Invariant subspaces and exact solutions of a class of dispersive evolution equations

The invariant subspace method is used to classify a class of systems of nonlinear dispersive evolution equations and determine their invariant subspaces and exact solutions. A crucial step is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that systems of evolution equations admit. A few examples of presenting exact solutions with generalized separated variables illustrate the effectiveness of the invariant subspace method in solving systems of nonlinear evolution equations.     

Conditional Lie‐Bäcklund Symmetry of Evolution System and Application for Reaction‐Diffusion System

Conditional Lie‐Bäcklund symmetry (CLBS) method is developed to study system of evolution equations. It is shown that reducibility of a system of evolution equations to a system of ordinary differential equations can be fully characterized by the CLBS of the considered system. As an application of the approach, a class of two‐component nonlinear diffusion equations is studied. The governing system and the admitted CLBS can be identified. As a consequence, exact solutions defined on the polynomial, exponential, trigonometric, and mixed invariant subspaces are constructed due to the corresponding symmetry reductions.     

一类二阶二次变系数微分算子的不变子空间及其应用

研究了一类二阶二次变系数微分算子的不变子空间,讨论了这类微分算子不变子空间的应用,并给出了具体应用的一些例子.在这些例子中,构造了大量变系数非线性演化方程的精确解.     

A unified ansätze for the solution of nonlinear differential equations

The aim of the paper is to propose a generalized ansätze for constructing exact solutions to nonlinear ordinary differential equations. This unified transformation is manipulated to acquire analytical solutions that are general solutions of simpler linear or nonlinear systems of ordinary differential equations that are either integrable or possess special solutions. The method is implemented to obtain several families of traveling wave solutions for a class of nonlinear evolution equations and for higher order wave equations of KdV type (I).     

Invariant subspaces and conditional Lie-Bäcklund symmetries of inhomogeneous nonlinear diffusion equations

The inhomogeneous nonlinear difusion equation is studied by invariant subspace and conditional Lie-Bcklund symmetry methods.It is shown that the equations admit a class of invariant subspaces governed by the nonlinear ordinary diferential equations,which is equivalent to a kind of higher-order conditional Lie-Bcklund symmetries of the equations.As a consequence,a number of new solutions to the inhomogeneous nonlinear difusion equations are constructed explicitly or reduced to solving fnite-dimensional dynamical systems.     

Optimal classification,exact solutions,and wave interactions of Euler system with large friction

We derive exact solutions of one-dimensional Euler system that accounts for gravity together with large friction. Certain optimal classes of subalgebra using Lie symmetry analysis are obtained for this system. We apply the reduction procedure to reduce the Euler system to a system of ordinary differential equations in terms of new similarity variable for each class of subalgebras leading to invariant solutions. The evolution of characteristic shock and its interaction with the weak discontinuity by using one of the invariant solutions is studied. Further, the properties of reflected and transmitted waves and jump in acceleration influenced by the incident wave have been characterized.     

非线性系统的精确解

Based on the computerized symbolic, a new generalized tanh functions method is used for constructing exact travelling wave solutions of nonlinear partial differential equations （PDES） in a unified way. The main idea of our method is to take full advantage of an auxiliary ordinary differential equation which has more new solutions. At the same time, we present a more general transformation, which is a generalized method for finding more types of travelling wave solutions of nonlinear evolution equations （NLEEs）. More new exact travelling wave solutions to two nonlinear systems are explicitly obtained.     

Conditional Lie–Bäcklund symmetries and functionally generalized separable solutions to the generalized porous medium equations with source

The functionally generalized separable solutions of the generalized porous medium equations with power law and exponential diffusivity are studied by using the conditional Lie–Bäcklund symmetry method. The variant forms of the considered equations, which admit the linear conditional Lie–Bäcklund symmetries, are identified. A number of examples are considered and some exact solutions, defined on the polynomial, trigonometric and exponential invariant subspaces determined by the linear conditional Lie–Bäcklund symmetries, are constructed.     

Involutive Distributions,Invariant Manifolds,and Defining Equations

We introduce the notion of an invariant solution relative to an involutive distribution. We give sufficient conditions for existence of such a solution to a system of differential equations. In the case of an evolution system of partial differential equations we describe how to construct auxiliary equations for functions determining differential constraints compatible with the original system. Using this theorem, we introduce linear and quasilinear defining equations which enable us to find some classes of involutive distributions, nonclassical symmetries, and differential constraints. We present examples of reductions and exact solutions to some partial differential equations stemming from applications.     

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

It is shown that Lie group analysis of differential equations provides the exact solutions of two-dimensional stratified rotating Boussinesq equations which are a basic model in geophysical fluid dynamics. The exact solutions are obtained as group invariant solutions corresponding to the translation and dilation generators of the group of transformations admitted by the equations. The comparison with the previous analytic studies and experimental observations confirms that the anisotropic nature of the wave motion allows to associate these invariant solutions with uni-directional internal wave beams propagating through the medium. It is also shown that the direction of internal wave beam propagation is in the transverse direction to one of the invariants which corresponds to a linear combination of the translation symmetries. Furthermore, the amplitudes of a linear superposition of wave-like invariant solutions forming the internal gravity wave beams are arbitrary functions of that invariant. Analytic examples of the latitude-dependent invariant solutions associated with internal gravity wave beams that have different general profiles along the obtained invariant and propagating in the transverse direction are considered. The behavior of the invariant solutions near the critical latitude is illustrated.     

Symmetry and exact solution of heat-mass transfer equations in thermonuclear plasma

For the nonlinear system of partial differential equations, which describes the evolution of temperature and density in TOKAMAK plasmas, multiparameter families of exact solutions are constructed. The solutions are constructed by the Lie-method reduction of initial systems of equations to a system of ordinary differential equations. Examples of non-Lie ansätze and exact solutions are also presented.     

New exact travelling wave solutions for some nonlinear evolution equations, Part II

By using new solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact travelling wave solutions for nonlinear evolution equation. By this method some nonlinear evolution equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.     

New exact travelling wave solutions for the Kawahara and modified Kawahara equations

By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact travelling wave solutions for nonlinear evolution equations. By this method the Kawahara and the modified Kawahara equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.     

Symmetry analysis and some exact solutions of cylindrically symmetric null fields in general relativity

The symmetry reduction method based on the Fréchet derivative of the differential operators is applied to investigate symmetries of the Field equations in general relativity corresponding to cylindrically symmetric space–time, that is a coupled system of nonlinear partial differential equations of second order. More specifically, this technique yields invariant transformation that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied for exact solutions.     

某些常微分方程的亚纯解表示与应用

本文运用Nevanlinna值分布理论研究了某些常微分方程亚纯解的存在性. 对于某些具有控制项的常系数常微分方程, 本文得到了亚纯解的表示, 并且给出了求相应偏微分方程精确解的一种方法.作为例子, 本文运用此方法得到了著名的KdV方程的所有亚纯行波精确解. 结果显示该方法比其他方法简单.     

New exact solutions for two generalized Hirota–Satsuma coupled KdV systems

By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact solutions for nonlinear evolution equations. By this method two generalized Hirota–Satsuma coupled KdV systems are investigated and new exact solutions are explicitly obtained with the aid of symbolic computation.     

Lie group analysis and propagation of weak discontinuity in one-dimensional ideal isentropic magnetogasdynamics

The aim of this paper is to carry out symmetry group analysis to obtain important classes of exact solutions from the given system of nonlinear partial differential equations (PDEs). Lie group analysis is employed to derive some exact solutions of one dimensional unsteady flow of an ideal isentropic, inviscid and perfectly conducting compressible fluid, subject to a transverse magnetic field for the magnetogasdynamics system. By using Lie group theory, the full one-parameter infinitesimal transformations group leaving the equations of motion invariant is derived. The symmetry generators are used for constructing similarity variables which leads the system of PDEs to a reduced system of ordinary differential equations; in some cases, it is possible to solve these equations exactly. Further, using the exact solution, we discuss the evolutionary behavior of weak discontinuity.     

Pseudo-almost Periodic and Pseudo-almost Automorphic Solutions to Evolution Equations in Hilbert Spaces

In this work we prove some existence and uniqueness results for pseudo-almost periodic and pseudo-almost automorphic solutions to a class of semi-linear differential equations in Hilbert spaces using theoretical measure theory. The main technique is based upon some appropriate composition theorems combined with the Banach contraction mapping principle and the method of the invariant subspaces for unbounded linear operators. A few illustrative examples will be discussed at the end of the paper.     

Some exact solutions of the ideal MHD equations through symmetry reduction method

We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3+1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1?r?4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation.     

Traveling solitary wave solutions to evolution equations with nonlinear terms of any order

Many physical phenomena in one- or higher-dimensional space can be described by nonlinear evolution equations, which can be reduced to ordinary differential equations such as the Lienard equation. Thus, to study those ordinary differential equations is of significance not only in mathematics itself, but also in physics. In this paper, a kind of explicit exact solutions to the Lienard equation is obtained. The applications of the solutions to the nonlinear RR-equation and the compound KdV-type equation are presented, which extend the results obtained in the previous literature.