Solitary wave solutions of the generalized two-component Hunter–Saxton system

The existence of solitary wave solutions of the generalized two-component Hunter–Saxton system is determined. It is also shown that there are peaked and cusped solitary waves with singularities among those smooth solitary wave solutions.     

On well-posedness of a dispersive system of the Whitham–Boussinesq type

The initial-value problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system was recently introduced in Dinvay (2018). It is numerically shown to be stable and a good approximation to the incompressible Euler equations. Here we prove local in time well-posedness. Our proof relies on an energy method and a compactness argument. In addition some numerical experiments, supporting the validity of the system as an asymptotic model for water waves, are carried out.     

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.     

Solitary-wave solutions to a dual equation of the Kaup–Boussinesq system

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the travelling-wave solutions to a dual equation of the Kaup–Boussinesq system. The expressions for smooth solitary-wave solutions are obtained.     

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.     

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.     

Solitary wave solution of the generalized Hirota–Satsuma coupled KdV system

In this research, we find the exact traveling wave solutions involving parameters of the generalized Hirota–Satsuma couple KdV system according to the modified simple equation method with the aid of Maple 16. When these parameters are taken special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the modified simple equation method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented.     

Solitary wave solutions of two nonlinear physical models by tanh–coth method

In this work, we established the exact solutions for some nonlinear physical models. The tanh–coth method was used to construct solitary wave solutions of nonlinear evolution equations. The tanh–coth method presents a wider applicability for handling nonlinear wave equations.     

Solitary wave solutions of higher-order NLSE with Raman and self-steepening effect in a cubic–quintic–septic medium

We find bright and dark solitary wave solutions for the higher-order nonlinear Schrodinger equation with cubic–quintic–septic terms adopting the ansatz solution of Li et al. [Li Z, Li L, Tian H, Zhou G. Phys. New types of solitary wave solutions for the higher-order nonlinear Schrödinger equation. Phys Rev Lett 2000;84(18):4096–99.] which may describe propagation of pulses upto the order of 10 fs in a non-Kerr media. In this context, we have taken into account both the Raman and the self-steepening effect. All the solitary wave parameters and the parametric conditions for the solitary wave solutions are worked out.     

Coinvasion–coexistence travelling wave solutions of an integro-difference competition system

This paper is concerned with the travelling wave solutions of an integro-difference competition system, of which the purpose is to model the coinvasion–coexistence process of two competitors with age structure. The existence of non-trivial travelling wave solutions is obtained by constructing generalized upper and lower solutions. The asymptotic and non-existence of travelling wave solutions are proved by combining the theory of asymptotic spreading with the idea of contracting rectangle.     

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.     

Hamiltonian Boussinesq formulation of wave–ship interactions

In this paper a new approach is described for the fully nonlinear treatment of the dynamic wave–ship interaction for potential flows. A major reduction of computational complexity is obtained by describing the fluid motion in horizontal variables only, the surface elevation and the potential at the surface. In such Boussinesq type of equations, the internal fluid motion is not calculated, but modeled in a consistent approximative way. The equations for the wave–ship interaction are based on a Lagrangian variational principle, leading to the formulation of the coupled system as a Hamiltonian system. With the ship position and orientation as canonical coordinates, the canonically conjugate momentum variables are the sum of the ship momemta and the fluid momenta. A beneficial consequence of this is that the momentum exchange between fluid and ship will be described without the need to calculate the pressure, which simplifies the numerical implementation of the equations considerably. Provided that the potentials with mixed Dirichlet–Neumann data can be calculated, the presented ship dynamics can be inserted in existing free surface flow solvers.     

Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled KdV equation by an analytical technique

In this work, we implement a relatively analytical technique, the homotopy perturbation method (HPM), for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo derivatives. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations which applied in engineering mathematics. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. He’s homotopy perturbation method (HPM) which does not need small parameter is implemented for solving the differential equations. It is predicted that HPM can be found widely applicable in engineering.     

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.     

Solitary wave solution of higher-order Korteweg–de Vries equation

An attempt has been made to obtain exact analytical traveling wave solution or simple wave solution of higher-order Korteweg–de Vries (KdV) equation by using tanh-method or hyperbolic method. The higher-order equation can be derived for magnetized plasmas by using the reductive perturbation technique. It is found that the exact solitary wave solution of higher-order KdV equation is obtained by tanh-method. Using this method, different kinds of nonlinear wave equations can be evaluated. The higher-order nonlinearity and higher-order dispersive effect can be observed from the solutions of the equations. The method is applicable for other nonlinear wave equations.     

Uniqueness of periodic solutions of a nonautonomous density-dependent predator–prey system

In this present paper, we investigate the uniqueness of periodic solutions of a nonautonomous density-dependent and ratio-dependent predator–prey system, where not only the prey density dependence but also the predator density dependence are considered, such that the studied predator–prey system conforms to the realistically biological environment. We start with a sufficient condition for the permanence of the system and then construct a weaker sufficient condition by introducing a specific set, denoted as Γ. Based on this Γ and the Brouwer fixed-point theorem, we obtain the existence condition of positive periodic solutions. Moreover, since the uniqueness of positive periodic solutions can be ensured by global attractiveness, we alternatively introduce a sufficient condition for global attractiveness. Similarly, we also provide a sufficient condition for the uniqueness of non-negative periodic solutions.     

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.