Regularity properties of the cubic nonlinear Schr "odinger equation on\n the half line

2015/09/11 by M. Burak Erdoğan, Erdogan, M. Burak, Nikolaos Tzirakis +1 · 2 citations
Mathematics · #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.1509.03546

Abstract

In this paper we study the local and global regularity properties of the\ncubic nonlinear Schr "odinger equation (NLS) on the half line with rough\ninitial data. These properties include local and global wellposedness results,\nlocal and global smoothing results and the behavior of higher order Sobolev\nnorms of the solutions. In particular, we prove that the nonlinear part of the\ncubic NLS on the half line is smoother than the initial data. The gain in\nregularity coincides with the gain that was observed for the periodic cubic NLS\n citeet2 and the cubic NLS on the line citeerin. We also prove that in the\ndefocusing case the norm of the solution grows at most polynomially-in-time\nwhile in the focusing case it grows exponentially-in-time. As a byproduct of\nour analysis we provide a different proof of an almost sharp local\nwellposedness in H^s( R⁺). Sharp L² local wellposedness was obtained in\n citeholmer and citebonaetal. Our methods simplify some ideas in the\nwellposedness theory of initial and boundary value problems that were developed\nin citecollianderkenig, holmer,holmer1,bonaetal.\n

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• Mathematical results for the nonlinear Winter's model
• Nonlinear smoothing implies improved lower bounds on the radius of spatial analyticity for nonlinear dispersive equations

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