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Title: Blow up of solutions to the nonlinear Schroedinger equations on manifolds

Abstract

In this paper, we partially settle down the long-standing open problem of the finite time blow-up property about the nonlinear Schroedinger equations on some Riemannian manifolds such as the standard 2-sphere S{sup 2} and the hyperbolic 2-space H{sup 2}(-1). Using the similar idea, we establish such blow-up results on higher dimensional standard sphere and hyperbolic n-space. Extensions to n-dimensional Riemannian warped product manifolds with n{>=}2 are also given.

Authors:
;  [1]
  1. Department of Mathematical Sciences, Tsinghua University, Peking 100084 (China)
Publication Date:
OSTI Identifier:
20929727
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 5; Other Information: DOI: 10.1063/1.2722560; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; MATHEMATICAL MANIFOLDS; MATHEMATICAL SOLUTIONS; MATHEMATICAL SPACE; NONLINEAR PROBLEMS; SCHROEDINGER EQUATION; SPHERES

Citation Formats

Ma Li, and Zhao Lin. Blow up of solutions to the nonlinear Schroedinger equations on manifolds. United States: N. p., 2007. Web. doi:10.1063/1.2722560.
Ma Li, & Zhao Lin. Blow up of solutions to the nonlinear Schroedinger equations on manifolds. United States. doi:10.1063/1.2722560.
Ma Li, and Zhao Lin. Tue . "Blow up of solutions to the nonlinear Schroedinger equations on manifolds". United States. doi:10.1063/1.2722560.
@article{osti_20929727,
title = {Blow up of solutions to the nonlinear Schroedinger equations on manifolds},
author = {Ma Li and Zhao Lin},
abstractNote = {In this paper, we partially settle down the long-standing open problem of the finite time blow-up property about the nonlinear Schroedinger equations on some Riemannian manifolds such as the standard 2-sphere S{sup 2} and the hyperbolic 2-space H{sup 2}(-1). Using the similar idea, we establish such blow-up results on higher dimensional standard sphere and hyperbolic n-space. Extensions to n-dimensional Riemannian warped product manifolds with n{>=}2 are also given.},
doi = {10.1063/1.2722560},
journal = {Journal of Mathematical Physics},
number = 5,
volume = 48,
place = {United States},
year = {Tue May 15 00:00:00 EDT 2007},
month = {Tue May 15 00:00:00 EDT 2007}
}
  • The model of the following two-coupled Schroedinger equations, i{sub t}+(1/2){delta}u=(g{sub 11}|u|{sup 2p}+g|u|{sup p-1}|v|{sup p+1})uu, (t,x)(set-membership sign)R{sub +}xR{sup N}, and iv{sub t}+(1/2){delta}v=(g|u|{sup p+1}|v|{sup p-1}+g{sub 22}|v|{sup 2p})v, (t,x)(set-membership sign)R{sub +}xR{sup N}, is proposed in the study of the Bose-Einstein condensates [Mitchell, et al., ''Self-traping of partially spatially incoherent light,'' Phys. Rev. Lett. 77, 490 (1996)]. We prove that for suitable initial data and p the solution blows up exactly like {delta} function. As a by-product, we prove that similar phenomenon occurs for the critical two-coupled Schroedinger equations with harmonic potential [Perez-Garcia, V. M. and Beitia, T. B., ''Sybiotic solitons in heteronuclear multicomponentmore » Bose-Einstein condensates,'' Phys. Rev. A 72, 033620 (2005)], iu{sub t}+(1/2){delta}u=({omega}/2)|x|{sup 2}u+(g{sub 11}|u|{sup 2p}+g|u|{sup p-1}|v|{sup p+1})u, x(set-membership sign)R{sup N}, and iv{sub t}+(1/2){delta}v=({omega}/2)|x|{sup 2}v+(g|u|{sup p+1}|v|{sup p-1}+g{sub 22}|v|{sup 2p})v, x(set-membership sign)R{sup N}.« less
  • We prove global existence, scattering, and blow up of solutions to systems of nonlinear Schroedinger equations with interactions of a particular type. Thus, we consider iu/sub t/ + ..delta..u = F(u), u(t/sub 0/,chi) = u/sub 0/(chi) where u is a function from R x R/sup n/ to C/sup m/ and F is a vector field on C/sup m/. We also determine the dependence of blow up and decay of solutions on a parameter lambda in nonlinear interactions of the general form F(u) = u absolute value u/sup p-1/ - lambda u absolute value u/sup q-1/ + eta u absolute valuemore » u/sup r-1/ for m = 1. We obtain global existence, scattering, and blow up if F is an exact interaction, i.e., an exact vector field on C/sup M/. For global existence, we allow F to be of the form u absolute value u/sup p-1/ + u absolute value u/sup q-1/, 2 less than or equal to p < q < n + 2/n-2 with restriction p > (n + 2)(2n - (n-2)q)/sup -1/. Using different techniques, we remove this restriction for n = 3 or 4 and prove global existence in H/sup 2/. For the case m = 1, these results allow more general interactions than (2) where global existence is proven in H/sup 1/. We also obtain global existence for perturbations of exact interactions of the form i ..beta..(t)u absolute value u/sup p-1/ for ..beta.. an odd function of t. If F is not an exact vector field, we obtain global existence provided that F satisfies a certain type of Lipshitz estimate.« less
  • Consider a nonlinear heat equation ..mu../sub t/-epsilon..delta..=f(..mu..) in a cylinder /xelement of..cap omega.., t>0/, with ..mu.. vanishing on the lateral boundary and ..mu..=phi/sub epsilon/(chi) initially (phi/sub epsilon/greater than or equal to0). Denote by T/sub epsilon/ the blow-up time for the solution. Asymptotic estimates are obtained for T/sub epsilon/ as epsilon..-->..0.
  • In this article, we prove that the equation of the Schroedinger maps from R{sup 2} to the hyperbolic 2-space H{sup 2} is SU(1,1)-gauge equivalent to the following 1+2 dimensional nonlinear Schroedinger-type system of three unknown complex functions p, q, r, and a real function u: iq{sub t}+q{sub zz}-2uq+2(pq){sub z}-2pq{sub z}-4|p|{sup 2}q=0, ir{sub t}-r{sub zz}+2ur+2(pr){sub z}-2pr{sub z}+4|p|{sup 2}r=0, ip{sub t}+(qr){sub z}-u{sub z}=0, p{sub z}+p{sub z}=-|q|{sup 2}+|r|{sup 2}, -r{sub z}+q{sub z}=-2(pr+pq), where z is a complex coordinate of the plane R{sup 2} and z is the complex conjugate of z. Although this nonlinear Schroedinger-type system looks complicated, it admits a class ofmore » explicit blow-up smooth solutions: p=0, q=(e{sup i(bzz/2(a+bt))}/a+bt){alpha}z, r=e{sup -i(bzz/2(a+bt))}/(a+bt){alpha}z, u=2{alpha}{sup 2}zz/(a+bt){sup 2}, where a and b are real numbers with ab<0 and {alpha} satisfies {alpha}{sup 2}=b{sup 2}/16. From these facts, we explicitly construct smooth solutions to the Schroedinger maps from R{sup 2} to the hyperbolic 2-space H{sup 2} by using the gauge transformations such that the absolute values of their gradients blow up in finite time. This reveals some blow-up phenomenon of Schroedinger maps.« less
  • The author's prove that for the parabolic initial value problem ..mu../sub t/=..delta mu..+deltaf(..mu..) there is a finite time blow-up of the solution, provided /e// is greater than the upper bound to the spectrum of the steady state problem and (f/f') is concave. An upper bound of the blow-up time is given. The proof is based on a comparison with a subsolution to the parabolic initial value problem.