On invariant sets and explicit solutions of nonlinear evolution equations with quadratic nonlinearities
PDEs of the form \(A(v)= R(v)+ p(x, t)\), where \[ A(v)= \sum_{i+ j\neq 0} a_{ij} {\partial^i v\over \partial t^i} {\partial^j v\over \partial t^j}+ \alpha vv_{xx}+ \beta(v_x)^2+ \gamma v^2+ \varepsilon vv_x, \] \[ R(v)= \sum^m_{k= 1} b_k {\partial^k v\over \partial t^k}+ \sum^k_{l= 0} d_l {\partial^l v\over \partial x^l}, \] and \(a_{ij}\), \(\alpha\), \(\beta\), \(\gamma\), \(\varepsilon\), \(b_k\), \(d_l\) are real numbers, are investigated. It is shown that under certain assumptions the operator \(A\) admits an invariant subset on a linear function space. This makes it possible to represent the initial PDE on the invariant subset as a finite-dimensional dynamical system obtaining nontrivial solutions. Thus a technique to construct new explicit solutions of nonlinear PDEs with quadratic nonlinearities is given in this paper. A generalization of this approach to equations with cubic nonlinearities are discussed, and examples illustrating the approach are given.
scientific article; zbMATH DE number 811757