Electron. J. Diff. Equ., Vol. 2016 (2016), No. 222, pp. 1-20.

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

Ahmet Batal, Turker Ozsari

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 $h\in H^{\frac{2s-1}{4}}(0,T)$.

Submitted March 2, 2016. Published August 17, 2016.
Math Subject Classifications: 35Q55, 35A01, 35A02, 35B30.
Key Words: Nonlinear Schrodinger equations; nonlinear boundary conditions; local well-posedness; inhomogeneous boundary conditions.

Show me the PDF file (334 KB), TEX file for this article.

Ahmet Batal
Department of Mathematics
Izmir Institute of Technology
Izmir, Turkey
email: ahmetbatal@iyte.edu.tr
Turker Ozsari
Department of Mathematics
Izmir Institute of Technology
Izmir, Turkey
email: turkerozsari@iyte.edu.tr

Return to the EJDE web page