Modulation Theory for Self-Focusing in the Nonlinear Schrdinger-Helmholtz Equation

Authors: Yanping Cao ^a; Ziad H. Musslimani ^b; Edriss S. Titi ^c^d

Affiliations:	^a Department of Mathematics, University of California, Irvine, California, USA
	^b Department of Mathematics, Florida State University, Tallahassee, Florida, USA
	^c Department of Mathematics, and Department of Mechanical and Aerospace Engineering, University of California, Irvine, California, USA
	^d Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel

DOI: 10.1080/01630560802679398

Publication Frequency: 12 issues per year

Published in: journal

Numerical Functional Analysis and Optimization, Volume 30, Issue 1 & 2 January 2009 , pages 46 - 69

Subjects: Functional Analysis; Inverse Problems; Mathematical Analysis; Mathematical Numerical Analysis; Optimization; Real Functions;

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Abstract

The nonlinear Schr ouml

dinger-Helmholtz (SH) equation in N space dimensions with 2σ nonlinear power was proposed as a regularization of the classic nonlinear Schr ouml

dinger (NLS) equation. It was shown that the SH equation has a larger regime $LNFA_A_368109_O_XML_IMAGES\LNFA_A_368109_O_ILM0001.gif$ of global existence and uniqueness of solutions compared with that of the classic NLS $LNFA_A_368109_O_XML_IMAGES\LNFA_A_368109_O_ILM0002.gif$ . In the limiting case where the Schr ouml

dinger-Helmholtz equation is viewed as a perturbed system of the classic NLS equation, we apply modulation theory to the classic critical case (σ = 1, N = 2) and show that the regularization prevents the formation of singularities of the NLS equation. Our theoretical results are supported by numerical simulations.

Keywords: Hamiltonian; Modulation theory; Perturbed critical nonlinear Schr ouml

dinger equation; Regularization of the nonlinear Schr ouml

dinger equation; Schr ouml

dinger-Helmholtz equation; Schr ouml

dinger-Newton equation

AMS Subject Classification: 35Q40; 35Q55; 78A60

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Modulation Theory for Self-Focusing in the Nonlinear Schrdinger-Helmholtz Equation

Abstract