KdV type asymptotics for solutions to higher-order nonlinear Schrödinger equations

Abstract

We consider the Cauchy problem for the higher-order nonlinear Schrödinger equation i∂tu - α/3 |∂x|3u - b/4 ∂4xu = λi∂x(|u|2u), (t, x) ∈ ℝ⁺ x ℝ, u(0, x) = u0(x), x ∈ ℝ, where α, b > 0, |∂x|α = F-1|ξ|α F and F is the Fourier transformation. Our purpose is to study the large time behavior of the solutions under the non-zero mass condition ∫ u0(x)dx ≠ 0.