Large Time Behavior of Solutions for Derivative Cubic Nonlinear Schrödinger Equations

Abstract

We study the asymptotic behavior in time and scattering problem for the solutions to the Cauchy problem for the derivative cubic nonlinear Schrödinger equations of the following form
(A)   iu_t+u_xx=\mathscr{N}(u, \bar{u}, u_x, \bar{u}_x),   t∈R, x∈R;   u(0, x)=u₀(x),   x∈R,
where
\mathscr{N}(u, \bar{u}, u_x, \bar{u}_x)=\mathscr{K}₁ | u | ²u+i\mathscr{K}₂ | u | ²u_x+i\mathscr{K}₃u²\bar{u}_x+\mathscr{K}₄ | u_x | ²u+\mathscr{K}₅\bar{u}u_x²+i\mathscr{K}₆ | u_x | ²u_x,
\mathscr{K}_j=\mathscr{K}_j(|u|²), \mathscr{K}_j(z)∈C³(R⁺); \mathscr{K}_j(z)=λ_j+O(z), as z→+0, \mathscr{K}₁, \mathscr{K}₆ are real valued functions. Here the parameters λ₁, λ₆∈R, and λ₂, λ₃, λ₄, λ₅∈C, are such that λ₁−λ₃∈R and λ₄−λ₅∈R. If \mathscr{K}₅(z)=\frac{λ₅}{1+μz} and λ₅=μ=±1, \mathscr{K}₁=\mathscr{K}₂=\mathscr{K}₃=\mathscr{K}₄=\mathscr{K}₆=0 equation (A) appears in the classical pseudospin magnet model [9]. We prove that if u₀∈H^{3, 0}∩H^{2, 1} and the norm ||u₀||_{3, 0}+||u₀||_{2, 1}=ε is sufficiently small, then the solution of (A) exists globally in time and satisfies the sharp time decay estimate ||u(t)||_{2, 0, ∞}≤Cε(1+|t|)^−1/2, where ||φ||_{m, s, p}=||(1+x²)^s/2(1−∂_x²)^m/2φ||_L^p, H_p^{m, s}={φ∈S';||φ||_{m, s, p}<∞}. Furthermore we prove existence of modified scattering states and nonexistence of nontrivial scattering states. Our method is based on a certain gauge transformation and an appropriate phase function.