On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. (English) Zbl 0753.35097

The author considers nonlinear Schrödinger equations with an additional linear potential \(V\) of class \((V)_ a\) in the sense of Kato and an attractive power law nonlinearity. He proves the existence of special solutions of type \(\exp(-iEt/h)\cdot v(x)\) (called multi-lump bound states) for each finite collection of nondegenerate critical points of \(V\); here \(v(x)\) is a real small perturbation of a sum of one-lump solutions concentrated near one critical point resp. of the potential \(V\).
This generalizes results of A. Floer and A. Weinstein on one- lump solutions for bounded potentials [J. Funct. Anal. 69, 397-408 (1986; Zbl 0613.35076)]. The crucial technical point is a new estimate of the norm of a certain Fredholm inverse which is not needed in the one-lump case.
Reviewer: H.Lange (Köln)


35Q55 NLS equations (nonlinear Schrödinger equations)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics


Zbl 0613.35076
Full Text: DOI


[1] [FW.a] Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential. J. Funct. Anal.69, 397–408 (1986) · Zbl 0613.35076
[2] [GlJa] Gilmm, J., Jaffe, A.: Quantum physics. Berlin, Heidelberg, New York: Springer 1981
[3] [Gr] Grillakis, M.: Linearized instability for nonlinear Schrödinger and Klein-Gordon equations. Commun. Pure Appl. Math.41, 747–774 (1988) · Zbl 0632.70015
[4] [GrSS] Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry, I. J. Funct. Anal.74, 160–197 (1987) · Zbl 0656.35122
[5] [JaT] Jaffe, A., Taubes, C. H.: Vortices and Monopoles, Boston: Birkhäuser 1980
[6] [Jo] Jones, C.: Instability of standing waves for nonlinear Schrödinger type equations. Ergodic Theory and Dynamical Systems8, 119–138 (1988) · Zbl 0636.35017
[7] [K] Kato, K.: Remarks on holomorphic families of Schrödinger and Dirac operators. In: Differential Equations. Knowles, I., Lewis, R. (eds.) pp. 341–352. Amsterdam: North Holland 1984 · Zbl 0565.47011
[8] [Kw] Kwong, M. K.: Uniqueness of positive solutions of {\(\Delta\)}u+u p=0 in \(\mathbb{R}\)n. Arch. Rational Mech. Anal.105, 243–266 (1989) · Zbl 0676.35032
[9] [O1] Oh, Y.-G.: Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class(V) a. Commun. Partial Diff. Eq.13, 1499–1519 (1988) · Zbl 0702.35228
[10] [O2] –: Correction to ”Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class(V) a”. Commun. Partial Diff. Eq.14, 833–834 (1989) · Zbl 0714.35078
[11] [O3] –: Stability of semi-classical bound states of nonlinear Schrödinger equations with potentials. Commun. Math. Phys.121, 11–33 (1989) · Zbl 0693.35132
[12] [ReS] Reed, M., Simon, B.: Methods of modern mathematical physics II, IV. New York: Academic Press 1978 · Zbl 0401.47001
[13] [RW.m] Rose, H., Weinstein, M.: On the bound states of the nonlinear Schrödinger equation with a linear potential. Physica D.30, 207–218 (1988) · Zbl 0694.35202
[14] [T] Taubes, C. H.: The existence of multi-monopole solutions to the non-abelian Yang-Mills-Higgs equations for arbitrary simple gauge groups. Commun. Math. Phys.80, 343–367 (1981) · Zbl 0486.35072
[15] [W.a] Weinstein, A.: Nonlinear stabilization of quasimodes. Proc. A.M.S. Symposium on Geometry of the Laplacian, Hawaii, 1979, AMS Colloq. Publ.36, 301–318 (1980)
[16] [W.m] Weinstein, M.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal.16, 567–576 (1985) · Zbl 0583.35028
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.