Bifurcations of traveling wave solutions for a two-component Fornberg–Whitham equation

A two-component Fornberg–Whitham equation is introduced as a model for water waves. The bifurcations of traveling wave solutions are studied. Parametric conditions to smooth soliton solution, kink solution, antikink solution and uncountable infinite many smooth periodic wave solutions are given. Some expressions for those solutions are presented.     

Bifurcations of travelling wave solutions for a class of nonlinear fourth order variant of a generalized Camassa–Holm equation

In this paper, by using the bifurcation theory of dynamical systems for a class of nonlinear fourth order variant of a generalized Camassa–Holm equation, the existence of solitary wave solutions, breaking bounded wave solutions, compacton solutions and non-smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

Exact traveling wave solutions of the Boussinesq–Burgers equation

The extended homogeneous balance method is used to construct exact traveling wave solutions of the Boussinesq–Burgers equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation. Many exact traveling wave solutions of the Boussinesq–Burgers equation are successfully obtained.     

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.     

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 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 of travelling wave solutions for two generalized Boussinesq systems

Using the methods of dynamical systems for two generalized Boussinesq systems, the existence of all possible solitary wave solutions and many uncountably infinite periodic wave solutions is obtained. Exact explicit parametric representations of the travelling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.     

On traveling wave solutions to combined KdV–mKdV equation and modified Burgers–KdV equation

In this work, we established exact solutions for some nonlinear evolution equations. The extended tanh method was used to construct solitary and soliton solutions of nonlinear evolution equations. The extended tanh method presents a wider applicability for handling nonlinear wave equations.     

Bifurcations and new exact travelling wave solutions for the Gerdjikov–Ivanov equation

In this paper, the Gerdjikov–Ivanov equation is investigated by using the bifurcation theory and the method of phase portraits analysis. The existence of every kind of travelling waves is proved, in some conditions, exact parametric representations of above travelling waves in explicit form are obtained.     

Nonexistence of the periodic peaked traveling wave solutions for rotation-Camassa–Holm equation

Recently, Zhu et al. (2020) proposed a kind of rotation-Camassa–Holm equation. In this paper, we study the question of nonexistence of periodic peaked traveling wave solution for rotation-Camassa–Holm equation. Indeed, rotation-Camassa–Holm equation has no nontrivial periodic Camassa–Holm peaked solution unlike Camassa–Holm equation, modified Camassa–Holm equation, Novikov equation.     

Rapidly oscillating solutions of a generalized Korteweg–de Vries equation

For a generalized Korteweg–de Vries equation, the existence of families of rapidly oscillating periodic solutions is proved and their asymptotic representation is found. The asymptotics of tori of different dimensions are examined. Formulas for solutions depending on all parameters of the problem are derived.     

Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions

We derive the Whitham modulation equations for the Zakharov–Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all periodic solutions traveling along the first spatial coordinate are linearly unstable with respect to purely transversal perturbations, and we obtain an explicit expression for the growth rate of perturbations in the long wave limit. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. We also calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches. Finally, we generalize the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and we determine the resulting domains of stability and instability.     

Global existence and asymptotic behavior of smooth solutions to a bipolar Euler–Poisson equation in a bound domain

In this paper, we present a bipolar hydrodynamic model from semiconductor devices and plasmas, which takes the form of bipolar isentropic Euler–Poisson with electric field and frictional damping added to the momentum equations. We firstly prove the existence of the stationary solutions. Next, we present the global existence and the asymptotic behavior of smooth solutions to the initial boundary value problem for a one-dimensional case in a bounded domain. The result is shown by an elementary energy method. Compared with the corresponding initial data case, we find that the asymptotic state is the stationary solution.     

Some exact and approximate solutions to a generalized Maxwell–Cattaneo equation

The simple heat conduction equation in one-space dimension does not have the property of a finite speed for information transfer. A partial resolution of this difficulty can be obtained within the context of heat conduction by the introduction of a partial differential equation (PDE ) called the Maxwell–Cattaneo (M-C) equation, elsewhere called the damped wave equation, a special case of the telegraph equation. We construct a generalization to the M-C equation by allowing the relaxation time parameter to be a function of temperature. In the balance of the paper, we present a variety of special exact and approximate solutions to this nonlinear PDE .     

Single peak solitary wave solutions for the generalized Camassa–Holm equation

This Letter presents all possible smooth, peaked and cusped solitary wave solutions for the generalized Camassa–Holm equation under the inhomogeneous boundary condition.The parametric conditions of existence of the smooth, peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for smooth, peaked and cusped solitary wave solutions of the generalized Camassa–Holm equation.     

Exact solutions of the generalized Sinh–Gordon equation

In this paper, we successfully derive a new exact traveling wave solutions of the generalized Sinh–Gordon equation by new application of the homogeneous balance method. This method could be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics.     

Blow-up of smooth solutions of the Korteweg–de Vries equation

The paper is devoted both to some initial–boundary value problems and to the Cauchy problem for the KdV equation.