Blow-up of the solution of a nonlinear Schr?dinger equation system with periodic boundary conditions

摘要

We consider a system of nonlinear Schr?dinger equations with periodic boundary conditions of the form i @uj @t + D2uj = ??fj(u; u); t > 0; x 2 (??2; 2); uj(0; x) = uj0(x); x 2 (??2; 2); Dkuj(t;??2) = Dkuj(t; 2); t > 0; k = 0; 1; where D = @=@x, j = 1; : : : ;m, fj(u; u) = @g(u; u)=@u, and @g=@uj = fj for some homogenous function g(u; u) such that g( u; u) = 6g(u; u). We obtain sufficient conditions for blow-up of solutions of this system in C1([0; t0);H2(??2; 2)).