Solitary and self-similar solutions of two-component system of nonlinear Schrِdinger equations

Conventionally, to learn wave collapse and optical turbulence, one must study finite-time blow-up solutions of one-component self-focusing nonlinear Schrِdinger equations (NLSE). Here we consider simultaneous blow-up solutions of two-component systems of self-focusing NLSE. By studying the associated self-similar solutions, we prove two components of solutions blow up at the same time. These self-similar solutions may come from solitary wave solutions with multi-bumps forming abundant geometric patterns which cannot be found in one-component self-focusing NLSE. Our results may provide the first step to investigate optical turbulence in two-component systems of NLSE.