Dressing for Fokker-Planck Equations: the Cases of $1+1$ and $1+\ell$ Dimensions
DOI:
https://doi.org/10.15407/mag19.02.503Ключові слова:
скорочене рiвняння Фоккера–Планка, матрична система Фоккера–Планка, процедура одягнення, перетворення Дарбу, матрична тотожнiсть, явний розв’язокАнотація
Ми розглядаємо процедуру одягнення та явнi розв’язки скороченого скалярного рiвняння Фоккера–Планка у випадку розмiрностi $(1+1)$ i матричної системи Фоккера–Планка у випадку розмiрностi $(1+\ell)$. Для цього ми використовуємо наше узагальнене перетворення Беклунда–Дарбу (УПБД). Є лише декiлька праць щодо процедури одягнення для важливого рiвняння Фоккера–Планка i цi працi стосуються випадкiв розмiрностей $1+1$ та $1+2$.
Mathematical Subject Classification 2020: 35A30, 35Q84, 15A16
Посилання
M. Bergvelt, M. Gekhtman, and A. Kasman, Spin Calogero particles and bispectral solutions of the matrix KP hierarchy, Math. Phys. Anal. Geom. 12 (2009), 181--200. https://doi.org/10.1007/s11040-009-9058-y
A. Boutet de Monvel and V.A. Marchenko, Generalization of the Darboux transform, Mat. Fiz. Anal. Geom. 1 (1994), 479--504.
J.L. Cieśliński, Algebraic construction of the Darboux matrix revisited, J. Phys. A 42 (2009), 404003. https://doi.org/10.1088/1751-8113/42/40/404003
M.M. Crum, Associated Sturm-Liouville systems, Quart. J. Math. Oxford(2) 6 (1955), 121--127. https://doi.org/10.1093/qmath/6.1.121
P.A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), 267--310. https://doi.org/10.1215/S0012-7094-78-04516-7
C.R. Doering and J.C. Gadoua, Resonant Activation over a Fluctuating Barrier, Phys. Rev. Lett. 69 (1992), 2318--2321. https://doi.org/10.1103/PhysRevLett.69.2318
F.R. Gantmacher, The Theory of Matrices, Chelsea, New York, 1959.
F. Gesztesy, A complete spectral characterization of the double commutation method, J. Funct. Anal. 117 (1993), 401--446. https://doi.org/10.1006/jfan.1993.1132
F. Gesztesy and G. Teschl, On the double commutation method, Proc. Amer. Math. Soc. 124 (1996), 1831--1840. https://doi.org/10.1090/S0002-9939-96-03299-6
F. Gesztesy and H. Holden, Soliton equations and their algebro-geometric solutions, Cambridge University Press, Cambridge, 2003. https://doi.org/10.1017/CBO9780511546723
I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Sturm-Liouville systems with rational Weyl functions: explicit formulas and applications, Integral Equations Operator Theory 30 (1998), 338--377. https://doi.org/10.1007/BF01195588
C.H. Gu, H. Hu, and Z. Zhou, Darboux Transformations in Integrable Systems, Springer-Verlag, Dordrecht, 2005.
Choon-Lin Ho, Time-dependent Darboux transformation and supersymmetric hierarchy of Fokker-Planck equations, Chinese J. Phys. 77 (2022), 1903--1911. https://doi.org/10.1016/j.cjph.2021.12.028
M.A. Kaashoek and A.L. Sakhnovich, Discrete skew self-adjoint canonical system and the isotropic Heisenberg magnet model, J. Funct. Anal. 228 (2005), 207--233. https://doi.org/10.1016/j.jfa.2004.10.022
A. Kasman and M. Gekhtman, Solitons and almost-intertwining matrices, J. Math. Phys. 42 (2001), 3540--3551 . https://doi.org/10.1063/1.1379313
A. Kostenko, A.L. Sakhnovich, and G. Teschl, Commutation methods for Schrödinger operators with strongly singular potentials, Math. Nachr. 285 (2012), 392--410. https://doi.org/10.1002/mana.201000108
S.D. Koval, A. Bihlo, and R.O. Popovych, Extended symmetry analysis of remarkable $(1+2)$-dimensional Fokke-Planck equation, https://doi.org/10.1017/S0956792523000074
D.Sh. Lundina and V.A. Marchenko, Limits of multisoliton solutions of the nonlinear Schrödinger operator, Dopov. Nats. Akad. Nauk Ukrainy No. 8 (1992), 21--24 (Russian).
V.A. Marchenko, Nonlinear Equations and Operator Algebras, Reidel Publishing Co., Dordrecht, 1988. https://doi.org/10.1007/978-94-009-2887-9
V.B. Matveev and M.A. Salle, Darboux Transformations and Solitons, Springer-Verlag, Berlin, 1991. https://doi.org/10.1007/978-3-662-00922-2
T. Novotny and P. Chvosta, Resonant activation phenomenon for non-Markovian potential-fluctuation processes, Phys. Rev. E 63 (2000), 012102. https://doi.org/10.1103/PhysRevE.63.012102
C. Rogers, Fokker-Planck equations chain-reducible to canonical form via Bäcklund transformations, Phys. Lett. A 99 (1983), No. 4, 136--142. https://doi.org/10.1016/0375-9601(83)90961-1
H.C. Rosu, Short Survey of Darboux Transformations, preprint, https://arxiv.org/abs/quant-ph/9809056.
A.L. Sakhnovich, Dressing procedure for solutions of nonlinear equations and the method of operator identities, Inverse Problems 10 (1994), 699--710. https://doi.org/10.1088/0266-5611/10/3/013
A.L. Sakhnovich, Generalized Bäcklund-Darboux transformation: spectral properties and nonlinear equations, J. Math. Anal. Appl. 262 (2001), 274--306. https://doi.org/10.1006/jmaa.2001.7577
A.L. Sakhnovich, Dynamical canonical systems and their explicit solutions, Discrete Contin. Dyn. Syst. 37 (2017), 1679--1689. https://doi.org/10.3934/dcds.2017069
A.L. Sakhnovich, Hamiltonian systems and Sturm-Liouville equations: Darboux transformation and applications, Integral Equations Operator Theory 88 (2017), 535--557. https://doi.org/10.1007/s00020-017-2385-7
A.L. Sakhnovich, Explicit solutions of matrix and dynamical Schrödinger equations and of KdV equation in terms of square roots of the generalised matrix eigenvalues, Oper. Matrices 16 (2022), 1175--1184. https://doi.org/10.7153/oam-2022-16-76
A.L. Sakhnovich, Dressing for generalised linear Hamiltonian systems depending rationally on the spectral parameter and some applications, Discrete Contin. Dyn. Syst. 43 (2023), 807--820. https://doi.org/10.3934/dcds.2022170
A.L. Sakhnovich, L.A. Sakhnovich, and I.Ya. Roitberg, Inverse Problems and Nonlinear Evolution Equations. Solutions, Darboux Matrices and Weyl-Titchmarsh Functions, De Gruyter, Berlin, 2013. https://doi.org/10.1515/9783110258615
L.A. Sakhnovich, On the factorization of the transfer matrix function, Sov. Math. Dokl. 17 (1976), 203--207.
L.A. Sakhnovich, Factorisation problems and operator identities, Russian Math. Surveys 41 (1986), 1--64. https://doi.org/10.1070/RM1986v041n01ABEH003200
L.A. Sakhnovich, Spectral Theory of Canonical Differential Systems, Method of Operator Identities, Birkhäuser-Verlag, Basel-Boston, 1999. https://doi.org/10.1007/978-3-0348-8713-7
A. Schulze-Halberg, Darboux transformations for $(1+2)$-dimensional Fokker-Planck equations with constant diffusion matrix, J. Math. Phys. 53 (2012), 103519.https://doi.org/10.1063/1.4759118
I.V. Verevkin, Euler-Darboux transformation for the Fokker-Planck equation, Theoret. and Math. Phys. 166 (2011), 58--65. https://doi.org/10.1007/s11232-011-0005-2
V.E. Zakharov and A.V. Mikhailov, On the integrability of classical spinor models in two-dimensional space-time, Comm. Math. Phys. 74 (1980), 21--40. https://doi.org/10.1007/BF01197576