Breather wave,periodic, and cross-kink solutions to the generalized Bogoyavlensky-Konopelchenko equation

The present article deals with multi-waves and breather wave solutions of the generalized Bogoyavlensky-Konopelchenko equation by virtue of the Hirota bilinear operator method and the semi-inverse variational principle. The obtained solutions for solving the current equation represent some localized waves including soliton, periodic, and cross-kink solutions in which have been investigated by the approach of the bilinear method. With certain parameter constraints in the multi-waves and breather, all cases of the periodic and cross-kink solutions can be captured from the one and two soliton(s). The obtained solutions are extended with numerical simulation to analyze graphically, which results into 1- and 2-soliton solutions and also periodic and cross-kink solutions profiles, that will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on.     

New kinds of solitons and periodic solutions to the generalized KdV equation

In this work, the sine‐cosine method, the tanh method, and specific schemes that involve hyperbolic functions are used to study solitons and periodic solutions governed by the generalized KdV equation. New solutions are determined by using the hyperbolic functions schemes. The study introduces new approaches to handle nonlinear PDEs in the solitary wave theory. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 247–255, 2007     

Explicit exact periodic wave solutions and their limit forms for a long waves-short waves model

A long waves-short waves model is studied by using the approach of dynamical systems. The sufficient conditions to guarantee the existence of solitary wave, kink and anti-kink waves, and periodic wave in different regions of the parametric space are given. All possible explicit exact parametric representations of above traveling waves are presented. When the energy of Hamiltonian system corresponding to this model varies, we also show the convergence of the periodic wave solutions, such as the periodic wave solutions converge to the solitary wave solutions, kink and anti-kink wave solutions, and periodic wave solutions, respectively.     

The breather wave solutions, M-lump solutions and semi-rational solutions to a (2+1)-dimensional generalized Korteweg-de Vries equation

Under investigation in this work is a (2+1)-dimensional generalized Korteweg-de Vries equation, which can be used to describe many nonlinear phenomena in plasma physics. By using the properties of Bell"s polynomial, we obtain the bilinear formalism of this equation. The expression of $N$-soliton solution is established in terms of the Hirota"s bilinear method. Based on the resulting $N$-soliton solutions, we succinctly show its breather wave solutions. Furthermore, with the aid of the corresponding soliton solutions, the $M$-lump solutions are well presented by taking a long wave limit. Two types of hybrid solutions are also represented in detail. Finally, some graphic analysis are provided in order to better understand the propagation characteristics of the obtained solutions.     

Abundant explicit exact solutions to the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law nonlinearities

This paper is concerned with the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law. Abundant explicit and exact solutions of the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law are derived uniformly by using the first integral method. These exact solutions are include that of extended hyperbolic function solutions, periodic wave solutions of triangle functions type, exponential form solution, and complex hyperbolic trigonometric function solutions and so on. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial DEs. Copyright © 2014 John Wiley & Sons, Ltd.     

New kink solutions for the van der Waals p‐system

The simple equation method and modified simple equation method are employed to seek exact traveling wave solutions to the (1 + 1)‐dimensional van der Waals gas system in the viscosity‐capillarity regularization form. Under the help of Mathematica, new classes of kink solutions are derived. Numerical simulations with special choices of the free parameters are displayed by three‐ and two‐dimensional plots. The two methods demonstrate simplicity, reliability, and efficiency.     

Gauss‐Lobatto‐Legendre‐Birkhoff pseudospectral approximations for the multi‐term time fractional diffusion‐wave equation with Neumann boundaryconditions

A second‐order finite difference/pseudospectral scheme is proposed for numerical approximation of multi‐term time fractional diffusion‐wave equation with Neumann boundary conditions. The scheme is based upon the weighted and shifted Grünwald difference operators approximation of the time fractional calculus and Gauss‐Lobatto‐Legendre‐Birkhoff (GLLB) pseudospectral method for spatial discretization. The unconditionally stability and convergence of the scheme are rigorously proved. Numerical examples are carried out to verify theoretical results.     

Periodic wave solutions and asymptotic analysis of the Hirota–Satsuma shallow water wave equation

In this paper, we devise a simple way to explicitly construct the Riemann theta function periodic wave solution of the nonlinear partial differential equation. The resulting theory is applied to the Hirota–Satsuma shallow water wave equation. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function. We obtain the one‐periodic and two‐periodic wave solutions of the equation. The relations between the periodic wave solutions and soliton solutions are rigorously established. Copyright © 2014 John Wiley & Sons, Ltd.     

Multi‐periodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n‐torus

In this paper, we study the solutions to the generalized Helmholtz equation with complex parameter on some conformally flat cylinders and on the n‐torus. Using the Clifford algebra calculus, the solutions can be expressed as multi‐periodic eigensolutions to the Dirac operator associated with a complex parameter λ∈?. Physically, these can be interpreted as the solutions to the time‐harmonic Maxwell equations on these manifolds. We study their fundamental properties and give an explicit representation theorem of all these solutions and develop some integral representation formulas. In particular, we set up Green‐type formulas for the cylindrical and toroidal Helmholtz operator. As a concrete application, we explicitly solve the Dirichlet problem for the cylindrical Helmholtz operator on the half cylinder. Finally, we introduce hypercomplex integral operators on these manifolds, which allow us to represent the solutions to the inhomogeneous Helmholtz equation with given boundary data on cylinders and on the n‐torus. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite difference spectral approximation for the time‐space fractional telegraph equation and its parameter estimation

The object of this paper is to present the numerical solution of the time‐space fractional telegraph equation. The proposed method is based on the finite difference scheme in temporal direction and Fourier spectral method in spatial direction. The fast Fourier transform (FFT) technique is applied to practical computation. The stability and convergence analysis are strictly proven, which shows that this method is stable and convergent with (2?α) order accuracy in time and spectral accuracy in space. Moreover, the Levenberg‐Marquardt (L‐M) iterative method is employed for the parameter estimation. Finally, some numerical examples are given to confirm the theoretical analysis.     

Wave solutions for variants of the KdV–Burger and the K(n,n)–Burger equations by the generalized G′/G‐expansion method

An application of the ‐expansion method to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for variants of the Korteweg–de Vries–Burger and the K(n,n)–Burger equations. The generalized ‐expansion method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. It is shown that the generalized ‐expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems. Copyright © 2017 John Wiley & Sons, Ltd.     

Application of the exp‐function method to nonlinear lattice differential equations for multi‐wave and rational solutions

In this paper, we extend the basic Exp‐function method to nonlinear lattice differential equations for constructing multi‐wave and rational solutions for the first time. We consider a differential‐difference analogue of the Korteweg–de Vries equation to elucidate the solution procedure. Our approach is direct and unifying in the sense that the bilinear formalism of the equation studied becomes redundant. Copyright © 2011 John Wiley & Sons, Ltd.     

Dispersive estimates for solutions to the perturbed one‐dimensional Klein‐Gordon equation with and without a one‐gap periodic potential

The knowledge about the stability properties of spatially localized structures in linear periodic media with and without defects is fundamental for many fields in nature. Its importance for the design of photonic crystals is, for example, described in 5 and 30 . Against this background, we consider a one‐dimensional linear Klein‐Gordon equation to which both a spatially periodic Lamé potential and a spatially localized perturbation are added. Given the dispersive character of the underlying equation, it is the purpose of this paper to deduce time‐decay rates for its solutions. We show that, generically, the part of the solution which is orthogonal to possible eigenfunctions of the perturbed Hill operator associated to the problem decays with a rate of w.r.t. the norm. In weighted L² norms, we even get a time decay of . Furthermore, we consider the situation of a perturbing potential that is only made up of a spatially localized part which, now, can be slightly more general. It is well‐known that, in general, it is not possible to obtain the endpoint estimate in one space dimension by means of the wave operators drawn from scattering theory. For this reason, we proceed directly and prove, along the lines of 17 , the expected decay rate of .