Sukeyuki Kumei

New classes of symmetries for partial differential equations

New classes of symmetries for partial differential equations are introduced. By writing a given partial differential equation S in a conserved form, a related system T with potentials as additional dependent variables is obtained. The Lie group of point transformations admitted by T induces a symmetry group of S. New symmetries may be obtained for S that are neither point nor Lie–Backlund symmetries. They are determined by a completely algorithmic procedure. Significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation.

On invariance properties of the wave equation

A complete group classification is given of both the wave equation c2(x)uxx−utt=0 (I) and its equivalent system vt=ux, c2(x)vx=ut (II) when the wave speed c(x)≠const. Equations (I) and (II) admit either a two‐ or four‐parameter group. For the exceptional case, c(x)=(Ax+B)2, equation (I) admits an infinite group. Equations (I) and (II) do not always admit the same group for a given c(x): The group for (I) can have more parameters or fewer parameters than that for (II); moreover, the groups can be different with the same number of parameters. Separately for (I) and (II), all possible c(x) that admit a four‐parameter group are found explicitly. The corresponding invariant (similarity) solutions are considered. Some of these wave speeds have realistic physical properties: c(x) varies monotonically from one positive constant to another positive constant as x goes from −∞ to +∞.

Symmetry-based algorithms to relate partial differential equations: II. Linearization by nonlocal symmetries

An algorithm is presented to linearize nonlinear partial differential equations by non-invertible mappings. The algorithm depends on finding nonlocal symmetries of the given equations which are realized as appropriate local symmetries of a related auxiliary system. Examples include the Hopf-Cole transformation and the linearizations of a nonlinear heat conduction equation, a nonlinear telegraph equation, and the Thomas equations.

Symmetry-based algorithms to relate partial differential equations: I. Local symmetries

Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical Kd V and the Kd V equations.

Exact solutions for wave equations of two‐layered media with smooth transition

The wave equation c2(x)uxx−utt=0 is solved for wave speeds c(x) corresponding to two‐layered media with smooth transition from layer to layer. The wave speed c(x) has four free parameters to fit a given medium. Solutions are constructed from invariant solutions of a related system of first‐order partial differential equations that admit a four‐parameter symmetry group. These solutions are superposed to solve general initial value problems for data with compact support; the computation of the superposition coefficients uses elementary Fourier analysis. Solutions are illustrated for various initial conditions.

Lie Groups of Transformations and Infinitesimal Transformations

In dimensional analysis the scaling transformations of the fundamental dimensions (1.86), the induced scaling transformations of the measurable quantities (1.88), the induced scaling transformations of all quantities (1.88), (1.89), and the induced scaling transformations preserving all constants (1.88), (1.91), are all examples of Lie groups of transformations. Scaling transformations are easily described in terms of their global properties as seen in Chapter 1. From the point of view of finding solutions to differential equations a general theory of Lie groups of transformations is unnecessary if transformations are restricted to scalings, translations, or rotations. However it turns out that much wider classes of transformations leave differential equations invariant including transformations composed of scalings, translations, and rotations. For the use and discovery of such transformations the notion of a Lie group of transformations is crucial—in particular the characterization of such transformations in terms of infinitesimal generators (which form a Lie algebra). This chapter introduces the basic ideas of Lie groups of transformations necessary in later chapters for the study of invariance properties of differential equations.

Dimensional Analysis, Modelling, and Invariance

In this chapter we introduce the ideas of invariance concretely through a thorough treatment of dimensional analysis. We show how dimensional analysis is connected to modelling and the construction of solutions obtained through invariance for boundary value problems for partial differential equations.

Noether’s Theorem and Lie-Bäcklund Symmetries

In the preceding chapters we established the algorithm to determine Lie groups of point transformations of differential equations and developed methods to solve differential equations using such symmetries. In this chapter we study one of the most important applications of symmetries to physical problems, namely, the construction of conservation laws.

Use of Group Analysis in Solving Overdetermined Systems of Ordinary Differential Equations

Abstract Group analysis is applied to overdetermined systems of ODEs. If each ODE of the system admits the same r -parameter solvable Lie group, then the use of the corresponding differential invariants greatly simplifies the analysis of the system. Moreover this can even lead to its explicit solution. Examples are given.