Classification and reduction of some systems of quasilinear partial differential equations
By the generalized conditional symmetry (GCS) method of Fokas and Liu the author investigates a class of systems of quasilinear partial differential equations (PDEs), which have applications in many areas of science and engineering. Reduction to systems of ordinary differential equations (ODEs) is made under GCS: \(\{ \sigma_1 , \sigma_2 \} = \{ u_{xxx} \pm u_{x} , v_{xxx} \pm v_{x} \}\) and \( \{ \sigma_1 , \sigma_2 \} = \{ u_{xxx} , v_{xxx} \}\). The relationship of the GCS method and the approach of invariant linear subspaces is analyzed. For systems of quasilinear hyperbolic PDEs exact solutions and reductions to ODE are constructed. There are considered three particular cases. For an illustration of the GCS method the author presents the following examples: parabolic system of equations with source and absorption; a system with quadratic nonlinearities; a system with nonlinear connection terms. Finally, explicit exact solutions are obtained.
scientific article; zbMATH DE number 1491654