George W. Bluman

Applications of symmetry methods to partial differential equations

Local Transformations and Conservation Laws.- Construction of Mappings Relating Differential Equations.- Nonlocally Related PDE Systems.- Applications of Nonlocally Related PDE Systems.- Further Applications of Symmetry Methods: Miscellaneous Extensions.

Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications

An eective algorithmic method is presented for nding the local conservation laws for partial dierential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for nding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the rst of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classication results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given. In the study of dierential equations, conservation laws have many signicant uses, particularly with regard to integrability and linearization, constants of motion, analysis of solutions, and numerical solution methods. Consequently, an important problem is how to calculate all of the conservation laws for given dierential equations. For a dierential equation with a variational principle, Noether’s theorem [12, 4, 6, 5, 14] gives a formula for obtaining the local conservation laws by use of symmetries of the action. One usually attempts to nd these symmetries by noting that any symmetry of the action leaves invariant the extremals of the action and hence gives rise to a symmetry of the dierential equation. However, all symmetries of a dierential equation do not necessarily arise from symmetries of the action when there is a variational principle. For example, if a dierential equation is scaling invariant, then the action is often not invariant. Indeed it is often computationally awkward to determine the symmetries of the action and carry out the calculation with the formula to obtain a conservation law. Moreover, in general a dierential equation need not have a variational principle even allowing for a change of variables. Therefore, it is more eective to seek a direct, algorithmic method without involving an action principle to nd the conservation laws of a given dierential equation.

Direct construction method for conservation laws of partial differential equations Part II: General treatment

This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.

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 the remarkable nonlinear diffusion equation (∂/∂x)[a (u+b)−2(∂u/∂x)]−(∂u/∂t)=0

We study the invariance properties (in the sense of Lie–Backlund groups) of the nonlinear diffusion equation (∂/∂x)[C (u)(∂u/∂x)]−(∂u/∂t) =0. We show that an infinite number of one‐parameter Lie–Backlund groups are admitted if and only if the conductivity C (u) =a (u+b)−2. In this special case a one‐to‐one transformation maps such an equation into the linear diffusion equation with constant conductivity, (∂2ū/∂x2)−(∂ū/∂t) =0. We show some interesting properties of this mapping for the solution of boundary value problems.

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 +∞.

Biology and Chemistry

This chapter considers elementary calculus problems related to the fields of biology and chemistry. Most of these problems are concerned with growth, decay and chemical reactions.

ON THE TRANSFORMATION OF DIFFUSION PROCESSES INTO THE WIENER PROCESS

Necessary and sufficient conditions are given for transforming (constructively) a one-dimensional diffusion process described by a Kolmogorov equation into the Wiener process. These conditions are shown to be equivalent to invariance of a parabolic partial differential equation under a six-parameter Lie group of point transformations. Moreover these conditions correspond to a significant generalization of Cherkasov’s result; i.e., a much wider class of Kolmogorov equations can be derived from the Wiener process than was previously realized.

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.

Use and construction of potential symmetries

Group-theoretic methods based on local symmetries are useful to construct invariant solutions of PDEs and to linearize nonlinear PDEs by invertible mappings. Local symmetries include point symmetries, contact symmetries and, more generally, Lie-Backlund symmetries. An obvious limitation in their utility for particular PDEs is the non-existence of local symmetries. A given system of PDEs with a conserved form can be embedded in a related auxiliary system of PDEs. A local symmetry of the auxiliary system can yield a nonlocal symmetry (potential symmetry) of the given system. The existence of potential symmetries leads to the construction of corresponding invariant solutions as well as to the linearization of nonlinear PDEs by non-invertible mappings. Recent work considers the problem of finding all potential symmetries of given systems of PDEs. Examples include linear wave equations with variable wave speeds as well as nonlinear diffusion, reaction-diffusion, and gas dynamics equations.