Complutense University Library

Localizing estimates of the support of solutions of some nonlinear Schrodinger equations - The stationary case

Díaz Díaz, Jesús Ildefonso and Begout , Pascal (2012) Localizing estimates of the support of solutions of some nonlinear Schrodinger equations - The stationary case. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 29 (1). pp. 35-38. ISSN 0294-1449

[img] PDF
Restricted to Repository staff only until 31 December 2020.

1MB

Official URL: http://www.sciencedirect.com/science/article/pii/S0294144911000837

View download statistics for this eprint

==>>> Export to other formats

Abstract

The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schrodinger equations, mainly the compactness of the support and its spatial localization. This question touches the very foundations underlying the derivation of the Schrodinger equation, since it is well-known a solution of a linear Schrodinger equation perturbed by a regular potential never vanishes on a set of positive measure. A fact, which reflects the impossibility of locating the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution is a compact set, and so any estimate on its spatial localization implies very rich information on places not accessible by the particle. Our results are obtained by the application of certain energy methods which connect the compactness of the support with the local vanishing of a suitable "energy function" which satisfies a nonlinear differential inequality with an exponent less than one. The results improve and extend a previous short presentation by the authors published in 2006.


Item Type:Article
Uncontrolled Keywords:singular complex potentials; operators; Nonlinear Schrodinger equation; Compact support; Energy method
Subjects:Sciences > Mathematics > Differential equations
ID Code:15006
References:

[1]S.N. Antontsev, J.I. Díaz, S. Shmarev. Energy methods for free boundary problems. Applications to Nonlinear PDEs and Fluid Mechanics, Progress in Nonlinear Differential Equations and Their Applications, vol. 48, Birkhäuser Boston Inc., Boston, MA (2002)

[2]P. Bégout, J.I. Díaz, Localizingestimates of the support of solutions of some nonlinearSchrödingerequations – the evolution case, in preparation.

[3]P. Bégout, J.I. Díaz, Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations, in preparation.

[4]P. Bégout, J.I. Díaz. On a nonlinearSchrödingerequation with a localizing effect. C. R. Math. Acad. Sci. Paris, 342 (7) (2006), pp. 459–463

[5]P. Bégout, V. Torri. Numerical computations of the support of solutions of some localizing stationary nonlinear Schrödinger equations, in preparation.

[6]J.A. Belmonte-Beitia. Varias cuestiones sobre la ecuación de Schrödinger no lineal con coeficientes dependientes del espacio. Bol. Soc. Esp. Mat. Apl. SMA, 52 (2010), pp. 97–128

[7]H. Brezis, T. Kato. Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pures Appl. (9), 58 (2) (1979), pp. 137–151

[8]R. Carles, C. Gallo. Finite time extinction by nonlinear damping for the Schrödingerequation. Comm. Partial Differential Equations, 36 (6) (2011), pp. 961–975

[9]T. Cazenave. Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10New York University Courant Institute of Mathematical Sciences, New York (2003)

[10]T. Cazenave. An Introduction to Semilinear Elliptic Equations. Editora do Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro (2006)

[11]J.I. Díaz. Nonlinear Partial Differential Equations and Free Boundaries, vol. I, Elliptic Equations. Research Notes in Mathematics, vol. 106Pitman (Advanced Publishing Program), Boston, MA (1985)

[12]J.I. Díaz, L. Véron. Local vanishing properties of solutions of elliptic and parabolic quasilinear equations. Trans. Amer. Math. Soc., 290 (2) (1985), pp. 787–814

[13]I. Ekeland, R. Temam. Convex Analysis and Variational Problems. Classics in Applied Mathematics, vol. 28Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999) (English edition, translated from the French)

[14]D. Gilbarg, N.S. Trudinger. Elliptic Partial Differential Equations of Second Order. Classics in MathematicsSpringer-Verlag, Berlin (2001) (Reprint of the 1998 edition)

[15]P. Grisvard. Elliptic Problems in Nonsmooth Domains.

Monographs and Studies in Mathematics, vol. 24Pitman (Advanced Publishing Program), Boston, MA (1985)

[16]L. Hörmander. Definitions of maximal differential operators. Ark. Mat., 3 (1958), pp. 501–504

[17]A. Jensen. Propagation estimates for Schrödinger-type operators. Trans. Amer. Math. Soc., 291 (1) (1985), pp. 129–144

[18]E. Kashdan, P. Rosenau. Compactification of nonlinear patterns and waves. Phys. Rev. Lett., 101 (26) (2008), p. 2616024

[19]B.J. LeMesurier. Dissipation at singularities of the nonlinearSchrödingerequation through limits of regularisations. Phys. D, 138 (3–4) (2000), pp. 334–343

[20]J.-L. Lions, E. Magenes. Problèmes aux limites non homogènes, II. Ann. Inst. Fourier (Grenoble), 11 (1961), pp. 137–178

[21]J.-L. Lions, E. Magenes. Problemi ai limiti non omogenei, III. Ann. Sc. Norm. Super. Pisa (3), 15 (1961), pp. 41–103

[22]V. Liskevich, P. Stollmann. Schrödinger operators with singular complex potentials as generators: existence and stability. Semigroup Forum, 60 (3) (2000), pp. 337–343

[23]P. Rosenau, S. Schochet. Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit. Chaos, 15 (1) (2005), p. 015111 18

[24]W.A. Strauss. Partial Differential Equations. An Introduction. John Wiley & Sons Inc., New York (1992)

[25]C. Sulem, P.-L. Sulem.The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Applied Mathematical Sciences, vol. 139Springer-Verlag, New York (1999)

Deposited On:25 Apr 2012 08:44
Last Modified:06 Feb 2014 10:14

Repository Staff Only: item control page