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

Abstract

Conventionally, to learn wave collapse and optical turbulence, one must study finite-time blow-up solutions of one-component self-focusing nonlinear Schrödinger equations (NLSE). Here we consider simultaneous blow-up solutions of two-component system 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 system of NLSE.

Multiple Existence of Nonradial Positive Solutions for a Coupled Nonlinear Schrödinger System

“…We will show the multiple existence of nonradial solutions of (P) in this case with μ 1 = μ 2 . Our results improve the results in [12]. (See Remark 1 and Remark 3).…”

“…It is known that the solutions of (P) is not necessarily radial [12]. We show that problem (P) has multiple nonradial solutions in case that |β| is sufficiently small.…”

“…In [12], the existence of positive solutions of (P ) of the form (1.5) was established in the case that {x 1 , x 2 , . .…”

“…The existence of positive solutions of (P ) of the form (1.7) was proved in [12] in the case that the spacial dimension is 2 and μ 1 , μ 2 satisfy…”

“…In case that β < 0 and |β| is small, there are positive solutions with one component concentrating on the origin and the other component concentrating around a regular polygon(cf. [12]). The existence of non radial positive solutions was also considered in [21] for the case that β < 0 and μ 1 = μ 2 .…”