The nonlinear Schrödinger equation on the half-line

Fokas, A. S.; Himonas, A. Alexandrou; Mantzavinos, Dionyssios

doi:10.1090/tran/6734

Cited by 120 publications

(198 citation statements)

References 29 publications

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“…Under the assumption of infinite compatibility of the initial and boundary data at the origin, wellposedness for (2.1) in the quarter plane is proved in reference [9] in the smooth setting. However, it is possible to prove analogous results under weaker assumptions [10]. …”

Section: Definition 21 a Solution Of The Free Schrödinger Equation mentioning

confidence: 87%

The nonlinear Schrödinger equation with t -periodic data: II. Perturbative results

Lenells

Fokas

2015

Proc. R. Soc. A.

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We consider the nonlinear Schrödinger equation on the half-line with a given Dirichlet boundary datum which for large t tends to a periodic function. We assume that this function is sufficiently small, namely that it can be expressed in the form αg b 0 (t), where α is a small constant. Assuming that the Neumann boundary value tends for large t to the periodic function g b 1 (t), we show that g b 1 (t) can be expressed in terms of a perturbation series in α which can be constructed explicitly to any desired order. As an illustration, we compute g b 1 (t) to order α 8 for the particular case that g b 0 (t) is the sum of two exponentials. We also show that there exist particular functions g b 0 (t) for which the above series can be summed up, and therefore, for these functions, g b 1 (t) can be obtained in closed form. The simplest such function is exp(iωt), where ω is a real constant.

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Section: Definition 21 a Solution Of The Free Schrödinger Equation mentioning

confidence: 87%

The nonlinear Schrödinger equation with t -periodic data: II. Perturbative results

Lenells

Fokas

2015

Proc. R. Soc. A.

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“…The case when a(k) and d(k) possess a finite number of distinct zeros, can be treated by augmenting the RH problem with a set of residue conditions [25]. The possible soliton contributions generated by these residues can easily be computed and added to the formula for the long-time asymptotics; the steps are similar to the analogous procedure for the NLS equation (see Appendix B of [13]). …”

Section: Preliminariesmentioning

confidence: 99%

Long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line

Arruda

Lenells

2017

Nonlinearity

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“…These arguments 5 suggest making a further modification of the first step of the proposed iteration algorithm: we will reformulate the inverse problem by:…”

Section: Formulation Of the Inverse Problemmentioning

confidence: 99%

Initial‐Boundary Value Problems for the Defocusing Nonlinear Schrödinger Equation in the Semiclassical Limit

Miller

Qin

2015

Stud Appl Math

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Initial-boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so-called unified transform techniques. The main obstruction to applying these methods in practice is that calculation of the spectral transforms of the initial and boundary data requires knowledge of too many boundary conditions, more than are required to make the problem well-posed. The elimination of the unknown boundary values is frequently addressed in the spectral domain via the so-called global relation, and types of boundary conditions for which the global relation can be solved are called linearizable. For the defocusing nonlinear Schrödinger equation, the global relation is only known to be explicitly solvable in rather restrictive situations, namely homogeneous boundary conditions of Dirichlet, Neumann, and Robin (mixed) type. General nonhomogeneous boundary conditions are not known to be linearizable. In this paper, we propose an explicit approximation for the nonlinear Dirichlet-to-Neumann map supplied by the defocusing nonlinear Schrödinger equation and use it to provide approximate solutions of general nonhomogeneous boundary value problems for this equation posed as an initial-boundary value problem on the half-line. Our method sidesteps entirely the solution of the global relation. The accuracy of our method is proven in the semiclassical limit, and we provide explicit asymptotics for the solution in the interior of the quarter-plane space-time domain.

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The nonlinear Schrödinger equation on the half-line

Cited by 120 publications

References 29 publications

The nonlinear Schrödinger equation with t -periodic data: II. Perturbative results

The nonlinear Schrödinger equation with t -periodic data: II. Perturbative results

Long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line

Initial‐Boundary Value Problems for the Defocusing Nonlinear Schrödinger Equation in the Semiclassical Limit

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