Nonlinear Schrodinger equations on the half-line with nonlinear boundary conditions

Abstract

In this article, we study the initial boundary value problem for nonlinear Schrodinger equations on the half-line with nonlinear boundary conditions $$ u_x(0,t)+\lambda|u(0,t)|^ru(0,t)=0,\quad \lambda\in\mathbb{R}-\{0\},\; r> 0. $$ We discuss the local well-posedness when the initial data $u_0=u(x,0)$ belongs to an $L^2$-based inhomogeneous Sobolev space $H^s(\mathbb{R}_+)$ with $s\in (\frac{1}{2},\frac{7}{2})-\{\frac{3}{2}\}$. We deal with the nonlinear boundary condition by first studying the linear Schrodinger equation with a time-dependent inhomogeneous Neumann boundary condition $u_x(0,t)=h(t)$ where $h\in H^{\frac{2s-1}{4}}(0,T)$.

Keywords

• Nonlinear Schrodinger equations
• nonlinear boundary conditions
• local well-posedness
• inhomogeneous boundary conditions