Blow-up of the solution of a nonlinear Schrödinger equation system with periodic boundary conditions

Abstract

We consider a system of nonlinear Schrödinger equations with periodic boundary conditions of the form
i ∂u_j/∂t + D²u_j = −f_j (u, ū),    t ≥ 0,  x ∈ (−2,2),
u_j (0, x) = u_j0(x),    x ∈ (−2,2),
D^ku_j (t, −2) = D^ku_j (t, 2),     t ≥ 0,   k = 0, 1,
where D = ∂/∂x,  j = 1,..., m,  f_j (u, ū) = ∂g(u, ū)/∂ū, and ∂g/∂u_j = ^–f _j for some homogenous function g(u, ū) such that g(λu, λū) = λ⁶g(u, ū). We obtain sufficient conditions for blow-up of solutions of this system in C¹([0,t₀);H²(−2,2)).