On the Unique Continuation Property for a Coupled System of Third-order Nonlinear Schrodinger Equations

Abstract

In this paper, we study the unique continuation properties for a coupled system of third-order nonlinear Schrodinger equations and show the Carleman estimates of L2 and Lp (p > 2) types, as well as exponential decay properties of the solutions. As a consequence we obtain that if (\(\mathit{u}\), \(\mathit{w}\)) = (\(\mathit{u}\)(\(\mathit{x}\), \(\mathit{t}\)), \(\mathit{w}\)(\(\mathit{x}\), \(\mathit{t}\))) is a suffciently smooth solution of the system such that there exists \(\mathit{l}\) \(\in\) \(\mathbb{R}\) with supp \(\mathit{u}\)(.,tj) \(\subseteq\)(\(\mathit{l}\), \(\infty\)) (\(\mathit{or}\)(-\(\infty\), \(\mathit{l}\))) and supp \(\mathit{w}\)(.,tj) \(\subseteq\)(\(\mathit{l}\), \(\infty\)) (\(\mathit{or}\)(-\(\infty\), \(\mathit{l}\))), for \(\mathit{j}\) = 1,2 (t1 \(\neq\) t2), then \(\mathit{u}\) \(\equiv\) 0 and \(\mathit{w}\) \(\equiv\) 0.