Smooth Periodic Wave Solutions Research Articles

Existence and bifurcation of traveling wave solutions to a generalized Boussinesq equation with nonlinear dispersion

This paper is devoted to the analytical study of traveling wave solutions to a generalized Boussinesq equation with nonlinear dispersion. Utilizing the bifurcation method of dynamical systems, the existence of compacton and peakon solutions are established. Under certain parameter conditions, the compacton and peakon solutions can be bifurcated by smooth periodic wave solutions, smooth solitary wave solutions, and singular cusp solutions. Numerical simulations are supplied to corroborate the analytical results. Some previous results are extended.

Solitary wave solutions for the originating waves that propagate of the fractional Wazwaz-Benjamin-Bona-Mahony system

In this article, we discuss various solitary wave solutions that are extremely important in applied mathematics. In order to construct accurate solution to nonlinear fractional PDEs, we have employed the Khater technique to nonlinear equation for 3-dimensional fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) using conformable fractional dervatives. We obtain novel sets of solutions by employing this method including both periodic kink and periodic singular kink solution, singular periodic waves solution, dark solitons, bell shaped soliton solution, bell shaped singular solution, bright solution and smooth periodic wave solutions. In order to comprehend the physical principles and significance of the technique, solutions have been graphically represented. The results acquired to demonstrate the efficiency of the computational technique for the WBBM equation, our findings unambiguously demonstrate that the suggested approach is a useful, potent and simple way for experimental result for various kinds of non-integer order differential equations in the engineering and applied sciences.

Bifurcations and Exact Traveling Wave Solutions of Two Shallow Water Two-Component Systems

This paper studies two two-component shallow water wave models. From the dynamical systems approach and using the singular traveling wave theory developed by Li and Chen [2007], all possible bounded solutions (solitary wave solutions, pseudo-peakons, periodic peakons, as well as smooth periodic wave solutions) are obtained under different parameter conditions. More than six explicit exact parametric representations are derived. More interestingly, it was found that, for the two-component Camassa–Holm equations with constant vorticity, its [Formula: see text]-traveling wave system has a pseudo-peakon wave solution. In addition, its [Formula: see text]-traveling wave system has four families of uncountably infinitely many solitary wave solutions. The new results complete a recent study of Dutykh and Ionescu-Kruse [2019].

Bifurcations and Dynamics of Traveling Wave Solutions for the Regularized Saint-Venant Equation

This paper studies the bifurcations of phase portraits for the regularized Saint-Venant equation (a two-component system), which appears in shallow water theory, by using the theory of dynamical systems and singular traveling wave techniques developed in [Li & Chen, 2007] under different parameter conditions in the two-parameter space. Some explicit exact parametric representations of the solitary wave solutions, smooth periodic wave solutions, periodic peakons, as well as peakon solutions, are obtained. More interestingly, it is found that the so-called [Formula: see text]-traveling wave system has a family of pseudo-peakon wave solutions, and their limiting solution is a peakon solution. In addition, it is found that the [Formula: see text]-traveling wave system has two families of uncountably infinitely many solitary wave solutions and compacton solutions.

Completing the Study of Traveling Wave Solutions for Three Two-Component Shallow Water Wave Models

For three two-component shallow water wave models, from the approach of dynamical systems and the singular traveling wave theory developed in [Li & Chen, 2007], under different parameter conditions, all possible bounded solutions (solitary wave solutions, pseudo-peakons, periodic peakons, as well as smooth periodic wave solutions) are derived. More than 19 explicit exact parametric representations are obtained. Of more interest is that, for the integrable two-component generalization of the Camassa–Holm equation, it is found that its [Formula: see text]-traveling wave system has a family of pseudo-peakon wave solutions. In addition, its [Formula: see text]-traveling wave system has two families of uncountably infinitely many solitary wave solutions. The new results complete a recent study by Dutykh and Ionescu-Kruse [2016].

Smooth Exact Traveling Wave Solutions Determined by Singular Nonlinear Traveling Wave Systems: Two Models

For a singular nonlinear traveling wave system of the first class, if there exist two node points of the associated regular system in the singular straight line, then the dynamics of the solutions of the singular system will be very complex. In this paper, two representative nonlinear traveling wave system models (namely, the traveling wave system of Green–Naghdi equations and the traveling wave system of the Raman soliton model for optical metamaterials) are investigated. It is shown that, if there exist two node points of the associated regular system in the singular straight line, then the singular system has no peakon, periodic peakon and compacton solutions, but rather, it has smooth periodic wave, solitary wave and kink wave solutions.

PEAKON AND CUSPON SOLUTIONS OF A GENERALIZED CAMASSA-HOLM-NOVIKOV EQUATION

The bounded traveling wave solutions of a generalized CamassaHolm-Novikov equation with <i>p</i>=2 and <i>p</i>=3 are derived via the dynamical system approach. The singular wave solutions including peakons and cuspons are obtained by the bifurcation analysis of the corresponding singular dynamical system and the orbits intersecting with or approaching the singular lines. The results show that the generalized Camassa-Holm-Novikov equation with <i>p</i>=2 and <i>p</i>=3 both admit smooth solitary wave, smooth periodic wave solutions, solitary peakons, periodic peakons, solitary cuspons and periodic cuspons as well. It is worth pointing out that the Novikov equation has no bounded traveling wave solutions with negative wave speed, but has a family of new periodic cuspons which are distinguished with the normal periodic cuspons for their discontinuous first-order derivatives at both maximum and minimum.

Periodic traveling wave solutions and their limit forms for the (n + 1)-dimensional double sinh-Gordon equation

In this paper, the (n+1)-dimensional double sinh-Gordon equation∑j=1nuxjxj−utt−αsinh(u)−βsinh(2u)=0is studied by using the bifurcation method of dynamical systems. The periodic traveling wave solutions and their limit forms are investigated. When the first integral varies, we show the convergence of the smooth periodic wave solutions and the periodic blow-up wave solutions, such as the smooth periodic wave solutions converge to the solitary wave solution, the smooth periodic wave solutions converge to the smooth periodic wave solution, the periodic blow-up wave solutions converge to the blow-up wave solution and the periodic blow-up wave solutions converge to the periodic blow-up wave solution. All possible explicit exact parametric representations of various nonlinear traveling waves also are given.

Exact Traveling Wave Solutions and Bifurcations for a Shallow Water Equation Modeling Surface Waves of Moderate Amplitude

In this paper, we consider the traveling wave solutions for a shallow water equation. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. On the basis of the theory of the singular traveling wave systems, we obtain the bifurcations of phase portraits and explicit exact parametric representations for solitary wave solutions and smooth periodic wave solutions, as well as periodic peakon solutions. We show the existence of compacton solutions of the equation under different parameter conditions.

Smooth soliton and kink solutions for a new integrable soliton equation

We establish a relationship between a new integrable soliton equation and Gardner’s equation by a transformation. Then, we use this transformation and solutions of Gardner’s equation to derive some exact solutions of the new integrable soliton equation. The obtained solutions include smooth soliton solution, kink solution and periodic wave solution. In particular, the hyperbolic type kink and the rational type kink waves are given.

Logarithm transformation together with the integral bifurcation method for investigating exact solutions of the CDF equation with mKdV type

In this paper, under a logarithm transformation, by using the integral bifurcation method, we study the Cologero–Degasperis–Fokas equation with mKdV type, which was introduced by Cologero, Degasperis and Fokas. Some exact traveling wave solutions such as smooth solitary wave solutions, smooth periodic wave solutions, kink wave solutions, singular kink wave solution, singular periodic wave solutions, blowup soliton solution and singular blowup wave solution are obtained. We show their profiles and discuss their dynamic properties aim at some typical solutions.

Three kinds of periodic wave solutions and their limit forms for a modified KdV-type equation

A modified KdV-type equation is studied by using the bifurcation theory of dynamical system. By investigating the dynamical behavior with phase space analysis, all possible explicit exact traveling wave solutions including peakon solutions, kink and anti-kink wave solutions, blow-up wave solutions, smooth periodic wave solutions, periodic cusp wave solutions, and periodic blow-up wave solutions are obtained. When the first integral varies, we also show the convergence of the periodic wave solutions, such as the smooth periodic wave solutions converge to the kink and anti-kink wave solutions, the periodic cusp wave solutions converge to the peakon solution, the periodic blow-up wave solutions converge to the blow-up wave solution, the blow-up wave solutions converge to the blow-up wave solution, and the periodic blow-up wave solutions converge to the periodic blow-up wave solution.

The explicit periodic wave solutions and their limit forms for a generalized b-equation

In this paper, for b ∈ (−∞,∞) and b ≠ −1,−2, we investigate the explicit periodic wave solutions for the generalized b-equation ut + 2kux − uxxt + (1+b)u2ux = buxuxx + uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable α. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for α. When α tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = −1 or b = −2, we guess that the equation does not have any one of above solutions.

Regular and Singular Pulse and Front Solutions and Possible Isochronous Behavior in the Extended-Reduced Ostrovsky Equation: Phase-Plane, Multi-Infinite Series and Variational Formulations

In this paper we employ three recent analytical approaches to investigate several classes of traveling wave solutions of the so-called extendedreduced Ostrovsky Equation (exROE). A recent extension of phase-plane analysis is first employed to show the existence of breaking kink wave solutions and smooth periodic wave (compacton) solutions. Next, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of the travelingwave equations for the exROE equation. These correspond to pulse solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding traveling-wave 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. And finally, variational methods are employed to generate families of both regular and embedded solitary wave solutions for the exROE PDE. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and it is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the assumed ansatz for the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that not much is known about solutions of the family of generalized exROE equations considered here, the results obtained are both new and timely.

Exact traveling wave solutions and bifurcations of the dual Ito equation

In this paper, we consider the traveling wave solutions for the dual Ito equation. The corresponding traveling wave systems are two singular planar dynamical systems with one and two singular straight lines, respectively. On the basis of the theory of the singular traveling wave systems, we obtain the bifurcations of phase portraits and all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic cusp wave solutions, smooth periodic wave solutions, compactons, etc.) under different parameter conditions.

Periodic wave solutions and their limits for the modified KdV-KP equations

In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the modified KdV-KP equations. Some explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain solitary wave solutions, periodic wave solutions, kink wave solutions and unbounded solutions.

On the wave length of smooth periodic traveling waves of the Camassa–Holm equation

This paper is concerned with the wave length λ of smooth periodic traveling wave solutions of the Camassa–Holm equation. The set of these solutions can be parametrized using the wave height a (or “peak-to-peak amplitude”). Our main result establishes monotonicity properties of the map a⟼λ(a), i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated with the equation, which distinguish between the two possible qualitative behaviors of λ(a), namely monotonicity and unimodality. The key point is to relate λ(a) to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems.

Bifurcations and Exact Solutions of a Modulated Equation in a Discrete Nonlinear Electrical Transmission Line (II)

In this paper, we consider a model which is the modulated equation in a discrete nonlinear electrical transmission line. This model is an integrable planar dynamical system having three singular straight lines. By using the theory of singular systems and investigating the dynamical behavior, we obtain bifurcations of the phase portraits of the system under different parameter conditions. Corresponding to some special level curves, we derive possible exact explicit parametric representations of solutions (including smooth solitary wave and periodic wave solutions, periodic cusp wave solutions) under different parameter conditions.

Exact Solutions and Bifurcations of a Modulated Equation in a Discrete Nonlinear Electrical Transmission Line (I)

In this paper, we consider a model which is a modulated equation in a discrete nonlinear electrical transmission line. By investigating the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we derive all explicit exact parametric representations of solutions (including smooth solitary wave solutions, smooth periodic wave solutions, peakons, compactons, periodic cusp wave solutions, etc.) under different parameter conditions.

Bifurcations of traveling wave solutions for the (2 + 1)-dimensional generalized asymmetric Nizhnik–Novikov–Veselov equation

By using the bifurcation theory of planar dynamical systems to the (2+1)-dimensional generalized asymmetric Nizhnik–Novikov–Veselov equation, the existence for solitary wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of these solutions mentioned are given. Furthermore, some exact explicit parametric expressions of these bounded traveling waves are obtained.