Nonlinear Steepest Descent Method Research Articles

Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data: Announcement of Results

We consider the one-dimensional focusing nonlinear Schr\"odinger equation (NLS) with a delta potential and even initial data. The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin boundary conditions at the origin. We follow the method of Bikbaev and Tarasov which utilizes a B\"acklund transformation to extend the solution on the half-line to a solution of the NLS equation on the whole line. We study the asymptotic stability of the stationary 1-soliton solution of the equation under perturbation by applying the nonlinear steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou. Our work strengthens, and extends, earlier work on the problem by Holmer and Zworski.

Initial-Boundary Value Problem for the Camassa–Holm Equation with Linearizable Boundary Condition

We present a Riemann–Hilbert problem formalism for the initial boundary value problem for the Camassa–Holm equation on the half-line x > 0 with homogeneous Dirichlet boundary condition at x = 0. We show that, similarly to the problem on the whole line, the solution of this problem can be obtained in parametric form via the solution of a Riemann–Hilbert problem determined by the initial data via associated spectral functions. This allows us to apply the non-linear steepest descent method and to describe the large-time asymptotics of the solution.

Nonlinear Steepest Descent Asymptotics for Semiclassical Limit of Integrable Systems: Continuation in the Parameter Space

The initial value problem for an integrable system, such as the Nonlinear Schrödinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent ([4,5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface \({\mathcal {R} = \mathcal {R}(x,t)}\) in the spectral variable, where the space-time variables (x, t) play the role of external parameters. The curves in the x, t plane, separating regions of different genuses of \({\mathcal {R}(x,t)}\), are called breaking curves or nonlinear caustics. The genus of \({\mathcal {R}(x,t)}\) is related to the number of oscillatory phases in the asymptotic solution of the integrable system at the point x, t. The evolution theorem ([10]) guarantees continuous evolution of the asymptotic solution in the space-time away from the breaking curves.In the case of the analytic scattering data f(z; x, t) (in the NLS case, f is a normalized logarithm of the reflection coefficient with time evolution included), the primary role in the breaking mechanism is played by a phase function \({{\Im\,h(z;x,t)}}\), which is closely related to the g function. Namely, a break can be caused ([10]) either through the change of topology of zero level curves of \({\Im\,h(z;x,t)}\) (regular break), or through the interaction of zero level curves of \({{\Im\,h(z;x,t)}}\) with singularities of f (singular break). Every time a breaking curve in the x, t plane is reached, one has to prove the validity of the nonlinear steepest descent asymptotics in the region across the curve.In this paper we prove that in the case of a regular break, the nonlinear steepest descent asymptotics can be “automatically” continued through the breaking curve (however, the expressions for the asymptotic solution will be different on the different sides of the curve). Our proof is based on the determinantal formula for h(z; x, t) and its space and time derivatives, obtained in [8,9]. Although the results are stated and proven for the focusing NLS equation, it is clear ([9]) that they can be reformulated for AKNS systems, as well as for the nonlinear steepest descend method in a more general setting.

Universality in the Profile of the Semiclassical Limit Solutions to the Focusing Nonlinear Schrodinger Equation at the First Breaking Curve

We consider the semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schrodinger equation (NLS) with decaying potentials. If a potential is a simple rapidly oscillating wave (the period has the order of the semiclassical parameter e) with modulated amplitude and phase, the space-time plane subdivides into regions of qualitatively different behaviors, with the boundary between them consisting typically of collection of piecewise smooth arcs (breaking curve(s)). In the first region, the evolution of the potential is ruled by modulation equations (Whitham equations), but for every value of the space variable x there is a moment of transition (breaking), where the solution develops fast, quasi-periodic behavior, that is, the amplitude becomes also fastly oscillating at scales of order e. The very first point of such transition is called the point of gradient catastrophe. We study the detailed asymptotic behavior of the left and right edges of the interface between these two regions at any time after the gradient catastrophe. The main finding is that the first oscillations in the amplitude are of nonzero asymptotic size even as e tends to zero, and they display two separate natural scales: of order in the parallel direction to the breaking curve in the (x,t) plane, and of order in a transversal direction. The study is based upon the inverse-scattering method and the nonlinear steepest descent method.

Long-Time Asymptotics for the Korteweg–de Vries Equation via Nonlinear Steepest Descent

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg–de Vries equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction to this method.

Stability of the Periodic Toda Lattice in the Soliton Region

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the periodic (and slightly more generally of the quasi-periodic finite-gap) Toda lattice for decaying initial data in the soliton region. In addition, we show how to reduce the problem in the remaining region to the known case without solitons.

LONG-TIME ASYMPTOTICS OF THE TODA LATTICE FOR DECAYING INITIAL DATA REVISITED

The purpose of this article is to give a streamlined and self-contained treatment of the long-time asymptotics of the Toda lattice for decaying initial data in the soliton and in the similarity region via the method of nonlinear steepest descent.

Long-time Asymptotics for the Camassa–Holm Equation

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Camassa–Holm equation for decaying initial data, completing previous results by Boutet de Monvel and Shepelsky.

Rational Functions with a General Distribution of Poles on the Real Line Orthogonal with Respect to Varying Exponential Weights: I

Orthogonal rational functions are characterized in terms of a family of matrix Riemann–Hilbert problems on ℝ, and a related family of energy minimisation problems is presented. Existence, uniqueness, and regularity properties of the equilibrium measures which solve the energy minimisation problems are established. These measures are used to derive a family of ‘model’ matrix Riemann–Hilbert problems which are amenable to asymptotic analysis via the Deift–Zhou non-linear steepest-descent method.

Long-time asymptotics for the Toda lattice in the soliton region

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice for decaying initial data in the soliton region. In addition, we point out how to reduce the problem in the remaining region to the known case without solitons.

Semiclassical nonlinear Schrödinger on the half line

We are studying the semiclassical limit of the 1+1 dimensional integrable nonlinear Schrödinger equation with defocusing cubic nonlinearity on the half line. Our analysis relies on the recent theory of Fokas et al., which reduces boundary value problems for soliton equations to Riemann–Hilbert factorization problems. We employ the method of nonlinear steepest descent to asymptotically deform the given Riemann–Hilbert problem to an explicilty solvable one.