Symmetry reduction for nonlinear relativistically invariant equations

Symmetry reduction is studied for the relativistically invariant scalar partial differential equation H(⧠u,(∇u)2,u)=0 in (n+1)‐dimensional Minkowski space M(n,1). The introduction of k symmetry variables ξ1, ... ,ξk as invariants of a subgroup G of the Poincare group P(n,1), having generic orbits of codimension k≤n in M(n,1), reduces the equation to a PDE in k variables. All codimension‐1 symmetry variables in M(n,1) (n arbitrary), reducing the equation studied to an ODE are found, as well as all codimension‐2 and ‐3 variables for the low‐dimensional cases n=2,3. The type of equation studied includes many cases of physical interest, in particular nonlinear Klein–Gordon equations (such as the sine–Gordon equation) and Hamilton–Jacobi equations.

[1]  P. Winternitz,et al.  Pseudopotentials and Lie symmetries for the generalized nonlinear Schrödinger equation , 1982 .

[2]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[3]  L. Vinet,et al.  Group actions on principal bundles and invariance conditions for gauge fields , 1980 .

[4]  M. Ablowitz,et al.  A connection between nonlinear evolution equations and ordinary differential equations of P‐type. II , 1980 .

[5]  J. Harnad,et al.  Group actions on principal bundles and dimensional reduction , 1980 .

[6]  M. Ablowitz,et al.  Nonlinear evolution equations and ordinary differential equations of painlevè type , 1978 .

[7]  L. Vinet,et al.  On the U(2) invariant solutions to Yang-Mills equations in compactified Minkowski space , 1978 .

[8]  C. Tracy,et al.  Painlevé functions of the third kind , 1977 .

[9]  R. T. Sharp,et al.  Symmetry breaking interactions for the time dependent Schrödinger equation , 1976 .

[10]  R. T. Sharp,et al.  Subgroups of the similitude group of three-dimensional Minkowski space , 1976 .

[11]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[12]  Jiří Patera,et al.  Continuous subgroups of the fundamental groups of physics. II. The similitude group , 1975 .

[13]  Jiří Patera,et al.  Continuous subgroups of the fundamental groups of physics. I. General method and the Poincaré group , 1975 .

[14]  H. Zassenhaus,et al.  The maximal solvable subgroups of SO(p,q) groups , 1974 .

[15]  A. Scott,et al.  The soliton: A new concept in applied science , 1973 .

[16]  F. Esposito,et al.  Theory and applications of the sine-gordon equation , 1971 .

[17]  P. Hartman.  Ordinary Differential Equations , 1965 .

[18]  B. Gambier,et al.  Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes , 1910 .

[19]  P. Painlevé,et al.  Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme , 1902 .

[20]  S. Lie.  Allgemeine Untersuchungen über Differentialgleichungen, die eine continuirliche, endliche Gruppe gestatten , 1885 .

[21]  H. Zassenhaus.  Lie groups, lie algebras and representation theory , 1981 .

[22]  G. Lamb.  Elements of soliton theory , 1980 .

[23]  E. Hille,et al.  Ordinary di?erential equations in the complex domain , 1976 .

[24]  J. Cole,et al.  Similarity methods for differential equations , 1974 .

[25]  F. J. Ernst.  NEW FORMULATION OF THE AXIALLY SYMMETRIC GRAVITATIONAL FIELD PROBLEM. II. , 1968 .

[26]  J. Gillis,et al.  Nonlinear Partial Differential Equations in Engineering , 1967 .

[27]  E. Vessiot.  Sur une classe d'équations différentielles , 1893 .

[28]  S. Lie,et al.  Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen / Sophus Lie ; bearbeitet und herausgegeben von Georg Scheffers. , 1893 .

[29]  S. Lie,et al.  Vorlesungen über Differentialgleichungen, mit bekannten infinitesimalen Transformationen , 1891 .