The bifurcation and exact peakons,solitary and periodic wave solutions for the Kudryashov–Sinelshchikov equation

In this paper, the Kudryashov–Sinelshchikov equation is studied by using the bifurcation method of dynamical systems and the method of phase portraits analysis. From dynamic point of view, the existence of peakon, solitary wave, smooth and non-smooth periodic waves is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, some new exact travelling wave solutions are presented through some special phase orbits.     

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.

     

Exact solutions for a variable coefficient and Wick-type stochastic long–short wave resonance equation

A Wick-type stochastic long–short wave resonance equation is studied. Using white noise analysis, Hermite transform, we obtained a number of Wick versions of period-like and soliton-like solutions.     

New peakon,solitary wave and periodic wave solutions for the modified Camassa–Holm equation

In this paper, an independent variable transformation is introduced to solve the modified Camassa–Holm equation using the bifurcation theory and the method of phase portrait analysis. Some peakons, solitary waves and periodic waves are found and their exact parametric representations in explicit form and in implicit form are obtained.     

Exact loop solutions,cusp solutions,solitary wave solutions and periodic wave solutions for the special CH–DP equation

By using the method of dynamical systems, the travelling wave solutions of a special CH–DP equation are studied. Exact explicit parametric representations of smooth solitary waves, solitary cusp waves, breaking waves and uncountably infinitely many smooth periodic wave solutions are given. In different regions of the parametric plane, different phase portraits are determined. The so called loop soliton solution is discussed.     

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.     

Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schrödinger equation

We study the periodic traveling wave solutions of the derivative nonlinear Schrödinger equation (DNLS). It is known that DNLS has two types of solitons on the whole line; one has exponential decay and the other has algebraic decay. The latter corresponds to the soliton for the massless case. In the new global results recently obtained by Fukaya, Hayashi and Inui [15], the properties of two-parameter of the solitons are essentially used in the proof, and especially the soliton for the massless case plays an important role. To investigate further properties of the solitons, we construct exact periodic traveling wave solutions which yield the solitons on the whole line including the massless case in the long-period limit. Moreover, we study the regularity of the convergence of these exact solutions in the long-period limit. Throughout the paper, the theory of elliptic functions and elliptic integrals is used in the calculation.     

Single peak solitary wave solutions for the Fornberg–Whitham equation

The qualitative theory of differential equations is applied to the Fornberg–Whitham equation. Smooth, peaked and cusped solitary wave solutions of the Fornberg–Whitham equation under inhomogeneous boundary condition are obtained. The 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 Fornberg–Whitham equation. The results presented in this article extend and improve the previous results.     

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.     

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.     

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.     

The approximate manifolds for the generalized (2+1)-dimensional long–short wave equations

In this paper, the long time behavior of the dissipative generalized (2+1)-dimensional long–short wave equations was studied in dynamics. By applying projecting operator and the eigenvalue methods, the approximate inertial manifolds were constructed. And it is proved that arbitrary trajectory of the dissipative generalized (2+1)-dimensional long–short wave equations goes into a small neighborhood of the approximate inertial manifolds after long time.     

Time periodic traveling wave solutions for periodic advection–reaction–diffusion systems

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a class of periodic advection–reaction–diffusion systems. Under certain conditions, we prove that there exists a maximal wave speed c^?$c^{?}$ such that for each wave speed c≤c^?$c\le c^{?}$, there is a time periodic traveling wave connecting two periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c≤c^?$c\le c^{?}$ are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves with speed c>c^?$c>c^{?}$.     

Existence of periodic and subharmonic solutions for second-order superlinear difference equations

By critical point theory, a new approach is provided to study the existence and multiplicity results of periodic and subharmonic solutions for difference equations. For secord-order difference equations△2xn-1+f(n,xn)=0some new results are obtained for the above problems when f(t, z) has superlinear growth at zero and at infinityin z.     

Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance

In this paper, we deal with the existence of unbounded orbits of the mapping {θ1 = θ 2nπ 1/ρμ(θ) o(ρ-1),ρ1=ρ c-μ′(θ) o(1), ρ→∞,where n is a positive integer, c is a constant and μ(θ) is a 2π-periodic function. We prove that if c ＞ 0 and μ(θ) ≠ 0, θ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the future for ρ large enough; if c ＜ 0 and μ(θ) ≠ 0, θ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the past for ρ large enough. By using this result, we prove that the equation x″ f(x)x′ ax -bx- φ(x) =p(t) has unbounded solutions provided that a, b satisfy 1/√a 1/√b = 2/n and F(x)(= ∫x0 f(s)ds),and φ(x) satisfies some limit conditions. At the same time, we obtain the existence of 2π-periodic solutions of this equation.     

Existence of periodic travelling wave solutions of non-autonomous reaction–diffusion equations with lambda–omega type

Periodic travelling wave solutions of reaction–diffusion equations were studied by many authors. The λ–ω$\lambda \text{–}\omega$ type reaction–diffusion system is a notable special model that admits explicit periodic travelling wave solutions and was introduced by Kopell and Howard in 1973. There are now similar systems which are investigated by means of autonomous dynamics. In contrast, there are few papers which are concerned with non-autonomous cases. For this reason, we apply Mawhin’s continuation theorem to derive the existence of periodic travelling wave solutions for non-autonomous λ–ω$\lambda \text{–}\omega$ systems, and we describe the ‘disappearance’ of periodic travelling wave solutions under special situations. Our main result is also illustrated by examples.     

Traveling wave solutions for Generalized Drinfeld–Sokolov equations

In this paper, solitary waves and periodic waves for Generalized Drinfeld–Sokolov equations are studied, by using the theory of dynamical systems. Bifurcation parameter sets are shown. Under given parameter conditions, explicit formulas of solitary wave, kink (anti-kink) wave and periodic wave solutions are obtained.     

Stability and long time evolution of the periodic solutions to the two coupled nonlinear Schrödinger equations

The system of two coupled nonlinear Schrödinger equations has wide applications in physics. In the past, the main attention has been their solitary waves. Here we turn our attention to their periodic wave solutions. In this paper, the stability of the periodic solutions is studied analytically and the criteria for the stability are obtained. The long time evolution of the solutions to the coupled system is studied numerically for the unstable case emphasizing wave–wave interactions in nonlinear optics. Different kinds of evolution are observed depending on the coefficients of the system and the parameters of the unperturbed waves and perturbation. For certain ranges of parameters, the evolution appears to be periodic, while for some other ranges of parameters, solitary wave or solitary wave pairs can be excited among the irregular background although often the evolution is completely chaotic.