Non-smooth Traveling Wave Solutions Research Articles

Bifurcation and the exact smooth, cusp solitary and periodic wave solutions of the generalized Kudryashov–Sinelshchikov equation

The main aim of this paper is to study the exact traveling wave solutions of the generalized Kudryashov–Sinelshchikov equation by using the auxiliary equation method based on the conclusion of qualitative analysis. The advantage of this method is to choose the effective and proper auxiliary equation on the base of the behaviors and traits of solutions revealed by analysis of phase portraits to study the solution of differential equations. By applying the proposed approach to the generalized Kudryashov–Sinelshchikov equation, the number, behavior and existence of smooth and non-smooth traveling wave solutions are gained, at the same time, the new exact smooth solitary, periodic wave solutions and cusp solitary, periodic wave solutions are obtained. From the dynamic point of view, the behavior of traveling wave solutions is analyzed. The profile,type and the form of exact expression of traveling wave solutions are influenced by the order of nonlinear term and nonlinear terms.

Bifurcations and exact traveling wave solutions for the KdV-like equation

Applying the combination of the qualitative theory of differential equation and bifurcation theory of planar dynamical systems to a combined form of KdV-like equation, the bifurcations of phase portraits to the corresponding traveling system of this equation are presented. The exact representations of smooth and non-smooth traveling wave solutions are obtained under different regions of parametric space. Moreover, numerical simulations are provided for some solutions of the equation.

Bifurcation on Smooth and Non-smooth Traveling Wave Solutions for a Generalized Two-component Camassa-Holm Equation

A generalized two-component Camassa-Holm equation is introduced as a model for shallow water waves moving over a linear shear flow. Bifurcations of traveling wave solutions are studied. Phase portraits of the traveling wave system are given. By using the method of planar dynamical systems, the existence of solitary wave solutions, smooth and non-smooth periodic traveling wave solutions is presented in different parametric conditions. Numerical simulations are made to agree the theoretical analysis. It shows that the existence of singular straight lines for the generalized two-component Camassa-Holm equation is the original cause of the non-smooth solutions. The existence of uncountably infinitely many breaking traveling wave solutions are given.

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.

Bifurcation of traveling wave solutions for the BBM-like B(2, 2) equation

In this paper, we employ the bifurcation method of dynamical systems to investigate the BBM-like B(2,2) equation. The phase portrait bifurcation of the traveling wave system corresponding to the equation is given. Under different parametric conditions, various sufficient conditions to guarantee the existence of smooth and non-smooth traveling wave solutions are given. Through some special phase orbits, Some solitary wave solutions expressed by implicit functions, periodic cusp wave solution, compacton solution and peakon solution are obtained.

Bifurcation of travelling wave solutions for the generalized KP-MEW equations

By using the theory of bifurcations of planar dynamical systems to the generalized KP-MEW equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are obtained.

Bifurcations and highly nonlinear traveling waves in periodic dimer granular chains

In this study, the highly nonlinear waves in periodic dimer granular chains were investigated by the theory of dynamical system and the method of phase diagram analysis. The bifurcations of the different traveling waves in parameter space and those different traveling waves and its phase diagram were given. In addition, the existence of smooth and non-smooth traveling wave solutions are shown and various sufficient conditions to guarantee the existence of the above solutions were listed. Copyright © 2011 John Wiley & Sons, Ltd.

Smooth and non-smooth traveling wave solutions of the Fornberg–Whitham equation with linear dispersion term

In this paper, the Fornberg–Whitham equation with linear dispersion term is investigated by employing the bifurcation method of dynamical systems. As a result, the existence of smooth and non-smooth traveling wave solutions is obtained. And the analytic expressions of solitary wave solutions, periodic cusp wave solutions and peakons are given under some parameter conditions.

Bifurcation of travelling wave solutions for the generalized Camassa–Holm–KP equations

By using the bifurcation theory of planar dynamical systems to the generalized Camassa–Holm–KP equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.

Bifurcation of travelling wave solutions for the generalized ZK-BBM equations

By using the bifurcation theory of planar dynamical systems to the generalized ZK-BBM equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.

Smooth and non-smooth traveling wave solutions of a class of nonlinear dispersive equation

There is the widespread existence of wave phenomena in physics, mechanics. This clearly necessitates a study of traveling waves in depth and of the modeling and analysis involved. In this paper, we study a nonlinear dispersive K( n,− n,2 n) equation, which can be regarded as a generalized K( n, n) equation. Applying the bifurcation theory and the method of phase portraits analysis, we obtain the dynamical behavior and special exact solutions of the K( n,− n,2 n) equation. As a result, the conditions under which peakon and compacton solutions appear are also given and the analytic expressions of peakon solutions, compacton and periodic cusp wave solutions are obtained.

Bifurcations of travelling wave solutions for the Gilson–Pickering equation

In this paper, the qualitative behavior and exact travelling wave solutions of the Gilson–Pickering equation are studied by using the qualitative theory of polynomial differential system. The phase portraits of the system are given under different parametric conditions. Some exact travelling wave solutions of the Gilson–Pickering equation are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of smooth and non-smooth travelling wave solutions are given.

Bifurcation of travelling wave solutions for the generalized ZK equations

By using the bifurcation theory of planar dynamical systems to the generalized ZK equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.

Bifurcations of travelling wave solutions for the mK(n, n) equation

Using the method of planar dynamical systems to the mK(n,n) equation, the existence of uncountably infinite many smooth and non-smooth periodic wave solutions, solitary wave solutions and kink and anti-kink wave solutions is proved. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All possible exact explicit parametric representations of smooth and non-smooth travelling wave solutions are obtain.

Bifurcations of traveling wave solutions for Zhiber–Shabat equation

In this paper, we investigate the dynamical behavior of traveling wave solutions in the Zhiber–Shabat equation by using the bifurcation theory and the method of phase portraits analysis. As a result, we obtain the conditions under which smooth and non-smooth traveling wave solutions exist, and give some exact explicit solutions for some special cases.

Smooth and non-smooth travelling wave solutions of generalized Degasperis–Procesi equation

The dynamical behavior and special exact solutions of the generalized Degasperis–Procesi equation u t + c 0 u x − u txx + auu x = 3 u x u xx + uu xxx is analyzed by using bifurcation theory and the method of phase portraits analysis. As a result, the analytic expressions of peakon solutions, compacton solutions and periodic cusp wave solutions are obtained. And the condition under which peakon and compacton solutions appear are also given. In the meantime, the reason that solitary cusp wave and periodic cusp wave appear is given.

Bifurcations of smooth and nonsmooth traveling wave solutions in a generalized degasperis–procesi equation

In this paper, we employ the bifurcation theory of planar dynamical systems to study the smooth and nonsmooth traveling wave solutions of the generalized Degasperis-Procesi equation u t - u xxt + 4 u m u x = 3 u x u xx + uu xxx . The parameter condition under which peakons, compactons and periodic cusp wave solutions exist is given. The numerical simulation results show the consistence with the theoretical analysis at the same time.

Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa–Holm equation

The dynamical behavior of travelling wave solutions in the Generalized Camassa–Holm equation u t + 2 ku x − u xxt + au m u x = 2 u x u xx + uu xxx is analyzed by using the bifurcation theory and the method of phase portraits analysis. The condition under which compactons and cusp waves appear are also given. In addition, the reason for solitary cusp wave and periodic cusp wave to exist is highlighted.

Bifurcations of Smooth and Non-Smooth Travelling Wave Solutions of the Degasperis-Procesi Equation

Bifurcations of Smooth and Non-Smooth Travelling Wave Solutions of the Degasperis-Procesi Equation

The smooth and nonsmooth travelling wave solutions in a nonlinear wave equation

The travelling wave solutions (TWS) in a class of P.D.E. is studied. The travelling wave equation of this P.D.E. is a planar cubic polynomial system in three-parameter space. The study for TWS became the topological classifications of bifurcations of phase portraits defined by the planar system. By using the theory of planar dynamical systems to do qualitative analysis, all topological classifications of the cubic polynomial system can be obtained. Returning the results of the phase plane analysis to TWS, u(ξ), and considering discontinuity of the right side of the equation of TWS when ξ=x−ct is varied along a phase orbit and passing through a singular curve, all conditions of existence of smooth and nonsmooth travelling waves are given.