New ray (characteristic) equations for the equations of physics; Hamilton - Jacobi theory in more-dimensional space

Abstract

The geometrical optical concept of rays is generalized for fields generated by any system of coupled non-linear partial differential equations of arbitrary order like the vector wave equations for (non)linear media, the Maxwell equations, the equations of magneto-hydrodynamics, the system of continuity (transport) equations for an inhomogeneous semiconductor, etc. It is shown that the pertinent solutions of such a system of partial differential equations can be obtained exactly from a set of ordinary non-linear differential equations, the so-called characteristic or ray equations. These characteristic (ray) equations are connected with the key equation of the theory, the generalized Hamilton - Jacobi equation. The generalized Hamilton - Jacobi equation is a first-order non-linear partial differential equation, a special case of which is known to be of great importance for classical mechanics. This equivalence between partial and ordinary differential equations implies an enormous simplification for the numerical evaluation of the original problem, as solving partial differential equations is much more difficult than solving ordinary differential equations. From the theoretical point of view this equivalence between partial and ordinary differential equations is also interesting for various reasons. We mention, for example, that interesting properties like stability, soliton behaviour, decay, growth, etc of the solutions of the field equations of physics can be directly deduced from the relatively easily obtained behaviour of the solutions of the ordinary non-linear differential equations for the `rays'. As an example of the potential powers of this new method we analyse the non-linear Schrodinger equation and derive its soliton solutions.