Symmetry Analysis of Nonlinear PDE with A “Mathematica” Program SYMMAN

Abstract

Computer-aided symbolic and graphic computation allows to make significantly easier both theoretical and applied symmetry analysis of PDE. This idea is illustrated by applying a special “Mathematica” package for obtaining conditional symmetries of the nonlinear wave equation u _t = (u u _x)_x invariant or partially invariant under its classical Lie symmetries.     

Existence of solitary waves in higher dimensions

The elliptic equation u=F(u) possesses non-trivial solutions inR ⁿ which are exponentially small at infinity, for a large class of functionsF. Each of them provides a solitary wave of the nonlinear Klein-Gordon equation.This work was supported in part by NSF Grant MCS 75-08827     

On Integration of the Nonlinear d’Alembert-Eikonal System and Conditional Symmetry of Nonlinear Wave Equations

Abstract

We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear d’Alembert equation □u = F (u) and nonlinear eikonal equation u _xµ u _x µ = G(u) in the complex Minkowski space R(1, 3). A method suggested makes it possible to establish necessary and sufficient compatibility conditions and construct a general solution of the d’Alembert-eikonal system for all cases when it is compatible. The results obtained can be applied, in particular, to construct principally new (non-Lie, non-similarity) solutions of the non-linear d’Alembert, Dirac, and Yang-Mills equations. Solutions found in this way are shown to correspond to conditional symmetry of the equations enumerated above. Using the said approach, we study in detail conditional symmetry of the nonlinear wave equation □w = F ₀(w) in the four-dimensional Minkowski space. A number of new (non-Lie) reductions of the above equation are obtained giving rise to its new exact solutions which contain arbitrary functions.     

Lie-symmetry vector fields for linear and nonlinear wave equations

We derive the Lie symmetry vector fields for the linear wave equation u=0 and nonlinear wave equation u=u ³. The conformal vector fields for the underlying metric tensor fieldg are also given. We construct the conservation laws and derive similarity solutions. Furthermore, we perform a Painlevé test for the nonlinear wave equation and discuss whether Lie-Bäcklund vector fields exist.     

Symmetry Analysis of the Multidimensional Polywave Equation

Abstract

We present symmetry classification of the polywave equation □ ^l u = F (u). It is established that the equation in question is invariant under the conformal group C(1, n) iff F (u) = λe ^u, n + 1 = 2l or F (u) = λu ^{(n+1+2l)/(n+1?2l)}, n + 1 6= 2l. Symmetry reduction for the biwave equation □ ² u = λu ^k is carried out. Some exact solutions are obtained.     

Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation

We prove that the set of solitary wave solutions of a generalized Kadomtsev-Petviashvili equation in two dimensions, (u _t+(u^m+1)_x+u_xxx)_x=u_yy is stable for 0<m<4/3.     

Algebraic and Geometric Properties of Matrix Solutions of Nonlinear Wave Equations

We improve the construction of exact matrix solutions for nonlinear wave equations by using unitary anti-Hermitian and anticommuting matrices. We prove the theorem that constructs the matrix functions u _n satisfying the nonlinear wave equation for a set of special potentials. In this case, the graph of complex solution u ₁ has a soliton-like form with a finite number of coils. Exponential representation of matrix solutions u _n is associated with continuous rotations that can be used for describing intrinsic rotations and state changes of elementary particles. We also prove the theorem on the decomposition of continuous rotation (described by solution u ₂) onto three simultaneous rotations about coordinate vectors. Each of the three constructed matrix solutions u ₃ is also decomposed into the triplet of elementary matrix solutions.     

Conservative Solutions to a Nonlinear Variational Wave Equation

We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation u _tt − c(u)(c(u)u _x ) _x =0, for initial data of finite energy. Here c(·) is any smooth function with uniformly positive bounded values.     

非线性波动方程的孤波解

用平衡法并结合吴消元法得到了一类较广泛非线性波动方程u_tt-a₁u_xx+a₂u_t+a₃u+a₄u³=0的若干孤波解公式，从而物理学上许多著名的方程，如φ⁴方程、Klein-Gordon方程、Landau-Ginzburg-Higgs方程、非线性电报方程等都可作为该方程的特殊情形得到相应的孤波解 关键词：     

A Strong Regularity Result for Parabolic Equations

We consider a parabolic equation with a drift term u+bu–u _t =0. Under the condition div b=0, we prove that solutions possess dramatically better regularity than those provided by standard theory. For example, we prove continuity of solutions when not even boundedness is expected.     

Zeitliche Entwicklung des Reflexionskoeffizienten einer Plasmaschicht bei starker elektromagnetischer Einstrahlung

We have calculated numerically the temporal evolution of the nonlinear reflection coefficient R of an overdense plasma layer by solving the system of partial differential equations consisting of the wave equation for the slowly varying amplitude of the electric field and the hydrodynamic equations for the ion motion, including a ponderomotive force term. In dependence of the (normalized) amplitude of the incident wave u_A two regimes exist: Below a critical amplitude u_A^* ? 1 the reflection coefficient is approximately independent on the amplitude u_A and temporally constant. In the opposite case u_A>u_A^*, on the other hand, ‖R‖² decreases slowly with time down to a minimum value and after that it increases rapidly to the initial value. We think, that our results are important to interpret the anomalous reflectivity observed in some experiments when strong electromagnetic waves are incident on an overdense plasma.     

Perfect-fluid space-times with type-D Weyl tensor

We consider solutions of the Einstein field equations for which the Weyl tensor is of Petrov typeD, and whose source is a perfect fluid with equation of statep=p(w), wherep andw are the energy density and pressure of the fluid, respectively. We also impose two additional restrictions which are satisfied by most of the known solutions, namely, that the fluid 4-velocityu lies in the 2-space spanned by the two repeated principal null directions of the Weyl tensor, and that the Weyl tensor has zero magnetic part relative tou. Our main result is that for this class of solutions, the equation of state satisfies eitherdp/dw=0 ordp/dw= 1, or else the solution admits three or more Killing vector fields.     

Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation

Abstract

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f(u) for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived.     

Equations for the description of wave scattering by multi-valued random surfaces

We consider a scattering theory for multi-valued rough surfaces which cannot be described by the conventional equation of the type z = ζ(x,y). Both Dirichlet and Neumann problems are analyzed. Starting with Green's theorem we obtain a representation of the scattered field, the surface integral equation, and the extinction theorem for such surfaces. In contrast to conventional theory, these equations contain three random functions x = x(u ₁,u ₂), y = y(u ₁,u ₂), and z = z(u ₁,u ₂), where u ₁ and u ₂ are the parameters describing the surface. We introduce two scattering amplitudes S _± for describing the scattered wave above and below the surface. The extinction theorem, if formulated in terms of S _?, allows us to determine S _? for an arbitrary multi-valued surface and after this it becomes possible to derive a simple integral equation for surface sources. Knowledge of the surface sources allows us to find S ₊ by integration.     

Symmetry Classification of the One–Dimensional Second Order Equation of a Hydrodynamic Type

Abstract

The paper contains a symmetry classification of the one–dimensional second order equation of a hydrodynamical type L(Lu)+λLu=F (u), where L ≡ ? _t+u? _x. Some classes of exact solutions of this equation are given.     

Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains

Consider the diffusive Hamilton-Jacobi equation u _t = Δu + |?u| ^p , p > 2, on a bounded domain Ω with zero-Dirichlet boundary conditions, which arises in the KPZ model of growing interfaces. It is known that u remains bounded and that ?u may blow up only on the boundary ?Ω. In this paper, under suitable assumptions on ${\Omega\subset \mathbb{R}^2}Consider the diffusive Hamilton-Jacobi equation u _t = Δu + |∇u| ^p , p > 2, on a bounded domain Ω with zero-Dirichlet boundary conditions, which arises in the KPZ model of growing interfaces. It is known that u remains bounded and that ∇u may blow up only on the boundary ∂Ω. In this paper, under suitable assumptions on W ì \mathbbR²{\Omega\subset \mathbb{R}^2} and on the initial data, we show that the gradient blow-up singularity occurs only at a single point x₀ ? ?W{x_0\in\partial\Omega}. This is the first result of this kind in the study of problems involving gradient blow-up phenomena. In general domains of \mathbbRⁿ{\mathbb{R}^n}, we also obtain results on nondegeneracy and localization of blow-up points.     

Blow-up,Zero <Emphasis Type="Italic">α</Emphasis> Limit and the Liouville Type Theorem for the Euler-Poincaré Equations

In this paper we study the Euler-Poincaré equations in . We prove local existence of weak solutions in , and local existence of unique classical solutions in , k > N/2 + 3, as well as a blow-up criterion. For the zero dispersion equation (α = 0) we prove a finite time blow-up of the classical solution. We also prove that as the dispersion parameter vanishes, the weak solution converges to a solution of the zero dispersion equation with sharp rate as α → 0, provided that the limiting solution belongs to with k > N/2 + 3. For the stationary weak solutions of the Euler-Poincaré equations we prove a Liouville type theorem. Namely, for α > 0 any weak solution is u=0; for α= 0 any weak solution is u=0.     

A NOTE ON TRAVELING WAVE SOLUTIONS TO THE TWO COMPONENT CAMASSA–HOLM EQUATION

In this paper we show that non-smooth functions which are distributional traveling wave solutions to the two component Camassa–Holm equation are distributional traveling wave solutions to the Camassa–Holm equation provided that the set u^-1(c), where c is the speed of the wave, is of measure zero. In particular there are no new peakon or cuspon solutions beyond those already satisfying the Camassa–Holm equation. However, the two component Camassa–Holm equation has distinct from Camassa–Holm equation smooth traveling wave solutions as well as new distributional solutions when the measure of u^-1(c) is not zero. We provide examples of such solutions.     

A New Integrable Hierarchy,Parametric Solutions and Traveling Wave Solutions

In this paper we give a new integrable hierarchy. In the hierarchy there are the following representatives: