On soliton solutions for Boussinesq–Burgers equations

The periodic wave solutions for Boussinesq–Burgers equations are obtained by using of Jacobi elliptic function method, in the limit cases, the multiple soliton solutions are also obtained. The properties of some periodic and soliton solution for this system are shown by some figures.     

On soliton solutions for the Fitzhugh–Nagumo equation with time-dependent coefficients

We consider a generalized Fitzhugh–Nagumo equation exhibiting time-varying coefficients and linear dispersion term. By means of specific solitary wave ansatz and the tanh method, a new variety of soliton solutions are derived. The physical parameters in the soliton solutions are obtained as function of the time-dependent model coefficients. The conditions of existence and uniqueness of solitons are presented. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous media that is described by the variable coefficients Fitzhugh–Nagumo equation. Clearly, adaptive methods are straightforward and concise and their applications for the Fitzhugh–Nagumo equation with t-dependent coefficients enable one to construct soliton-like solutions.     

Exact periodic kink-wave and degenerative soliton solutions for potential Kadomtsev–Petviashvili equation

Exact periodic kink-wave solution, periodic soliton and doubly periodic solutions for the potential Kadomtsev–Petviashvii (PKP) equation are obtained using homoclinic test technique and extended homoclinic test technique, respectively. It is investigated that periodic soliton is degenerated into doubly periodic wave varying with direction of wave propagation.     

On positive periodic solutions of the time–space periodic Lotka–Volterra cooperating system in multi-dimensional media

This work is devoted to the study of the time–space periodic reaction–diffusion–advection Lotka–Volterra cooperating system in multi-dimensional media. By using the method of sub-super solutions and its associated iterations, we prove the existence and uniqueness of the positive periodic solution under appropriate conditions. Finally, we are able to derive the asymptotic behavior of the solutions to the associated Cauchy problem.     

Permanence and periodic solutions for a diffusive ratio-dependent predator–prey system

This paper considers a diffusive predator–prey model, in which there is a ratio-dependent functional response with Holling III type. We establish some sufficient conditions for the ultimate boundedness of solutions and permanence of this system. The existence of a unique globally stable periodic solution is also presented.     

The stability problem and special solutions for the 5-components Maxwell–Bloch equations

For the 5-components Maxwell–Bloch system the stability problem for the isolated equilibria is completely solved. Using the geometry of the symplectic leaves, a detailed construction of the homoclinic orbits is given. Studying the problem of invariant sets for the system, we discover a rich family of periodic solutions in explicit form.     

Permanence and periodic solutions for an impulsive reaction–diffusion food-chain system with ratio-dependent functional response

An impulsive reaction–diffusion periodic food-chain system with ratio-dependent functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the food-chain system are established based on the comparison theory of differential equation and upper and lower solution method. By constructing appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given in the end of the paper.     

Multiple periodic solutions of a delayed predator–prey system with stage structure for the predator

By using the continuation theorem of coincidence degree theory, the existence of multiple positive periodic solutions for a delayed predator–prey system with stage structure for the predator is established.     

Solitons in the system of coupled Hirota–Maxwell–Bloch equations

We propose the system of coupled Hirota–Maxwell–Bloch equations which governs the propagation of optical pulses in an erbium doped nonlinear fibre with higher order dispersion, self-steepening and self induced transparency (SIT) effects. The Lax pair is explicitly constructed and the soliton solution is obtained using the Darboux–Bäcklund transformations. Hence, the system is found to admit soliton type lossless wave propagation.     

Positive ground state solutions for the critical Klein–Gordon–Maxwell system with potentials

This paper deals with the Klein–Gordon–Maxwell system when the nonlinearity exhibits critical growth. We prove the existence of positive ground state solutions for this system when a periodic potential V$V$ is introduced. The method combines the minimization of the corresponding Euler–Lagrange functional on the Nehari manifold with the Brézis and Nirenberg technique.     

Almost periodic solutions of a discrete Lotka–Volterra competition system with delays

In this paper, we consider a discrete almost periodic Lotka–Volterra competition system with delays. Sufficient conditions are obtained for the permanence and global attractivity of the system. Further, by means of an almost periodic functional hull theory, we show that the almost periodic system has a unique strictly positive almost periodic solution, which is globally attractive. Some examples are presented to verify our main results.     

An Ambrosetti–Prodi-type result for doubly periodic solutions of the telegraph system

This article deals with an Ambrosetti–Prodi-type result for doubly periodic solutions of the telegraph system by using Leray–Schauder degree theory, the method of upper and lower solutions.     

On the periodic Cauchy problem for a coupled Camassa–Holm system with peakons

In this paper, we first prove that the solution map of the Cauchy problem for a coupled Camassa–Holm system is not uniformly continuous in \({H^{s}(\mathbb{T}) \times H^{s}(\mathbb{T}),s > \frac{3}{2}}\), the proof of which is based on well posedness estimates and the method of approximate solutions. Then we study the continuity properties of its solution map further and show that it is Hölder continuous in the \({H^\sigma(\mathbb{T}) \times H^\sigma(\mathbb{T})}\) topology with \({\frac{1}{2} < \sigma < s}\). Our results can also be carried out on the nonperiodic case.     

Nonlocal symmetry and soliton–cnoidal wave solutions of the Bogoyavlenskii coupled KdV system

The truncated Painlevé expansion is developed to construct Bäcklund transformations and nonlocal symmetries of the Bogoyavlenskii coupled KdV (BcKdV) system. The Schwarzian form of BcKdV system is found while the nonlocal symmetry is localized to offer the corresponding nonlocal group. Furthermore, the BcKdV system is verified to have a consistent Riccati expansion (CRE). Stemming from the consistent tan-function expansion (CTE), which is a special form of CRE, the soliton–cnoidal wave solutions are explicitly studied.     

Almost periodic solutions for Lotka–Volterra systems with delays

This paper studies a general class of delayed almost periodic Lotka–Volterra system with time-varying delays and distributed delays. By using the definition of almost periodic function, the sufficient conditions for the existence and uniqueness of globally exponentially stable almost periodic solution are obtained. The conditions can be easily reduced to special cases of cooperative systems and competitive systems.     

Permanence of periodic predator–prey system with two predators and stage structure for prey

A stage-structured three-species predator–prey system with Beddington–DeAngelis and Holling IV functional response is proposed and analyzed. Based on the comparison theorem, some sufficient and necessary conditions are derived for permanence of the system. Finally, two examples are presented to illustrate the application of our main results.     

On the Painlevé integrability,periodic wave solutions and soliton solutions of generalized coupled higher-order nonlinear Schrödinger equations

It is proven that generalized coupled higher-order nonlinear Schrödinger equations possess the Painlevé property for two particular choices of parameters, using the Weiss–Tabor–Carnevale method and Kruskal’s simplification. Abundant families of periodic wave solutions are obtained by using the Jacobi elliptic function expansion method with the assistance of symbolic manipulation system, Maple. It is also shown that these solutions exactly degenerate to bright soliton, dark soliton and mixed dark and bright soliton solutions with physical interests.