Traveling Wave Reduction Research Articles

General solution of the traveling wave reduction for the perturbed Chen-Lee-Liu equation

We consider the perturbed Chen-Lee-Liu equation for describing propagation pulse in optical fiber. The Cauchy problem for this equation is not solved by the inverse scattering transform and we study this equation using the traveling wave reduction. We show that there are two first integrals for the system of equations corresponding to real and imaginary parts of the perturbed Chen-Lee-Liu equation. These first integrals are used to obtain the nonlinear first-order differential equation. The general solution of the first-order ordinary differential equation is found via the Weierstrass and Jacobi elliptic functions. Periodic and solitary waves of the perturbed Chen-Lee-Liu equation in the form of the traveling reduction are presented and illustrated.

General solution of traveling wave reduction for the Kundu–Mukherjee–Naskar model

Traveling wave reduction of the Kundu–Mukherjee–Naskar model for describing of the optical solitons is considered. Two first integrals for the system of equations are found. These first integrals are used to obtain the nonlinear second-order differential equation. The general solution of the second-order ordinary differential equation is found via the Weierstrass and Jacobi elliptic functions.

First integrals and general solution of the traveling wave reduction for Schrödinger equation with anti-cubic nonlinearity

The traveling wave reduction of the Schrödinger equation with anti-cubic nonlinearity for describing of the optical solitons is studied. We find two first integrals of the system of equations corresponding to the real and imaginary part of the optical pulse. Using the first integrals we obtain the general and special solutions of the Schrödinger equation with anti-cubic nonlinearity.

First integrals and solutions of the traveling wave reduction for the Triki–Biswas equation

The integrability of the traveling wave reduction for the Triki–Biswas equation is considered. Two first integrals for this equation are found. It is shown that the Triki–Biswas equation is transformed to the nonlinear ordinary differential equation of the first order. The general solution of the traveling wave reduction for the Triki–Biswas equation is written using the quadrature. The general solution of the traveling wave reduction for the Kaup–Newell equation is given. Special solutions of the Triki–Biswas equation are presented.

Traveling wave reduction of the modified KdV hierarchy: The Lax pair and the first integrals

The traveling wave reduction of the modified Korteweg-de Vries hierarchy is considered. The linear system of equations associated with this hierarchy is found in the general form. The Lax pair is used to obtain the first integrals for the traveling wave reduction of this hierarchy. The first three members of the mKdV hierarchy are considered in more detail. Exact solutions for the traveling wave reduction for the mKdV hierarchy and its first integrals are given. The obtained first integrals can be considered as the expansion of the list for the nonlinear integrable differential equations.

Lax pair and first integrals of the traveling wave reduction for the KdV hierarchy

The traveling wave reduction of the Korteweg-de Vries hierarchy is considered. The linear system of equations associated with this hierarchy is found. The Lax pair is used to obtain the first integrals of the traveling wave reduction for the Korteweg-de Vries hierarchy. Exact formulas for the first integrals of the hierarchy are given. The first three members of the hierarchy are considered in more detail. These first integrals are the examples of the integrable nonlinear differential equations with the Painlevé property. Exact solutions in the form of the soliton for the traveling wave reduction of the Korteweg-de Vries hierarchy and its first integrals are presented.

Poisson structures for difference equations

We study the existence of log-canonical Poisson structures that are preserved by difference equations of a special form. We also study the inverse problem, given a log-canonical Poisson structure to find a difference equation preserving this structure. We give examples of quadratic Poisson structures that arise for the Kadomtsev–Petviashvili (KP) type maps which follow from a travelling-wave reduction of the corresponding integrable partial difference equation.

Analytical Solutions for Nonlinear Convection–Diffusion Equations with Nonlinear Sources

Nonlinear equations of the convection–diffusion type with nonlinear sources are used for the description of many processes and phenomena in physics, mechanics, and biology. In this work, we consider the family of nonlinear differential equations which are the traveling wave reductions of the nonlinear convection–diffusion equation with polynomial sources. The question of the construction of the general analytical solutions to these equations is studied. The steady-state and nonstationary cases with and without account of convection are considered. The approach based on the application of nonlocal transformations generalizing the Sundman transformation is applied for construction of the analytical solutions. It is demonstrated that in the steady-state case without account of convection, the general analytical solution can be found without any constraints on the equation parameters and it is expressed in terms of the Weierstrass elliptic function. In the general case, this solution has a cumbersome form; the constraints on the parameters for which it has a simple form are found, and the corresponding analytical solutions are obtained. It is shown that in the nonstationary case, both with and without account of convection, the general solution to the studied equations can be constructed under certain constraints on the parameters. The integrability criteria for the Lienard-type equations are used for this purpose. The corresponding general analytical solutions to the studied equations are explicitly constructed in terms of exponential or elliptic functions.

Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation

AbstractFor a generalized KdV-Burgers-Kuramoto equation we have studied conservation laws by using the multiplier method, and investigated its first-level and second-level potential systems. Furthermore, the Lie point symmetries of the equation and the Lie point symmetries associated with the conserved vectors are determined. We obtain travelling wave reductions depending on the form of an arbitrary function. We present some explicit solutions: soliton solutions, kinks and antikinks.

Elliptic solutions and solitary waves of a higher order KdV–BBM long wave equation

We provide conditions for the existence of hyperbolic, unbounded periodic and elliptic solutions in terms of Weierstrass ℘ functions of both third and fifth-order KdV–BBM (Korteweg-de Vries–Benjamin-Bona-Mahony) regularized long wave equation. An analysis for the initial value problem is developed together with a local and global well-posedness theory for the third-order KdV–BBM equation. Traveling wave reduction is used together with zero boundary conditions to yield solitons and periodic unbounded solutions, while for nonzero boundary conditions we find solutions in terms of Weierstrass elliptic ℘ functions. For the fifth-order KdV–BBM equation we show that the parameter γ=112, which leads to a Hamiltonian, represents a restriction in where there are constraint curves that never intersect a region of unbounded solitary waves. This in turn shows that only dark or bright solitons and no unbounded solutions exist. Motivated by the lack of a Hamiltonian structure for γ≠112 we develop Hk bounds, and we show for the non-Hamiltonian system that dark and bright solitons coexist together with unbounded periodic solutions. For nonzero boundary conditions, due to the complexity of the nonlinear algebraic system of coefficients of the elliptic equation we construct Weierstrass solutions for a particular set of parameters only.

Nonlinear waves described by the generalized Swift-Hohenberg equation

We study the wave processes described by the generalized Swift-Hohenberg equation. We show that the traveling wave reduction of this equation does not pass the Kovalevskaya test. Some solitary wave solutions and kink solutions of the generalized Swift-Hohenberg equation are found. We use the pseudo-spectral algorithm to perform the numerical simulation of the wave processes described by the mixed boundary value problem for the generalized Swift-Hohenberg equation. This algorithm was tested on the obtained solutions. Some features of the nonlinear waves evolution described by the generalized Swift-Hohenberg equation are studied.

On the general traveling wave solutions of some nonlinear diffusion equations

We consider a family of nonlinear diffusion equations with nonlinear sources. We assume that all nonlinearities are polynomials with respect to a dependent variable. The traveling wave reduction of this family of equations is an equation of the Lienard-type. Applying recently obtained criteria for integrability of Lienard-type equations we find some new integrable families of traveling wave reductions of nonlinear diffusion equations as well as their general analytical solutions.

Poroacoustic waves under a mixture-theoretic based reformulation of the Jordan–Darcy–Cattaneo model

Using mixture theory, we present a reformulation of the Jordan–Darcy–Cattaneo (JDC) poroacoustic model. The one-dimensional (1D) version of the resulting system is then applied to the description of first ‘start-up’ acceleration waves, and then those exhibited under the traveling wave reduction, in a gas that saturates a rigid, porous solid. Results obtained are contrasted with those of the Darcy–Jordan model, the impact of τ, the model’s relaxation time parameter, is assessed, and issues stemming from the acceleration wave analysis performed by Ciarletta et al. (2013), who proposed the original JDC model, are resolved. Also, the weakly-nonlinear Darcy–Jordan model is generalized to include the case τ>0.

Analytical Solutions of a Nonlinear Convection-Diffusion Equation With Polynomial Sources

Nonlinear convection–diffusion equations are widely used for the description of various processes and phenomena in physics, mechanics and biology. In this work we consider a family of nonlinear ordinary differential equations which is a traveling wave reduction of a nonlinear convection–diffusion equation with a polynomial source. We study a question about integrability of this family of nonlinear ordinary differential equations. We consider both stationary and non–stationary cases of this equation with and without convection. In order to construct general analytical solutions of equations from this family we use an approach based on nonlocal transformations which generalize the Sundman transformations. We show that in the stationary case without convection the general analytical solution of the considered family of equations can be constructed without any constraints on its parameters and can be expressed via the Weierstrass elliptic function. Since in the general case this solution has a cumbersome form we find some correlations on the parameters which allow us to construct the general solution in the explicit form. We show that in the non–stationary case both with and without convection we can find a general analytical solution of the considered equation only imposing some correlation on the parameters. To this aim we use criteria for the integrability of the Lienard equation which have recently been obtained. We find explicit expressions in terms of exponential and elliptic functions for the corresponding analytical solutions.

一类KdV型方程的新解析解

我们首先利用行波约化法将一类KdV型方程约化为常微分方程，然后运用指数函数法，并借助于数学软件Mathematica，获得了该方程丰富的精确解析解，并绘制解的图像。 We first use traveling wave reduction method to transform a class of KdV-type equation to ordinary differential equation, and then apply Exp-Function method as well as symbolic software Mathematica to obtain new accurate analytical solutions of the equation under study. Moreover, we draw the graphs of such solutions.

Extended two dimensional equation for the description of nonlinear waves in gas–liquid mixture

We consider a system of equations for the description of nonlinear waves in a liquid with gas bubbles. Taking into account high order terms with respect to a small parameter, we derive a new nonlinear partial differential equation for the description of density perturbations of mixture in the two-dimensional case. We investigate integrability of this equation using the Painlevé approach. We show that traveling wave reduction of the equation is integrable under some conditions on parameters. Some exact solutions of the equation derived are constructed. We also perform numerical investigation of the nonlinear waves described by the derived equation.

Travelling Wave Solutions for the Unsteady Flow of a Third Grade Fluid Induced Due to Impulsive Motion of Flat Porous Plate Embedded in a Porous Medium

AbstractThis work describes the time-dependent flow of an incompressible third grade fluid filling the porous half space over an infinite porous plate. The flow is induced due to the motion of the porous plate in its own plane with an arbitrary velocityV(t). Translational type symmetries are employed to perform the travelling wave reduction into an ordinary differential equation of the governing nonlinear partial differential equation which arises from the laws of mass and momentum. The reduced ordinary differential equation is solved exactly, for a particular case, as well as by using the homotopy analysis method (HAM). The better solution from the physical point of view is argued to be the HAM solution. The essentials features of the various emerging parameters of the flow problem are presented and discussed.

Exact traveling wave solutions of some nonlinear evolution equations

Using a traveling wave reduction technique, we have shown that Maccari equation, (2?1)-dimensional nonlinear Schrodinger equation, medium equal width equation, (3?1)-dimensional modified KdV-Zakharov- Kuznetsev equation, (2?1)-dimensional long wave-short wave resonance interaction equation, perturbed nonlinear Schrodinger equation can be reduced to the same family of auxiliary elliptic-like equations. Then using extended F-expansion and projective Riccati equation methods, many types of exact traveling wave solutions are obtained. With the aid of solutions of the elliptic-like equation, more explicit traveling wave solutions expressed by the hyper- bolic functions, trigonometric functions and rational func- tions are found out. It is shown that these methods provide a powerful mathematical tool for solving nonlinear evolu- tion equations in mathematical physics. A variety of structures of the exact solutions of the elliptic-like equation are illustrated.

On some new forms of lattice integrable equations

AbstractInspired by the forms of delay-Painleve equations, we consider some new differential-discrete systems of KdV, mKdV and Sine-Gordon — type related by simple one way Miura transformations to classical ones. Using Hirota bilinear formalism we construct their new integrable discretizations, some of them having higher order. In particular, by this procedure, we show that the integrable discretization of intermediate sine-Gordon equation is exactly lattice mKdV and also we find a bilinear form of the recently proposed lattice Tzitzeica equation. Also the travelling wave reduction of these new lattice equations is studied and it is shown that all of them, including the higher order ones, can be integrated to Quispel-Roberts-Thomson (QRT) mappings.

Convergent analytic solutions for homoclinic orbits in reversible and non-reversible systems

In this paper, convergent, multi-infinite, series solutions are derived for the homoclinic orbits of a canonical fourth-order ODE system, in both reversible and non-reversible cases. This ODE includes traveling-wave reductions of many important nonlinear PDEs or PDE systems, for which these analytical solutions would correspond to regular or localized pulses of the PDE. As such, the homoclinic solutions derived here are clearly topical, and they are shown to match closely to earlier results obtained by homoclinic numerical shooting. In addition, the results for the non-reversible case go beyond those that have been typically considered in analyses conducted within bifurcation-theoretic settings. We also comment on generalizing the treatment here to parameter regimes where solutions homoclinic to exponentially small periodic orbits are known to exist, as well as another possible extension placing the solutions derived here within the framework of a comprehensive categorization of ALL possible traveling-wave solutions, both smooth and non-smooth, for our governing ODE.