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Semilinear stochastic equations with bilinear fractional noise
Linear approximation of nonlinear Schrödinger equations driven by cylindrical Wiener processes
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 |
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. |
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