Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems

被引：1

作者：

Shepelsky, Dmitry ^{[1
,2
]}

Karpenko, Iryna ^{[1
,3
]}

Bogdanov, Stepan ^{[4
]}

Prilepsky, Jaroslaw E. ^{[4
]}

机构：

[1] B Verkin Inst Low Temp Phys & Engn, Kharkiv, Ukraine

[2] Kharkov Natl Univ, Kharkiv, Ukraine

[3] Univ Vienna, Vienna, Austria

[4] Aston Univ, Aston Inst Photon Technol, Birmingham, England

来源：

PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES | 2024年 / 480卷 / 2286期

关键词：

Riemann-Hilbert problem; Fokas method; nonlinear Schrodinger equation; periodic finite-band solutions; RIEMANN-HILBERT PROBLEMS; FOURIER-TRANSFORM; SCHRODINGER-EQUATION; MODULATION;

D O I：

10.1098/rspa.2023.0828

中图分类号：

O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];

学科分类号：

07 ; 0710 ; 09 ;

摘要：

We consider the Riemann-Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schrodinger (NLS) equation. An RH problem for the solution of the finite-band problem has been recently derived via the Fokas method (Deconinck et al. 2021 Lett. Math. Phys. 111, 1-18. (doi:10.1007/s11005-021-01356-7); Fokas & Lenells. 2021 Proc. R. Soc. A 477, 20200605. (doi:10.1007/s11005-021-01356-7)) Building on this method, a finite-band solution to the NLS equation can be given in terms of the solution of an associated RH problem, the jump conditions for which are characterized by specifying the endpoints of the arcs defining the contour of the RH problem and the constants (so-called phases) involved in the jump matrices. In our work, we solve the problem of retrieving the phases given the solution of the NLS equation evaluated at a fixed time. Our findings are corroborated by numerical examples of phases computation, demonstrating the viability of the method proposed.

引用

页数：28

共 33 条

[1]

ABLOWITZ MJ, 1974, STUD APPL MATH, V53, P249

[2]

Ablowitz MJ., 1991, Solitons, Nonlinear Evolution Equation and Inverse Scattering, DOI 10.1017/CBO9780511623998

[3] Phase computation for the finite-genus solutions to the focusing nonlinear Schrodinger equation using convolutional neural networks [J].

Bogdanov, Stepan ;

Shepelsky, Dmitry ;

Vasylchenkova, Anastasiia ;

Sedov, Egor ;

Freire, Pedro J. ;

Turitsyn, Sergei K. ;

Prilepsky, Jaroslaw E. .

COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, 2023, 125

[4] The implementation of the unified transform to the nonlinear Schrodinger equation with periodic initial conditions [J].

Deconinck, B. ;

Fokas, A. S. ;

Lenells, J. .

LETTERS IN MATHEMATICAL PHYSICS, 2021, 111 (01)

[5] An extension of the steepest descent method for Riemann-Hilbert problems: The small dispersion limit of the Korteweg-de Vries (KdV) equation [J].

Deift, P ;

Venakides, S ;

Zhou, X .

PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, 1998, 95 (02) :450-454

[6] THE COLLISIONLESS SHOCK REGION FOR THE LONG-TIME BEHAVIOR OF SOLUTIONS OF THE KDV EQUATION [J].

DEIFT, P ;

VENAKIDES, S ;

ZHOU, X .

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1994, 47 (02) :199-206

[7] Capacity estimates for optical transmission based on the nonlinear Fourier transform [J].

Derevyanko, Stanislav A. ;

Prilepsky, Jaroslaw E. ;

Turitsyn, Sergei K. .

NATURE COMMUNICATIONS, 2016, 7

[8] Capacity Limits of Optical Fiber Networks [J].

Essiambre, Rene-Jean ;

Kramer, Gerhard ;

Winzer, Peter J. ;

Foschini, Gerard J. ;

Goebel, Bernhard .

JOURNAL OF LIGHTWAVE TECHNOLOGY, 2010, 28 (04) :662-701

[9] A new approach to integrable evolution equations on the circle [J].

Fokas, A. S. ;

Lenells, J. .

PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 2021, 477 (2245)

[10] The nonlinear Schrodinger equation on the interval [J].

Fokas, AS ;

Its, AR .

JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 2004, 37 (23) :6091-6114

← 1 2 3 4 →