Linear approximation of nonlinear Schrödinger equations driven by cylindrical Wiener processes

Previous Article
Semilinear stochastic equations with bilinear fractional noise
DCDS-B Home
This Issue
Next Article
Taylor schemes for rough differential equations and fractional diffusions

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
Related Papers

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 & 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, (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. 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. 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. 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. 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. doi: 10.1007/s00028-005-0195-x.
[8]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, University Press, (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, (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. doi: 10.1080/07362994.2011.581091.
[11]	W. Grecksch and C. Tudor, Stochastic Evolution Equations. A Hilbert Space Approach,, Akademie Verlag, (1995).
[12]	H. M. Itô, Optimal Gaussian solutions of nonlinear stochastic partial differential equations,, J. Stat. Phys., 37 (1984), 653. doi: 10.1007/BF01010500.
[13]	S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods,, Springer Verlag, (2003).
[14]	J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer Verlag, (1971).
[15]	R. Mikulevicius and B. Rozovskii, On martingale problem solutions for stochastic Navier Stokes equations,, In: Stochastic Partial Differential Equations and Applications, 227 (2002), 405.
[16]	C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations,, Lecture Notes in Mathematics Vol. 1905, (1905).
[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. 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, (1990). doi: 10.1007/978-94-011-3830-7.
[20]	E. Zeidler, Nonlinear Functional Analysis and Its Applications. I: Fixed-point Theorems,, Springer-Verlag, (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, (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, (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. 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. 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. 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. 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. doi: 10.1007/s00028-005-0195-x.
[8]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, University Press, (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, (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. doi: 10.1080/07362994.2011.581091.
[11]	W. Grecksch and C. Tudor, Stochastic Evolution Equations. A Hilbert Space Approach,, Akademie Verlag, (1995).
[12]	H. M. Itô, Optimal Gaussian solutions of nonlinear stochastic partial differential equations,, J. Stat. Phys., 37 (1984), 653. doi: 10.1007/BF01010500.
[13]	S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods,, Springer Verlag, (2003).
[14]	J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer Verlag, (1971).
[15]	R. Mikulevicius and B. Rozovskii, On martingale problem solutions for stochastic Navier Stokes equations,, In: Stochastic Partial Differential Equations and Applications, 227 (2002), 405.
[16]	C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations,, Lecture Notes in Mathematics Vol. 1905, (1905).
[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. 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, (1990). doi: 10.1007/978-94-011-3830-7.
[20]	E. Zeidler, Nonlinear Functional Analysis and Its Applications. I: Fixed-point Theorems,, Springer-Verlag, (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, (1990). doi: 10.1007/978-1-4612-0985-0.

[1]	Jing Cui, Shu-Ming Sun. Nonlinear Schrödinger equations on a finite interval with point dissipation. Mathematical Control & Related Fields, 2019, 9 (2) : 351-384. doi: 10.3934/mcrf.2019017
[2]	Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060
[3]	Veronica Felli, Alberto Ferrero, Susanna Terracini. On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3895-3956. doi: 10.3934/dcds.2012.32.3895
[4]	Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183
[5]	Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[6]	Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337
[7]	Alexander Pankov. Nonlinear Schrödinger Equations on Periodic Metric Graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 697-714. doi: 10.3934/dcds.2018030
[8]	Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359
[9]	Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565
[10]	Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703
[11]	Nobu Kishimoto. A remark on norm inflation for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1375-1402. doi: 10.3934/cpaa.2019067
[12]	Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692
[13]	Boling Guo, Yan Lv, Wei Wang. Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2795-2818. doi: 10.3934/dcds.2014.34.2795
[14]	Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379
[15]	Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253
[16]	Thierry Colin, Pierre Fabrie. Semidiscretization in time for nonlinear Schrödinger-waves equations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 671-690. doi: 10.3934/dcds.1998.4.671
[17]	Juan Belmonte-Beitia, Vladyslav Prytula. Existence of solitary waves in nonlinear equations of Schrödinger type. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1007-1017. doi: 10.3934/dcdss.2011.4.1007
[18]	Shuangjie Peng, Huirong Pi. Spike vector solutions for some coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2205-2227. doi: 10.3934/dcds.2016.36.2205
[19]	Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071
[20]	Masahito Ohta. Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1671-1680. doi: 10.3934/cpaa.2018080

2017 Impact Factor: 0.972

American Institute of Mathematical Sciences