Linear approximation of nonlinear Schrödinger equations driven by cylindrical Wiener processes

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November 2016, 21(9): 3095-3114. doi: 10.3934/dcdsb.2016089

Linear approximation of nonlinear Schrödinger equations driven by cylindrical Wiener processes

Wilfried Grecksch ^1, and Hannelore Lisei ^2,

1.	Martin-Luther-University Halle-Wittenberg, Faculty of Natural Sciences II, Institute of Mathematics, 06099 Halle (Saale)
2.	Babeş-Bolyai University, Faculty of Mathematics and Computer Science, Str. Kogălniceanu nr. 1, RO - 400084 Cluj-Napoca

Received December 2015 Revised February 2016 Published October 2016

Abstract
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In this paper the existence and uniqueness of the solution for a stochastic nonlinear Schrödinger equation, which is perturbed by a cylindrical Wiener process is investigated. The existence of the variational solution and of the generalized weak solution are proved by using sequences of successive approximations, which are the solutions of certain linear problems.

Keywords: fixed point, cylindrical Wiener process., Nonlinear stochastic Schrödinger equations.

Mathematics Subject Classification: Primary: 60H15, 60H35, 35Q41; Secondary: 81Q0.

Citation: Wilfried Grecksch, Hannelore Lisei. Linear approximation of nonlinear Schrödinger equations driven by cylindrical Wiener processes. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3095-3114. doi: 10.3934/dcdsb.2016089

References:

[1]	N. U. Ahmed, Semigroup Theory with Applications to Systems and Control, Longman Scientific & Technical, Harlow Essex, 1991.
[2]	N. N. Akhmeddiev, A. Ankiewicz and J. M. Soto-Crespo, Singularities and special soliton solutions of the cubic quintic complex Ginzburg-Landau equation, Phys. Rev. E, 53 (1996), 1190-1201, Available from: http://dx.doi.org/10.1103/PhysRevE.53.1190 doi: 10.1103/PhysRevE.53.1190.
[3]	A. De Bouard and A. Debussche, A stochastic nonlinear Schrödinger equation with multiplicative noise, Commun. Math. Phys., 205 (1999), 161-181. doi: 10.1007/s002200050672.
[4]	A. De Bouard and A. Debussche, A semi-discrete scheme for the stochastic nonlinear Schr\"odinger equation, Numer. Math., 96 (2004), 733-770. doi: 10.1007/s00211-003-0494-5.
[5]	A. Biswas and S. Konar, Introduction to Non-Kerr Law Optical Solitons, Chapman and Hall/CRC, 2007.
[6]	Y. Chen, $L^\infty(\mathbb R^n)$ decay for solutions to a class of Schrödinger equations, Appl. Anal., 39 (1990), 209-226. doi: 10.1080/00036819008839982.
[7]	A. Debussche and C. Odasso, Ergodicity for a weakly damped stochastic non-linear Schrödinger equation, J. Evol. Equ., 5 (2005), 317-356. doi: 10.1007/s00028-005-0195-x.
[8]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.
[9]	R. Dautray and J.-L. Lions, Mathematical Analysis And Numerical Methods For Science And Technology, Volume 5: Evolution Problems I, Springer Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1.
[10]	W. Grecksch and H. Lisei, Stochastic nonlinear equations of Schrödinger type, Stoch. Anal. Appl., 29 (2011), 631-653. doi: 10.1080/07362994.2011.581091.
[11]	W. Grecksch and C. Tudor, Stochastic Evolution Equations. A Hilbert Space Approach, Akademie Verlag, Berlin, 1995.
[12]	H. M. Itô, Optimal Gaussian solutions of nonlinear stochastic partial differential equations, J. Stat. Phys., 37 (1984), 653-671. doi: 10.1007/BF01010500.
[13]	S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods, Springer Verlag, Heidelberg, 2003.
[14]	J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer Verlag, Berlin-New York, 1971.
[15]	R. Mikulevicius and B. Rozovskii, On martingale problem solutions for stochastic Navier Stokes equations, In: Stochastic Partial Differential Equations and Applications, Edited by G. Da Prato, L. Tubaro. Marcel & Dekker, Inc. New York, Basel, 227 (2002), 405-415.
[16]	C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics Vol. 1905, Springer Verlag, 2007.
[17]	M. Reed and B. Simao, Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press San Diego, 1978.
[18]	J. J. Rasmussen and K. Rypdal, Blow-up in nonlinear Schroedinger equations-I. A general review, Phys. Scripta., 33 (1986), 481-497. doi: 10.1088/0031-8949/33/6/001.
[19]	B. L. Rozovskii, Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering, Kluwer Academic Publishers, Dordrecht, 1990. doi: 10.1007/978-94-011-3830-7.
[20]	E. Zeidler, Nonlinear Functional Analysis and Its Applications. I: Fixed-point Theorems, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.
[21]	E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A: Linear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

References:

[1]	N. U. Ahmed, Semigroup Theory with Applications to Systems and Control, Longman Scientific & Technical, Harlow Essex, 1991.
[2]	N. N. Akhmeddiev, A. Ankiewicz and J. M. Soto-Crespo, Singularities and special soliton solutions of the cubic quintic complex Ginzburg-Landau equation, Phys. Rev. E, 53 (1996), 1190-1201, Available from: http://dx.doi.org/10.1103/PhysRevE.53.1190 doi: 10.1103/PhysRevE.53.1190.
[3]	A. De Bouard and A. Debussche, A stochastic nonlinear Schrödinger equation with multiplicative noise, Commun. Math. Phys., 205 (1999), 161-181. doi: 10.1007/s002200050672.
[4]	A. De Bouard and A. Debussche, A semi-discrete scheme for the stochastic nonlinear Schr\"odinger equation, Numer. Math., 96 (2004), 733-770. doi: 10.1007/s00211-003-0494-5.
[5]	A. Biswas and S. Konar, Introduction to Non-Kerr Law Optical Solitons, Chapman and Hall/CRC, 2007.
[6]	Y. Chen, $L^\infty(\mathbb R^n)$ decay for solutions to a class of Schrödinger equations, Appl. Anal., 39 (1990), 209-226. doi: 10.1080/00036819008839982.
[7]	A. Debussche and C. Odasso, Ergodicity for a weakly damped stochastic non-linear Schrödinger equation, J. Evol. Equ., 5 (2005), 317-356. doi: 10.1007/s00028-005-0195-x.
[8]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.
[9]	R. Dautray and J.-L. Lions, Mathematical Analysis And Numerical Methods For Science And Technology, Volume 5: Evolution Problems I, Springer Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1.
[10]	W. Grecksch and H. Lisei, Stochastic nonlinear equations of Schrödinger type, Stoch. Anal. Appl., 29 (2011), 631-653. doi: 10.1080/07362994.2011.581091.
[11]	W. Grecksch and C. Tudor, Stochastic Evolution Equations. A Hilbert Space Approach, Akademie Verlag, Berlin, 1995.
[12]	H. M. Itô, Optimal Gaussian solutions of nonlinear stochastic partial differential equations, J. Stat. Phys., 37 (1984), 653-671. doi: 10.1007/BF01010500.
[13]	S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods, Springer Verlag, Heidelberg, 2003.
[14]	J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer Verlag, Berlin-New York, 1971.
[15]	R. Mikulevicius and B. Rozovskii, On martingale problem solutions for stochastic Navier Stokes equations, In: Stochastic Partial Differential Equations and Applications, Edited by G. Da Prato, L. Tubaro. Marcel & Dekker, Inc. New York, Basel, 227 (2002), 405-415.
[16]	C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics Vol. 1905, Springer Verlag, 2007.
[17]	M. Reed and B. Simao, Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press San Diego, 1978.
[18]	J. J. Rasmussen and K. Rypdal, Blow-up in nonlinear Schroedinger equations-I. A general review, Phys. Scripta., 33 (1986), 481-497. doi: 10.1088/0031-8949/33/6/001.
[19]	B. L. Rozovskii, Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering, Kluwer Academic Publishers, Dordrecht, 1990. doi: 10.1007/978-94-011-3830-7.
[20]	E. Zeidler, Nonlinear Functional Analysis and Its Applications. I: Fixed-point Theorems, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.
[21]	E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A: Linear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

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