Depencence of solutions of the nonlinear Schrödinger equation on dissipation parameters

Abstract

We consider an initial-boundary-value problem for the nonlinear Schrödinger equation in the complexvalued functionE=E(x,z):

$$\partial _z E + i\Delta E + i\alpha \left| E \right|^p E + \beta \left| E \right|^q E = 0, q > p \geqslant 0, \beta > 0,$$

((1))

$$\left. E \right|_{z = 0} = E_0 \in H^2 (\Omega ) \cap H_0^1 (\Omega ), \left. E \right|_{\partial \Omega } = 0, \Omega \subset R^2 , \partial \Omega \in C^2 .$$

((2))

We investigate the behavior of the solution of problem (1)–(2) as β→0 and its closeness to the solution of the degenerate equation (β=0). Given the consistency conditionq(β)=p+εln(1/β), 0<ɛ≤ɛ₀, we establish boundedness of the norm$\left\| E \right\|_{C([0,z_0 ]):H_0^1 (\Omega ))} + \left\| {\partial _z E} \right\|_{C([0,z_0 ]);L^2 (\Omega ))}$ for every finitez ₀>0 as β→0. For α≤0 and a fixedq, we prove uniform (in β) boundness of solutions of problem (1)–(2) on some interval [0,Z] and their convergence as β→0 to the solution of the degenerate problem (β=0) in the normC([0,Z];L ² (Ω)).