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This paper concerns with the existence problem of travelling wave solutions to a singularly perturbed Burgers-KdV equation. For this, we use the dynamical systems approach, specifically, the geometric singular perturbation theory and centre manifold theory. We also numerically show approximations, in particular, for kink-type waves.
In this paper, we implement the exp-function method to obtain the exact travelling wave solutions of (N + 1)-dimensional nonlinear evolution equations. Four models, the (N + 1)-dimensional generalized Boussinesq equation, (N + 1)-dimensional sine-cosine-Gordon equation, (N + 1)-double sinh-Gordon equation and (N + 1)-sinhcosinh-Gordon equation, are used as vehicles to conduct the analysis. New travelling...
The surface water waves in a water tunnel can be described by systems of the form [Bona and Chen, PhysicaD116, 191 (1998)] 1 $$ \label{BWE} \left\{ \begin{array}{l} v_t+u_x+(uv)_x+au_{xxx}-bv_{xxt}=0, \\ u_t+v_x+uu_x+cv_{xxx}-du_{xxt}=0, \end{array} \right. $$ where a, b, c and d are real constants. In general, the exact travelling wave solutions will be helpful in the theoretical and...
In this work, we present travelling wave solutions for the Burgers, Burgers–Huxley and modified Burgers–KdV equations. The (G′/G)-expansion method is used to determine travelling wave solutions of these sets of equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. It is shown that the proposed method is direct, effective...
In this paper, we establish exact solutions for some special nonlinear partial differential equations. The (G′/G)-expansion method is used to construct travelling wave solutions of the two-dimensional sine-Gordon equation, Dodd–Bullough–Mikhailov and Schrödinger–KdV equations, which appear in many fields such as, solid-state physics, nonlinear optics, fluid dynamics, fluid flow, quantum field theory,...
In this paper, coupled Higgs field equation and Hamiltonian amplitude equation are studied using the Lie classical method. Symmetry reductions and exact solutions are reported for Higgs equation and Hamiltonian amplitude equation. We also establish the travelling wave solutions involving parameters of the coupled Higgs equation and Hamiltonian amplitude equation using (G′/G)-expansion method, where...
In this paper, we investigate the bifurcations and dynamic behaviour of travelling wave solutions of the Klein–Gordon–Zakharov equations given in Shang et al, Comput. Math. Appl.56, 1441 (2008). Under different parameter conditions, we obtain some exact explicit parametric representations of travelling wave solutions by using the bifurcation method (Feng et al, Appl. Math. Comput.189, 271 (2007);...
The nonlinear stochastic evolution equations have a wide range of applications in physics, chemistry, biology, economics and finance from various points of view. In this paper, the $\left({G^{\prime}}/{G}\right)$ -expansion method is implemented for obtaining new travelling wave solutions of the nonlinear (2 + 1)-dimensional stochastic Broer–Kaup equation and stochastic coupled Korteweg–de Vries...
In this paper, we construct the travelling wave solutions to the perturbed nonlinear Schrödinger’s equation (NLSE) with Kerr law non-linearity by the extended (G′/G)-expansion method. Based on this method, we obtain abundant exact travelling wave solutions of NLSE with Kerr law nonlinearity with arbitrary parameters. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric...
In this article, a variety of solitary wave solutions are found for some nonlinear equations. In mathematical physics, we studied two complex systems, the Maccari system and the coupled Higgs field equation. We construct sufficient exact solutions for nonlinear evolution equations. To study travelling wave solutions, we used a fractional complex transform to convert the particular partial differential...
We examine the effect of dissipation on travelling waves in nonlinear dispersive systems modelled by Benjamin–Bona–Mahony (BBM)-like equations. In the absence of dissipation, the BBM-like equations are found to support soliton and compacton /anticompacton solutions depending on whether the dispersive term is linear or nonlinear. We study the influence of increasing nonlinearity of the medium on the...
A direct rational exponential scheme is introduced and applied to construct exact multisoliton solutions of the clannish random walker’s parabolic and the Vakhnenko–Parkes equations. We discuss the nature of soliton solutions before and after their interactions, and present their fusion (non-elastic) and elastic collisions of the soliton solutions. These soliton solutions of the equations are connected...
In this paper, the canonical-like transformation method and trial equation method are applied to $$(2+1)$$ (2+1) -dimensional Chaffee–Infante equation, and some exact solutions are obtained. In particular, a new solution in terms of elliptic functions is given.
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