long wave small amplitude water waves in a channel, where the four parameters (a,b,c,d) satisfy one constraint. In this paper we focus on the solitary wave solutions to such a system. In particular we work in two parameter regimes where the system does not admit a Hamiltonian structure (corresponding to b≠d). We prove via analytic global bifurcation techniques the existence of solitary waves in such parameter regimes. Some qualitative properties of the solutions are also derived, from which sharp results can be obtained for the global solution curves.Specifically, we first construct solutions bifurcating from the stationary waves, and obtain a global continuous curve of solutions that exhibits a loss of ellipticity in the limit. The second family of solutions bifurcate from the classical Boussinesq supercritical waves. We show that the curve associated to the second class either undergoes a loss of ellipticity in the limit or becomes arbitrarily close to having a stagnation point.

Traveling waves in a generalized Camassa–Holm equation involving dual-power law nonlinearities

This article focuses on traveling waves including peakons, periodic peakons, compactons and solitary waves as well as the persistence of some of these waves in a generalized Camassa–Holm (gCH) equation involving dual-power law nonlinearities. Firstly, we analyze the global phase portraits of the associated traveling wave system by a dynamical system-based approach. All the traveling waves are classified and their corresponding parameter conditions are obtained. It is found that the gCH equation with dual-power law nonlinearities can admit different peakons under different parameter conditions. Secondly, we pay attention to the persistence of solitary waves, i.e., the existence of solitary waves in a perturbed gCH equation by using geometric singular perturbation theory (GSPT) and an “explicit” Melnikov method. An elementary method to calculate the unperturbed homoclinic orbits and the associated Melnikov integral in an “explicit” way is developed in this article.

Modeling chemotaxis with nonstandard production/degradation mechanisms from Doebner–Goldin theory: Existence of solitary waves

In this paper we investigate various forms of the chemical production–degradation mechanism in a chemotactic system of Keller–Segel type under which the existence of solitary wave solutions is guaranteed. Specifically, the existence of sech-type or compactly supported solitary wave solutions as well as exponential traveling wave profiles for the cell concentration is shown under certain conditions on the physical parameters.

The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation

&lt;p style='text-indent:20px;'&gt;In this paper, we are concerned with the existence of solitary waves for a generalized Kawahara equation, which is a model equation describing solitary-wave propagation in media. We obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the generalized Kawahara equation without delay and perturbation by employing the phase space analysis. Furthermore the existence of solitary wave solutions for the equation with two types of special delay convolution kernels is proved by combining the geometric singular perturbation theory, invariant manifold theory and Fredholm orthogonality. We also discuss the asymptotic behaviors of traveling wave solutions by means of the asymptotic theory. Finally, some examples are given to illustrate our results.&lt;/p&gt;

Bifurcation of Gap Solitons in Coupled Mode Equations in d Dimensions

We consider a system of first order coupled mode equations in {mathbb {R}}^d describing the envelopes of wavepackets in nonlinear periodic media. Under the assumptions of a spectral gap and a generic assumption on the dispersion relation at the spectral edge, we prove the bifurcation of standing gap solitons of the coupled mode equations from the zero solution. The proof is based on a Lyapunov–Schmidt decomposition in Fourier variables and a nested Banach fixed point argument. The reduced bifurcation equation is a perturbed stationary nonlinear Schrödinger equation. The existence of solitary waves follows in a symmetric subspace thanks to a spectral stability result. A numerical example of gap solitons in {mathbb {R}}^2 is provided.

The domain of existence of solitary waves in fluid-filled thin elastic tubes

Under given prestress conditions, solitary waves in fluid-filled elastic tubes are confined to a rather narrow set of combinations of the background fluid velocity and the wave speed. This set, which we call the domain of existence, is bounded by several curves that represent various physical and mathematical restrictions. Remarkably, these restrictions can be cast as purely algebraic conditions to be imposed upon the governing system of differential equations. Paramount among the physical restrictions are the avoidance of wrinkles and the self-impenetrability of the wave profile. In particular, the existence of a critical wave speed of impending wrinkling, independent of the background fluid velocity, is established rigorously.

Nonlinear propagation of an intense Laguerre–Gaussian laser pulse in a plasma channel* *Project supported by the National Natural Science Foundation of China (Grant Nos. 61665006 and 61865011) and the Natural Science Foundation of Jiangxi Province of China (Grant Nos. 20171ACB21018, 20161BAB212041, and 20162BCB23012).

The nonlinear propagation of an intense Laguerre–Gaussian (LG) laser pulse in a parabolic preformed plasma channel is analyzed by means of the variational method. The evolution equation of the spot size is derived including the effects of relativistic self-focusing, preformed channel focusing, and ponderomotive self-channeling. The parametric conditions of the LG laser pulse and plasma channel for propagating with constant spot size, periodically focusing and defocusing oscillation, catastrophic focusing, and solitary waves are obtained. Compared with the laser pulse with fundamental Gaussian (FG) mode, it is found that the effect of vacuum diffraction is reduced by half and the effects of relativistic and wakefield focusing are decreased by a quarter due to the hollow transverse intensity profile of the LG laser pulse, while the effect of channel focusing is the same order of magnitude with that of the FG laser pulse. Thus, the matched condition for the intense LG laser pulse with constant spot size is released obviously, while the parameters of the laser and plasma for the existence of solitary waves nearly coincide with those of the FG laser pulse.

Solitary wave and periodic wave for a generalized Nizhnik-Novikov-Veselov equation with diffusion term

This paper is concerned with a generalized Nizhnik-Novikov-Veselov equation with weak backward diffusion term. The existence of solitary wave and periodic wave solutions are obtained by using the method of the dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. Furthermore, based on the ratio of the Abelian integral, the monotonicity of the wave speed is established. At the same time, the bounds of the limit wave speed and the properties of the periodic wave solutions to this equation are also given.

No solitary waves in 2D gravity and capillary waves in deep water

A fundamental question in the study of water waves is the existence and stability of solitary waves. Solitary waves have been proved to exist and have been studied in many interesting situations, and often arise from the balance of different forces/factors influencing the fluid dynamics, e.g. gravity, surface tension or the fluid bottom. However, the existence of solitary waves has remained an open problem in two of the simplest cases, namely for either pure gravity waves or pure capillary waves in infinite depth. In this article we settle both of these questions in two space dimensions. Precisely, we consider incompressible, irrotational, infinite depth water wave equation, either with gravity and without surface tension, or without gravity but with surface tension. In both of these cases we prove that there are no solitary waves.