Applied Mathematics and Computation, ISSN 0096-3003, 09/2018, Volume 332, pp. 148 - 159

In this work, we investigate the two-component modified Korteweg-de Vries (mKdV) equation, which is a complete integrable system, and accepts a generalization...

Initial-boundary value problem | Unified transform method | Two-component mKdV equation | Riemann–Hilbert problem | MATHEMATICS, APPLIED | Riemann-Hilbert problem | SOLITON-SOLUTIONS | EVOLUTION-EQUATIONS | NONLINEAR SCHRODINGER-EQUATION | KDV

Initial-boundary value problem | Unified transform method | Two-component mKdV equation | Riemann–Hilbert problem | MATHEMATICS, APPLIED | Riemann-Hilbert problem | SOLITON-SOLUTIONS | EVOLUTION-EQUATIONS | NONLINEAR SCHRODINGER-EQUATION | KDV

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 01/2017, Volume 262, Issue 1, pp. 506 - 558

Boundary value problems for integrable nonlinear differential equations can be analyzed via the Fokas method. In this paper, this method is employed in order...

Integrable system | Initial–boundary value problem | Dirichlet-to-Neumann map | Riemann–Hilbert problem | Coupled nonlinear Schrödinger equation | MATHEMATICS | Riemann-Hilbert problem | Coupled nonlinear Schrodinger equation | INTEGRABILITY | UNIFIED TRANSFORM METHOD | LONG-TIME ASYMPTOTICS | Initial-boundary value problem | EVOLUTION-EQUATIONS | Analysis | Methods | Differential equations

Integrable system | Initial–boundary value problem | Dirichlet-to-Neumann map | Riemann–Hilbert problem | Coupled nonlinear Schrödinger equation | MATHEMATICS | Riemann-Hilbert problem | Coupled nonlinear Schrodinger equation | INTEGRABILITY | UNIFIED TRANSFORM METHOD | LONG-TIME ASYMPTOTICS | Initial-boundary value problem | EVOLUTION-EQUATIONS | Analysis | Methods | Differential equations

Journal Article

2016, Graduate studies in mathematics, ISBN 1470430959, Volume 177, xxiii, 275 pages

Book

Applied Mathematics Letters, ISSN 0893-9659, 09/2019, Volume 95, pp. 65 - 71

The matrix Riemann–Hilbert problem of the Hirota equation with non-zero boundary conditions is investigated. Based on the matrix Riemann–Hilbert problem, the...

Darboux transform | Rogue waves | Hirota equation | Non-zero boundary conditions | Riemann–Hilbert problem | Breathers | MATHEMATICS, APPLIED | Riemann-Hilbert problem | Water waves

Darboux transform | Rogue waves | Hirota equation | Non-zero boundary conditions | Riemann–Hilbert problem | Breathers | MATHEMATICS, APPLIED | Riemann-Hilbert problem | Water waves

Journal Article

2016, Graduate studies in mathematics, ISBN 9780821848418, Volume 172, xi, 461

Random matrices (probabilistic aspects; for algebraic aspects see 15B52) | Equations of mathematical physics and other areas of application | Partial differential equations | Approximations and expansions | Probability theory and stochastic processes | Special matrices | Operator theory | Probability theory on algebraic and topological structures | Riemann-Hilbert problems | Exact enumeration problems, generating functions | Convex and discrete geometry | Special classes of linear operators | Combinatorics | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) | Time-dependent statistical mechanics (dynamic and nonequilibrium) | Enumerative combinatorics | Exactly solvable dynamic models | Linear and multilinear algebra; matrix theory | Special processes | Statistical mechanics, structure of matter | Toeplitz operators, Hankel operators, Wiener-Hopf operators | Tilings in $2$ dimensions | Interacting random processes; statistical mechanics type models; percolation theory | Discrete geometry | Random matrices | Combinatorial analysis

Book

Applied Mathematics Letters, ISSN 0893-9659, 01/2019, Volume 87, pp. 57 - 63

The solution of the initial value problem (IVP) for the Fokas–Lenells equation (FLE) was constructed in terms of the solution M(x,t,k) of a 2 × 2 matrix...

Fokas–Lenells equation | Initial value problem | Negative order Lax pair | Riemann–Hilbert problem | MATHEMATICS, APPLIED | Riemann-Hilbert problem | SOLITONS | Fokas-Lenells equation

Fokas–Lenells equation | Initial value problem | Negative order Lax pair | Riemann–Hilbert problem | MATHEMATICS, APPLIED | Riemann-Hilbert problem | SOLITONS | Fokas-Lenells equation

Journal Article

7.
Boundary value problems for analytic and harmonic functions in nonstandard banach function spaces

2012, ISBN 1619423014

Book

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, 11/2016, Volume 472, Issue 2195, p. 20160588

In this paper, we implement the Fokas method to study initial-boundary value problems of the mixed coupled nonlinear Schrdinger equation formulated on the...

Mixed coupled nonlinear Schrödinger equation | Riemann-Hilbert problem | Initial-boundary value problem | mixed coupled nonlinear Schrodinger equation | MULTIDISCIPLINARY SCIENCES | BOUNDARY-VALUE-PROBLEMS | ASYMPTOTICS | EVOLUTION-EQUATIONS | initial-boundary value problem | 1008 | mixed coupled nonlinear Schrödinger equation | 120 | Riemann–Hilbert problem

Mixed coupled nonlinear Schrödinger equation | Riemann-Hilbert problem | Initial-boundary value problem | mixed coupled nonlinear Schrodinger equation | MULTIDISCIPLINARY SCIENCES | BOUNDARY-VALUE-PROBLEMS | ASYMPTOTICS | EVOLUTION-EQUATIONS | initial-boundary value problem | 1008 | mixed coupled nonlinear Schrödinger equation | 120 | Riemann–Hilbert problem

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 08/2015, Volume 259, Issue 3, pp. 1098 - 1148

We use the Deift–Zhou method to obtain, in the solitonless sector, the leading order asymptotic of the solution to the Cauchy problem of the Fokas–Lenells...

Deift–Zhou method | Fokas–Lenells equation | Initial value problem | Riemann–Hilbert problem | Riemann-Hilbert problem | Fokas-Lenells equation | Deift-Zhou method | MATHEMATICS | NONLINEAR SCHRODINGER-EQUATION

Deift–Zhou method | Fokas–Lenells equation | Initial value problem | Riemann–Hilbert problem | Riemann-Hilbert problem | Fokas-Lenells equation | Deift-Zhou method | MATHEMATICS | NONLINEAR SCHRODINGER-EQUATION

Journal Article

Physica D: Nonlinear Phenomena, ISSN 0167-2789, 10/2016, Volume 333, pp. 171 - 184

The main goal of this paper is to put together: a) the Whitham theory applicable to slowly modulated N-phase nonlinear wave solutions to the focusing nonlinear...

Semiclassical limit of the focusing NLS | Whitham equations | Riemann–Hilbert Problem approach | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | PHYSICS, MULTIDISCIPLINARY | PHYSICS, FLUIDS & PLASMAS | Riemann-Hilbert Problem approach | SMALL DISPERSION LIMIT | PHYSICS, MATHEMATICAL | Physics - Exactly Solvable and Integrable Systems

Semiclassical limit of the focusing NLS | Whitham equations | Riemann–Hilbert Problem approach | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | PHYSICS, MULTIDISCIPLINARY | PHYSICS, FLUIDS & PLASMAS | Riemann-Hilbert Problem approach | SMALL DISPERSION LIMIT | PHYSICS, MATHEMATICAL | Physics - Exactly Solvable and Integrable Systems

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 02/2018, Volume 76, pp. 81 - 89

We present a Riemann–Hilbert problem formalism for the initial value problem of the Kundu–Eckhaus equation on the line. The long-time asymptotic for the...

Deift–Zhou nonlinear steepest descent method | Long-time asymptotic | Kundu–Eckhaus equation | Initial value problem | Riemann–Hilbert problem | Deift-Zhou nonlinear steepest descent method | MATHEMATICS, APPLIED | Riemann-Hilbert problem | SOLITONS | Kundu-Eckhaus equation | NONLINEAR SCHRODINGER-EQUATION

Deift–Zhou nonlinear steepest descent method | Long-time asymptotic | Kundu–Eckhaus equation | Initial value problem | Riemann–Hilbert problem | Deift-Zhou nonlinear steepest descent method | MATHEMATICS, APPLIED | Riemann-Hilbert problem | SOLITONS | Kundu-Eckhaus equation | NONLINEAR SCHRODINGER-EQUATION

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 04/2017, Volume 448, Issue 2, pp. 864 - 884

A complex harmonic function of finite Dirichlet energy on a Jordan domain has boundary values in a certain conformally invariant sense, by a construction of H....

Dirichlet space | Quasidisks | Limiting Cauchy integral | Harmonic reflection | Riemann–Hilbert problem | MATHEMATICS | MATHEMATICS, APPLIED | Riemann-Hilbert problem | THEOREM | Naturvetenskap | Matematisk analys | Mathematics | Natural Sciences | Matematik | Mathematical Analysis

Dirichlet space | Quasidisks | Limiting Cauchy integral | Harmonic reflection | Riemann–Hilbert problem | MATHEMATICS | MATHEMATICS, APPLIED | Riemann-Hilbert problem | THEOREM | Naturvetenskap | Matematisk analys | Mathematics | Natural Sciences | Matematik | Mathematical Analysis

Journal Article

13.
Full Text
On zeros of the scattering data of the initial value problem for the short pulse equation

Applied Mathematics Letters, ISSN 0893-9659, 08/2019, Volume 94, pp. 8 - 12

The short pulse equation is an important integrable equation, which can be solved by the so-called inverse scattering method with the help of a...

Short pulse equation | Linear initial value | Box-like initial value | Riemann–Hilbert problem | MATHEMATICS, APPLIED | Riemann-Hilbert problem | LONG-TIME ASYMPTOTICS | RIEMANN-HILBERT PROBLEMS

Short pulse equation | Linear initial value | Box-like initial value | Riemann–Hilbert problem | MATHEMATICS, APPLIED | Riemann-Hilbert problem | LONG-TIME ASYMPTOTICS | RIEMANN-HILBERT PROBLEMS

Journal Article

1994, Aspects of mathematics. E, ISBN 9783528064969, Volume 22., ix, 190

Book

Physica D: Nonlinear Phenomena, ISSN 0167-2789, 02/2018, Volume 364, pp. 27 - 61

We employ the Ablowitz–Ladik system as an illustrative example in order to demonstrate how to analyze initial–boundary value problems for integrable nonlinear...

Integrable system | Initial–boundary value problem | Riemann–Hilbert problem | MATHEMATICS, APPLIED | PHYSICS, MULTIDISCIPLINARY | PDES | EVOLUTION-EQUATIONS | NONLINEAR SCHRODINGER-EQUATION | PHYSICS, MATHEMATICAL | Riemann-Hilbert problem | HALF-LINE | Initial-boundary value problem | DIRICHLET | DIFFERENTIAL-DIFFERENCE EQUATIONS | X-3 LAX PAIRS | HIERARCHY | Physics - Exactly Solvable and Integrable Systems

Integrable system | Initial–boundary value problem | Riemann–Hilbert problem | MATHEMATICS, APPLIED | PHYSICS, MULTIDISCIPLINARY | PDES | EVOLUTION-EQUATIONS | NONLINEAR SCHRODINGER-EQUATION | PHYSICS, MATHEMATICAL | Riemann-Hilbert problem | HALF-LINE | Initial-boundary value problem | DIRICHLET | DIFFERENTIAL-DIFFERENCE EQUATIONS | X-3 LAX PAIRS | HIERARCHY | Physics - Exactly Solvable and Integrable Systems

Journal Article

Zeitschrift für Naturforschung A, ISSN 0932-0784, 01/2019, Volume 74, Issue 2, pp. 139 - 145

Under investigation in this article is the integrable spin-1 Gross–Pitaevskii (SGP) equations, which can be used to describe light transmission in bimodal...

Riemann–Hilbert Problem (RHP) | Inverse Scattering Transform | 02.30.Ik | Soliton Solution | 05.45.Yv | Integrable Spin-1 Gross–Pitaevskii (SGP) Equations | 02.30.Jr | 04.20.Jb | Integrable Spin-1 Gross-Pitaevskii (SGP) Equations | Riemann-Hilbert Problem (RHP) | PHYSICS, MULTIDISCIPLINARY | CHEMISTRY, PHYSICAL | NONLINEAR SCHRODINGER-EQUATION | N-SOLITON SOLUTIONS

Riemann–Hilbert Problem (RHP) | Inverse Scattering Transform | 02.30.Ik | Soliton Solution | 05.45.Yv | Integrable Spin-1 Gross–Pitaevskii (SGP) Equations | 02.30.Jr | 04.20.Jb | Integrable Spin-1 Gross-Pitaevskii (SGP) Equations | Riemann-Hilbert Problem (RHP) | PHYSICS, MULTIDISCIPLINARY | CHEMISTRY, PHYSICAL | NONLINEAR SCHRODINGER-EQUATION | N-SOLITON SOLUTIONS

Journal Article

1994, Operator theory, advances and applications, ISBN 3764329998, Volume 67, xi, 252

Book

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 11/2016, Volume 443, Issue 2, pp. 797 - 816

In this paper a theory is developed for obtaining families of solutions to the KdV equation by formulating a Riemann–Hilbert problem with an appropriate shift....

Factorization | Integrable systems | Riemann–Hilbert problems | Riemann-Hilbert problems | MATHEMATICS | MATHEMATICS, APPLIED | DE-VRIES EQUATION | Analysis | Numerical analysis

Factorization | Integrable systems | Riemann–Hilbert problems | Riemann-Hilbert problems | MATHEMATICS | MATHEMATICS, APPLIED | DE-VRIES EQUATION | Analysis | Numerical analysis

Journal Article

19.
Full Text
Asymptotics for the Sasa–Satsuma equation in terms of a modified Painlevé II transcendent

Journal of Differential Equations, ISSN 0022-0396, 06/2020, Volume 268, Issue 12, pp. 7480 - 7504

We consider the initial-value problem for the Sasa–Satsuma equation on the line with decaying initial data. Using a Riemann–Hilbert formulation and steepest...

Sasa–Satsuma equation | Asymptotics | Initial value problem | Riemann–Hilbert problem | MATHEMATICS | Riemann-Hilbert problem | STEEPEST DESCENT METHOD | RIEMANN-HILBERT PROBLEMS | Sasa-Satsuma equation

Sasa–Satsuma equation | Asymptotics | Initial value problem | Riemann–Hilbert problem | MATHEMATICS | Riemann-Hilbert problem | STEEPEST DESCENT METHOD | RIEMANN-HILBERT PROBLEMS | Sasa-Satsuma equation

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2019, Volume 2019, Issue 1, pp. 1 - 19

In this paper, we study methods of solution for some kinds of convolution type singular integral equations with Cauchy kernel. By means of the classical...

Integral operators | Singular integral equations | Analysis | 45E10 | Convolution type | Mathematics, general | Mathematics | Applications of Mathematics | 45E05 | 30E25 | Riemann–Hilbert problems | MATHEMATICS | MATHEMATICS, APPLIED | Riemann-Hilbert problems | CAUCHY KERNEL | Boundary value problems | Convolution | Analytic functions | Mathematical analysis | Integral equations

Integral operators | Singular integral equations | Analysis | 45E10 | Convolution type | Mathematics, general | Mathematics | Applications of Mathematics | 45E05 | 30E25 | Riemann–Hilbert problems | MATHEMATICS | MATHEMATICS, APPLIED | Riemann-Hilbert problems | CAUCHY KERNEL | Boundary value problems | Convolution | Analytic functions | Mathematical analysis | Integral equations

Journal Article

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