×

zbMATH — the first resource for mathematics

Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. (English) Zbl 0613.35076
The authors consider the problem of existence of stationary solutions to a one-dimensional nonlinear Schrödinger equation of the form \[ (1)\quad ihu_ t=-(h^ 2/2m)u_{xx}+Vu-a| u|^ 2\cdot u \] with a bounded smooth potential V, \(a>0\), h small. They show that for every critical point of V there is an \(h_ 0\) such that for all \(h\in (0,h_ 0)\) one has a nontrivial stationary solution to (1) which lies in a neighborhood of a translate around the critical point \(x_ 0\) of the ground state solution \(u_ 0\) of the free equation (that is equation (1) without potential); this means that the solution is concentrated near \(x_ 0\) and in some sense nonspreading. The method the authors use in their proof is inspired by procedures to find instanton solutions for the Yang-Mills equation, namely a Lyapunov-Schmidt reduction to finite dimensions combined with a crucial argument using the nondegeneracy of the ground state \(u_ 0\) (i.e. the kernel of the Frechet derivative at \(u_ 0\) of the nonlinear operator belonging to (1) should be spanned by the components of grad \(u_ 0)\).
Reviewer: H.Lange

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Berestycki, H; Lions, P.L, Nonlinear scalar field equations, I and II, Arch. rational mech. anal., 82, 313-379, (1983)
[2] Cantor, M, Some problems of global analysis on asymptotically simple manifolds, Compositio math., 38, 3-35, (1979) · Zbl 0402.58004
[3] Cazenave, T; Lions, P.L, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. math. phys., 85, 549-561, (1982) · Zbl 0513.35007
[4] Ebin, D.G, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Ann. of math., 105, 141-200, (1977) · Zbl 0373.76007
[5] Jaffe, A; Taubes, C, Vortices and monopoles: structure of static gauge theories, () · Zbl 0457.53034
[6] Laedke, E.W; Spatschek, K.H, Liapunov stability of generalized Langmuir solitons, Phys. fluids, 23, 44-51, (1980) · Zbl 0429.76017
[7] Landau, L.D; Lifschitz, E.M, Quantum mechanics, non-relativistic theory, (1968), Addison-Wesley Reading, Mass · Zbl 0997.70501
[8] Rubin, H; Ungar, P, Motion under a strong constraining force, Comm. pure appl. math., 10, 65-87, (1957) · Zbl 0077.17401
[9] Stanton, R.J; Weinstein, A, On the L4 norm of spherical harmonics, (), 343-358 · Zbl 0479.33010
[10] Strauss, W.A, Existence of solitary waves in higher dimensions, Comm. math. phys., 55, 149-162, (1977) · Zbl 0356.35028
[11] Takens, F, Motion under the influence of a strong constraining force, (), 425-445
[12] Taubes, C, Stability in Yang-Mills theories, Comm. math. phys., 91, 235-263, (1983) · Zbl 0524.58020
[13] Taubes, C, Self-dual connections on non-self-dual 4-manifolds, J. differential geom., 17, 139-170, (1982) · Zbl 0484.53026
[14] Weinstein, A, Nonlinear stabilization of quasimodes, (), 301-318
[15] Weinstein, M, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. pure appl. math., 39, 51-67, (1986) · Zbl 0594.35005
[16] Whitham, G.B, Linear and nonlinear waves, (1974), Wiley New York · Zbl 0373.76001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.