Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations
Summary: 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 \[ iu_t+ u_{xx}={\mathcal N}(u,\overline u, u_x, \overline u_x),\quad t\in\mathbb{R},\quad x\in\mathbb{R};\quad u(0,x)= u_0(x),\quad x\in\mathbb{R},\tag{1} \] where \[ {\mathcal N}(u,\overline u, u_x,\overline u_x)={\mathcal K}_1|u|^2u+ i{\mathcal K}_2|u|^2 u_x+ i{\mathcal K}_3 u^2\overline u_x+{\mathcal K}_4|u_x|^2u+{\mathcal K}_5\overline u u^2_x+i{\mathcal K}_6|u_x|^2 u_x, \] \({\mathcal K}_j={\mathcal K}_j(|u|^2)\), \({\mathcal K}_j(z)\in C^3(\mathbb{R}^+)\); \({\mathcal K}_j(z)= \lambda_j+ O(z)\), as \(z\to+ 0\), \({\mathcal K}_1\), \({\mathcal K}_6\) are real valued functions. Here the parameters \(\lambda_1,\lambda_6\in \mathbb{R}\), and \(\lambda_2,\lambda_3,\lambda_4,\lambda_5\in \mathbb{C}\) are such that \(\lambda_2- \lambda_3\in\mathbb{R}\) and \(\lambda_4- \lambda_5\in \mathbb{R}\). If \({\mathcal K}_5(z)= {\lambda_s\over 1+\mu z}\) and \(\lambda_5= \mu= \pm 1\), \({\mathcal K}_1={\mathcal K}_2={\mathcal K}_3={\mathcal K}_4={\mathcal K}_6= 0\) equation (1) appears in the classical pseudospin magnet model. We prove that if \(u_0\in H^{3,0}\cap H^{2,1}\) and the norm \(\|u_0\|_{3,0}+\|u_0\|_{2,1}= \varepsilon\) is sufficiently small, then the solution of (1) exists globally in time and satisfies the sharp time decay estimate \(\|u(t)\|_{2,0,\infty}\leq C\varepsilon(1+|t|)^{-1/2}\), where \(\|\varphi\|_{m,s,p}= \|(1+ x^2)^{s/2}(1- \partial^2_x)^{m/2} \varphi\|_{L^p}\), \(H^{m,s}_p= \{\varphi\in S'; \|\varphi \|_{m, s,p}< \infty\}\). 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.
scientific article; zbMATH DE number 1439858