Abstract
This paper investigates a Schrödinger problem with power-type nonlinearity and Lipschitz-continuous diffusion term on a bounded one-dimensional domain. Using the Galerkin method and a truncation, results from stochastic partial differential equations can be applied and uniform a priori estimates for the approximations are shown. Based on these boundedness results and the structure of the nonlinearity, it follows the unique existence of the variational solution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barbu V., Röckner M., Zhang D.: Stochastic nonlinear Schrödinger equations with linear multiplicative noise: rescaling approach. J. Nonlinear Sci. 24(3), 383–409 (2014)
Brézis H., Gallouet T.: Nonlinear Schrödinger evolution equations. Nonlinear Anal. 4(4), 677–681 (1980)
Cazenave, T.: Semilinear Schrödinger Equations, vol. 10. Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence (2003)
Cazenave T., Weissler F.B.: The Cauchy problem for the critical nonlinear Schrödinger equation in H s. Nonlinear Anal. 14(10), 807–836 (1990)
de Bouard A., Debussche A.: A stochastic nonlinear Schrödinger equation with multiplicative noise. Commun. Math. Phys. 205(1), 161–181 (1999)
de Bouard A., Debussche A.: The stochastic nonlinear Schrödinger equation in H 1. Stoch. Anal. Appl. 21(1), 97–126 (2003)
Diestel, J., Uhl, J.J. Jr.: Vector Measures, vol 15. Mathematical Surveys and Monographs. American Mathematical Society, Providence (1977)
Duren, P.: Invitation to Classical Analysis, vol 17. Pure and Applied Undergraduate Texts. American Mathematical Society, Providence (2012)
Fang D., Zhang L., Zhang T.: On the well-posedness for stochastic Schrödinger equations with quadratic potential. Chin. Ann. Math. Ser. B 32(5), 711–728 (2011)
Gajewski H.: Über Näherungsverfahren zur Lösung der nichtlinearen Schrödinger-Gleichung. Math. Nachr. 85, 283–302 (1978)
Gajewski H.: On an initial-boundary value problem for the nonlinear Schrödinger equation. Internat. J. Math. Math. Sci. 2(3), 503–522 (1979)
Ginibre J., Velo G.: On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 32(1), 1–32 (1979)
Grecksch W., Lisei H.: Stochastic nonlinear equations of Schrödinger type. Stoch. Anal. Appl. 29(4), 631–653 (2011)
Hayashi N.: Classical solutions of nonlinear Schrödinger equations. Manuscripta Math. 55(2), 171–190 (1986)
Jiu Q., Liu J.: Existence and uniqueness of global solutions of [the nonlinear Schrödinger equation] on \({{{\mathbf{R} }}^2}\). Acta Math. Appl. Sinica (English Ser.) 13(4), 414–424 (1997)
Kato T.: On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré. Phys. Théor. 46(1), 113–129 (1987)
Keller D., Lisei H.: Variational solution of stochastic Schrödinger equations with power-type nonlinearity. Stoch. Anal. Appl. 33(4), 653–672 (2015)
Lions, J.-L.: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod; Gauthier-Villars, Paris (1969)
Liu S., Fu Z., Liu S., Zhao Q.: Multi-order exact solutions of the complex Ginzburg–Landau equation. Phys. Lett. A 269(5–6), 319–324 (2000)
Liu X., Jia H.: Existence of suitable weak solutions of complex Ginzburg–Landau equations and properties of the set of singular points. J. Math. Phys. 44(11), 5185–5193 (2003)
Pecher H.: Solutions of semilinear Schrödinger equations in H s. Ann. Inst. H. Poincaré, Phys. Théor. 67(3), 259–296 (1997)
Pecher H., von Wahl W.: Time dependent nonlinear Schrödinger equations. Manuscripta Math. 27(2), 125–157 (1979)
Pinaud O.: A note on stochastic Schrödinger equations with fractional multiplicative noise. J. Differ. Equ. 256(4), 1467–1491 (2014)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1978)
Rozovskiĭ, B.L.: Stochastic Evolution Systems: Linear Theory and Applications to Nonlinear Filtering, vol. 35. Mathematics and its Applications (Soviet Ser.). Kluwer Academic Publishers Group, Dordrecht (1990)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68. of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, (1997)
Tsutsumi M.: On smooth solutions to the initial-boundary value problem for the nonlinear Schrödinger equation in two space dimensions. Nonlinear Anal. 13(9), 1051–1056 (1989)
Tsutsumi M., Hayashi N.: Classical solutions of nonlinear Schrödinger equations in higher dimensions. Math. Z. 177(2), 217–234 (1981)
Zeidler E.: Nonlinear Functional Analysis and its Applications. I: Fixed-Point Theorems. Springer, New York (1986)
Zeidler E.: Nonlinear Functional Analysis and its Applications. II/A: Linear Monotone Operators. Springer, New York (1990)
Zeidler E.: Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators. Springer, New York (1990)
Zhao, D., Yu, M.Y.: Generalized nonlinear Schrödinger equation as a model for turbulence, collapse, and inverse cascade. Phys. Rev. E (3), 83(3), 036405, 7 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lisei, H., Keller, D. A stochastic nonlinear Schrödinger problem in variational formulation. Nonlinear Differ. Equ. Appl. 23, 22 (2016). https://doi.org/10.1007/s00030-016-0374-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-016-0374-1