TY - JOUR

T1 - Backscattering and nonparaxiality arrest collapse of damped nonlinear waves

AU - Fibich, G.

AU - Ilan, B.

AU - Tsynkov, S.

PY - 2003/6

Y1 - 2003/6

N2 - The critical nonlinear Schrödinger equation (NLS) models the propagation of intense laser light in Kerr media. This equation is derived from the more comprehensive nonlinear Helmholtz equation (NLH) by employing the paraxial approximation and neglecting the backscattered waves. It is known that if the input power of the laser beam (i.e., L2 norm of the initial solution) is sufficiently high, then the NLS model predicts that the beam will self-focus to a point (i.e., collapse) at a finite propagation distance. Mathematically, this behavior corresponds to the formation of a singularity in the solution of the NLS. A key question which has been open for many years is whether the solution to the NLH, i.e., the "parent" equation, may nonetheless exist and remain regular everywhere, particularly for those initial conditions (input powers) that lead to blowup in the NLS. In the current study we address this question by introducing linear damping into both models and subsequently comparing the numerical solutions of the damped NLH (boundary-value problem) with the corresponding solutions of the damped NLS (initial-value problem) for the case of one transverse dimension. Linear damping is introduced in much the same way as is done when analyzing the classical constant-coefficient Helmholtz equation using the limiting absorption principle. Numerically, we have found that it provides a very efficient tool for controlling the solutions of both the NLH and NLS. In particular, we have been able to identify initial conditions for which the NLS solution does become singular, while the NLH solution still remains regular everywhere. We believe that our finding of a larger domain of existence for the NLH than for the NLS is accounted for by precisely those mechanisms that have been neglected when deriving the NLS from the NLH, i.e., nonparaxiality and backscattering.

AB - The critical nonlinear Schrödinger equation (NLS) models the propagation of intense laser light in Kerr media. This equation is derived from the more comprehensive nonlinear Helmholtz equation (NLH) by employing the paraxial approximation and neglecting the backscattered waves. It is known that if the input power of the laser beam (i.e., L2 norm of the initial solution) is sufficiently high, then the NLS model predicts that the beam will self-focus to a point (i.e., collapse) at a finite propagation distance. Mathematically, this behavior corresponds to the formation of a singularity in the solution of the NLS. A key question which has been open for many years is whether the solution to the NLH, i.e., the "parent" equation, may nonetheless exist and remain regular everywhere, particularly for those initial conditions (input powers) that lead to blowup in the NLS. In the current study we address this question by introducing linear damping into both models and subsequently comparing the numerical solutions of the damped NLH (boundary-value problem) with the corresponding solutions of the damped NLS (initial-value problem) for the case of one transverse dimension. Linear damping is introduced in much the same way as is done when analyzing the classical constant-coefficient Helmholtz equation using the limiting absorption principle. Numerically, we have found that it provides a very efficient tool for controlling the solutions of both the NLH and NLS. In particular, we have been able to identify initial conditions for which the NLS solution does become singular, while the NLH solution still remains regular everywhere. We believe that our finding of a larger domain of existence for the NLH than for the NLS is accounted for by precisely those mechanisms that have been neglected when deriving the NLS from the NLH, i.e., nonparaxiality and backscattering.

KW - Kerr medium

KW - Limiting absorption

KW - Linear damping

KW - Nonlinear wave propagation

KW - Self-focusing

KW - Singularity formation

KW - Two-way ABCs

UR - http://www.scopus.com/inward/record.url?scp=0345327613&partnerID=8YFLogxK

U2 - 10.1137/S0036139902411855

DO - 10.1137/S0036139902411855

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:0345327613

SN - 0036-1399

VL - 63

SP - 1718

EP - 1736

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

IS - 5

ER -