On double waves of equations with three independent variables (Q1374951)

This paper deals with double wave form solutions of four equations of three independent variables \[ \sum^3_{\alpha= 1}(\lambda_\alpha\cdot p^\alpha_j(\lambda, \mu)+ \mu_\alpha\cdot q^\alpha_j(\lambda,\mu))= 0,\quad j= 1,2,3,4. \] This system admits the transformation group \(G^4\) with a Lie algebra whose basis consists of the operators \(x_\alpha\cdot\partial_{x_\alpha}\), \(\partial_{x_i}\). \(i=1, 2,3\). We consider the case when, in examining compatibility, a homogeneous autonomous system that consists of four quasi-linear first-linear differential equations can be formed. For such systems, we present necessary existence conditions of solutions with functional arbitrariness that are irreducible to invariant solutions.