Symmetry properties, reduction, and exact solutions for a system of Korteweg-de Vries-like equations (Q2740288)

The problem considered by the authors is as follows: construct a system of two coupled KdV-like equations in such a way that it is invariant with respect to the extended Galilei algebra (i.e. Galilei algebra complemented by the subgroup of projective transformations). It is shown that such a system locally has the form NEWLINE\[NEWLINE{\mathbf u}_t+\langle {\lambda,{\mathbf u}}\rangle {\mathbf u}_x+{{\lambda}\over \|\lambda \|^2}\langle {\lambda}^{\perp},{\mathbf u}\rangle_{xx}=0,\tag{1}NEWLINE\]NEWLINE where \({\mathbf u}:\mathbb R^2 \mapsto \mathbb R^2\) is the unknown vector-function of independent variables \((t,x), {\lambda}\in \mathbb R^2\) is an arbitrary vector, and \({\lambda}^{\perp}\) is a vector orthogonal to \({\lambda}\). The maximal invariance algebra of the system (1) is found, and due to this fact 17 ansatzes, which reduce the system (1) to 2-D ODEs systems, are constructed. The proofs are omitted.