Classification of Traveling Waves for a Class of Nonlinear Wave Equations

We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification.     

New no-traveling wave solutions for the Liouville equation by Bäcklund transformation method

With the aid of the known Bäcklund transformation, starting from some given traveling solutions, we consider new exact no-traveling wave solutions to the Liouville equation, and a series of breather soliton solutions, doubly periodic solutions, two-soliton solutions as well as periodic-soliton solutions are obtained.     

On the traveling waves governed by the Camassa–Holm equation

The Camassa–Holm equation admits undistorted traveling waves that are either smooth or exhibit peaks or cusps. All three wave types can be periodic or solitary. Also waves of different types may be combined. In the present paper it is shown that, apart from peaks and cusps, the traveling waves governed by the Camassa–Holm equation can be found from some simpler equation. In the case of peaked solutions, this reduced equation is even linear. The governing equation of traveling waves in its original form can be interpreted as a nonlinear combination of the reduced equation and its first integral. For a small range of the integration constant, the reduced equation admits bounded solutions, which then are directly inherited by the Camassa–Holm equation. In general, the solutions of the reduced equation are unbounded and cannot be considered to represent traveling waves. The full equation, however, has a nonlinearity in the highest derivative, which is characteristic for the Camassa–Holm and some other equations. This nonlinear term offers the possibility of constructing bounded traveling waves from the unbounded solutions of the reduced equation. These waves necessarily have discontinuities in the slope and are, therefore, solutions only in a generalized sense.     

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term简

This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.     

The effects of horizontal singular straight line in a generalized nonlinear Klein–Gordon model equation

In this paper, we investigate bounded traveling waves of the generalized nonlinear Klein–Gordon model equations by using bifurcation theory of planar dynamical systems to study the effects of horizontal singular straight lines in nonlinear wave equations. Besides the well-known smooth traveling wave solutions and the non-smooth ones, four kinds of new bounded singular traveling wave solution are found for the first time. These singular traveling wave solutions are characterized by discontinuous second-order derivatives at some points, even though their first-order derivatives are continuous. Obviously, they are different from the singular traveling wave solutions such as compactons, cuspons, peakons. Their implicit expressions are also studied in this paper. These new interesting singular solutions, which are firstly founded, enrich the results on the traveling wave solutions of nonlinear equations. It is worth mentioning that the nonlinear equations with horizontal singular straight lines may have abundant and interesting new kinds of traveling wave solution.     

On the integrability and Riemann theta functions periodic wave solutions of the Benjamin Ono equation

In this paper, the complete integrability of the Benjamin Ono equation is systematically studied. Its bilinear equation, soliton solutions, bilinear Bäcklund transformation and Lax pair are successfully obtained, by virtue of generalized Bell’s polynomials scheme. Moreover, by using multidimensional Riemann theta functions, the periodic wave solutions of the Benjamin Ono equation are constructed. Further, the asymptotic behaviors of the periodic wave solutions are presented with a limiting procedure, which shows the relations between the periodic wave solutions and soliton solutions.     

Exact solutions for a perturbed nonlinear Schrödinger equation by using Bäcklund transformations

The Bäcklund transformation from the Riccati form of inverse method is presented for the Perturbed Nonlinear Schrödinger Equation. Consequently, the exact solutions for Perturbed Nonlinear Schrödinger equation can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS; that is, a linear eigenvalues problem in the form of a system of PDE. Moreover, we construct a new soliton solution from the old one and its wave function.     

Multi-soliton solutions of a variable-coefficient KdV hierarchy

In this paper, Hirota’s bilinear method is extended to a new KdV hierarchy with variable coefficients. As a result, one-soliton solution, two-soliton solution and three-soliton solutions are obtained, from which the uniform formula of \(N\) -soliton solution is derived. Thanks to the arbitrariness of the included functions, these obtained solutions possess rich local structural features like the ridge soliton and the concave column soliton. It is shown that the Hirota’s bilinear method can be used for constructing multi-soliton solutions of some other hierarchies of nonlinear partial differential equations with variable coefficients.     

Exact traveling wave solutions to 2D-generalized Benney-Luke equation

By using the dynamical system method to study the 2D-generalized Benney- Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parametric representations for solutions of kink wave, periodic wave and unbounded traveling wave are obtained.     

Bifurcations and exact solutions of an asymptotic rotation-Camassa–Holm equation

This paper investigates the rotation-Camassa–Holm equation, which appears in long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the earth’s rotation. The rotation-Camassa–Holm equation contains the famous Camassa–Holm equation and is a special case of the generalized Camassa–Holm equation. By using the approach of dynamical systems and singular traveling wave theory to its traveling wave system, in different parameter conditions of the five-parameter space, the bifurcations of phase portraits are studied. Some exact explicit parametric representations of the smooth solitary wave solutions, periodic wave solutions, peakons and anti-peakons, periodic peakons as well as compacton solutions are obtained.

     

Traveling Wave Solutions in a Generalized Theory for Macroscopic Capillarity

One-dimensional traveling wave solutions for imbibition processes into a homogeneous porous medium are found within a recent generalized theory of macroscopic capillarity. The generalized theory is based on the hydrodynamic differences between percolating and nonpercolating fluid parts. The traveling wave solutions are obtained using a dynamical systems approach. An exhaustive study of all smooth traveling wave solutions for primary and secondary imbibition processes is reported here. It is made possible by introducing two novel methods of reduced graphical representation. In the first method the integration constant of the dynamical system is related graphically to the boundary data and the wave velocity. In the second representation the wave velocity is plotted as a function of the boundary data. Each of these two graphical representations provides an exhaustive overview over all one-dimensional and smooth solutions of traveling wave type, that can arise in primary and secondary imbibition. Analogous representations are possible for other systems, solution classes, and processes.     

Painlevé integrability, similarity reductions, new soliton and soliton-like similarity solutions for the (2+1)-dimensional BKP equation

The (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation of B-type (BKP) is hereby investigated. New soliton solutions and soliton-like similarity solutions are constructed for the (2+1)-dimensional BKP equation. The similarity solutions are not travelling wave solutions when the arbitrary functions involved are chosen appropriately. Painlevé test shows that there are two solution branches, one of which has the resonance ?2. And four similarity reductions for the BKP equation are given out through nontrivial variable transformations. Moreover, abundant soliton behaviour modes of the solutions, such as soliton fusion and soliton reflection, are discussed in detail.     

The integral bifurcation method combined with factoring technique for investigating exact solutions and their dynamical properties of a generalized Gardner equation

It is well known that it is difficult to obtain exact solutions of some partial differential equations with highly nonlinear terms or high order terms because these kinds of equations are not integrable in usual conditions. In this paper, by using the integral bifurcation method and factoring technique, we studied a generalized Gardner equation which contains both highly nonlinear terms and high order terms, some exact traveling wave solutions such as non-smooth peakon solutions, smooth periodic solutions and hyperbolic function solutions to the considered equation are obtained. Moreover, we demonstrate the profiles of these exact traveling wave solutions and discuss their dynamic properties through numerical simulations.     

Traveling Waves in Diffusive Random Media

The current paper is devoted to the study of traveling waves in diffusive random media, including time and/or space recurrent, almost periodic, quasiperiodic, periodic ones as special cases. It first introduces a notion of traveling waves in general random media, which is a natural extension of the classical notion of traveling waves. Roughly speaking, a solution to a diffusive random equation is a traveling wave solution if both its propagating profile and its propagating speed are random variables. Then by adopting such a point of view that traveling wave solutions are limits of certain wave-like solutions, a general existence theory of traveling waves is established. It shows that the existence of a wave-like solution implies the existence of a critical traveling wave solution, which is the traveling wave solution with minimal propagating speed in many cases. When the media is ergodic, some deterministic \hbox{properties} of average propagating profile and average propagating speed of a traveling wave solution are derived. When the media is compact, certain continuity of the propagating profile of a critical traveling wave solution is obtained. Moreover, if the media is almost periodic, then a critical traveling wave solution is almost automorphic and if the media is periodic, then so is a critical traveling wave solution. Applications of the general theory to a bistable media are discussed. The results obtained in the paper generalize many existing ones on traveling waves. AMS Subject Classification: 35K55, 35K57, 35B50     

Solitary wave solutions and their interactions for fully nonlinear water waves with surface tension in the generalized Serre equations

Some effects of surface tension on fully nonlinear, long, surface water waves are studied by numerical means. The differences between various solitary waves and their interactions in subcritical and supercritical surface tension regimes are presented. Analytical expressions for new peaked traveling wave solutions are presented in the dispersionless case of critical surface tension. Numerical experiments are performed using a high-accurate finite element method based on smooth cubic splines and the four-stage, classical, explicit Runge–Kutta method of order 4.     

Nonlocal symmetry and similarity reductions for a $$\varvec{}$$-dimensional Korteweg–de Vries equation

Based on the Lax pair, the nonlocal symmetries to \((2+1)\)-dimensional Korteweg–de Vries equation are investigated, which are also constructed by the truncated Painlevé expansion method. Through introducing some internal spectrum parameters, infinitely many nonlocal symmetries are given. By choosing four suitable auxiliary variables, nonlocal symmetries are localized to a closed prolonged system. Via solving the initial-value problems, the finite symmetry transformations are obtained to generate new solutions. Moreover, rich explicit interaction solutions are presented by similarity reductions. In particular, bright soliton, dark soliton, bell-typed soliton and soliton interacting with elliptic solutions are found. Through computer numerical simulation, the dynamical phenomena of these interaction solutions are displayed in graphical way, which show meaningful structures.     

Traveling Wave Solutions for a Class of One-Dimensional Nonlinear Shallow Water Wave Models

In this paper we consider a class of one-dimensional nonlinear shallow water wave models that support weak solutions. We construct new traveling wave solutions for these models. Moreover, we show that these new traveling wave solutions are stable.     

Existence,Uniqueness, and Stability of Generalized Traveling Waves in Time Dependent Monostable Equations

The current paper is devoted to the study of traveling wave solutions of spatially homogeneous monostable reaction diffusion equations with ergodic or recurrent time dependence, which includes periodic and almost periodic time dependence as special cases. Such an equation has two spatially homogeneous and time recurrent solutions with one of them being stable and the other being unstable. Traveling wave solutions are a type of entire solutions connecting the two spatially homogeneous and time recurrent solutions. Recently, the author of the current paper proved that a spatially homogeneous time almost periodic monostable equation has a spreading speed in any given direction. This result can be easily extended to monostable equations with recurrent time dependence. In this paper, we introduce generalized traveling wave solutions for time recurrent monostable equations and show the existence of such solutions in any given direction with average propagating speed greater than or equal to the spreading speed in that direction and non-existence of such solutions of slower average propagating speed. We also show the uniqueness and stability of generalized traveling wave solutions in any given direction with average propagating speed greater than the spreading speed in that direction. Moreover, we show that a generalized traveling wave solution in a given direction with average propagating speed greater than the spreading speed in that direction is unique ergodic in the sense that its wave profile and wave speed are unique ergodic, and if the time dependence of the monostable equation is almost periodic, it is almost periodic in the sense that its wave profile and wave speed are almost periodic.     

Exact solutions,conservation laws,bifurcation of nonlinear and supernonlinear traveling waves for Sharma–Tasso–Olver equation

In the present work, we observe the dynamical behavior of nonlinear and supernonlinear traveling waves for Sharma–Tasso–Olver (STO) equation. Exact solutions are derived using \({1}/{G^{^{\prime }}}\) expansion and modified Kudryashov methods. The wave transformation is used to transform STO equation into an ordinary differential equation. Combining Runge–Kutta fourth-order and Fourier spectral technique, we use a mixed scheme for the numerical study of STO equation. Since spectral methods expand the solution in trigonometric series resulting into higher-order technique and Runge–Kutta produces improved accuracy, we extract these qualities for a mixed scheme. Results so produced are presented graphically which provide a useful information about the dynamical behavior. Bifurcation behavior of nonlinear and supernonlinear traveling waves of STO equation is studied with the help of bifurcation theory of planar dynamical systems. It is observed that STO equation supports nonlinear solitary wave, periodic wave, shock wave, stable oscillatory wave and most important supernonlinear periodic wave.