List of Figures2.1 The complex λ-plane with three eigenvalues λ k in the upper half-plane, their complex conjugates, and the contours C, C * , and domains D, D * . The orientation in the figure is ω = +1. 4.1 Examples of contours C and C * appropriate for studying a semiclassical soliton ensemble in the limit N ↑ ∞.

In the Toda shock problem (see [7], (1 11, [S], and also 131) one considers a driving particle moving with a fixed velocity 2a and impinging on a one-dimensional semi-infinite lattice of particles, initially equally spaced and at rest, and interacting with exponential forces. In this paper we consider the related Toda rarefaction problem in which the driving particle now moves away from the lattice at fixed speed, in analogy with a piston being withdrawn, as it were, from a container filled with gas.We make use of the Riemann-Hilbert factorization formulation of the related inverse scattering problem. In the case where the speed 21al of the driving particle is sufficiently large (la1 > l), we show that the particle escapes from the lattice, which then executes a free motion of the type studied, for example, in [5]. In other words, in analogy with a piston being withdrawn too rapidly from a container filled with gas, cavitation develops.

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We provide rigorous analysis of the long time behavior of the (doubly infinite) Toda lattice under initial data that decay at infinity, in the absence of solitons. We solve (approximately and for large times) the Riemann-Hilbert matrix factorization problem equivalent to the related inverse scattering problem with the help of the Beals-Coifman formula, by reducing it to a simpler one through a series of contour deformations in the spirit of the Deift-Zhou method.

A revision of the last appendix of the paper "Existence and Regularity for an Energy Maximization Problem in Two Dimensions" by S.Kamvissis and E.

Articles you may be interested inThe inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions J. Math. Phys. 55, 101505 (2014); 10.1063/1.4898768Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions We are studying the semiclassical limit of the 1ϩ1 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. I. AN INITIAL-BOUNDARY VALUE PROBLEM FOR THE NONLINEAR SCHRÖ DINGER EQUATIONIn recent years there has been a series of results by Fokas and others on boundary value problems for soliton equations ͑see Ref. 1 for a comprehensive review͒. The Fokas method goes beyond existence and uniqueness. In fact, it reduces such problems to Riemann-Hilbert factorization problems in the complex plane, thus generalizing the existing theory which reduces initial value problems to Riemann-Hilbert problems via the method of inverse scattering. One of the main advantages of the Riemann-Hilbert formulation is that one can use recent powerful results on the asymptotic behavior of solutions to these problems ͑as some parameter goes to infinity͒ to derive asymptotics for the solution of the associated soliton equation. Such methods were pioneered by Its and made rigorous and systematic by Deift and Zhou; the Deift-Zhou method is known as ''nolinear steepest descent'' in analogy with the linear steepest descent method which is applicable to asymptotic problems for Fourier-type integrals ͑see, e.g., Ref. 2͒. A generalization of the steepest descent method developed in Ref. 3 is able to give rigorous results for the so-called ''semiclassical'' or ''zero dispersion'' limit of the solution of the Cauchy problem for 1ϩ1 dimensional integrable evolution equations, in the case where the Lax operator is self-adjoint. The method has been further extended in Ref. 4 for the ''non-self-adjoint'' case, where in fact a ''steepest descent'' contour is, for the first time, introduced and its characterization and computation made systematic.In this paper we consider the most basic example, that is the defocusing nonlinear Schrödinger ͑NLS͒ equation. ͑In a recent paper 5 we dealt with the simple problem of so-called linearizable data, for both the defocusing NLS and Korteweg-de Vries equations.͒ We make use of the recent results of Ref. 6 in order to study the so-called ''semiclassical'' limit of a particular initialboundary value problem. More precisely we consider the 1ϩ1 dimensional, integrable, defocusing, nonlinear Schrödinger equation on the half-linewhere f 0 is assumed to be in the Schwartz space of the positive real line. We also assume that all a͒ Electronic We use the following convention...

We study the zero-dispersion limit for certain initial boundary value problems for the defocusing nonlinear Schrödinger (NLS) equationand for the Korteweg-de Vries (KdV)equation with dominant surface tension. These problems are formulated on the half-line and they involve linearisable boundaryconditions

We consider the effect of real spectral singularities on the long time behavior of the solutions of the focusing Nonlinear Schroedinger equation We find that for each spectral singularity // e IR, such an effect is limited to the region of the (x,t)-plane in which /! is close to the point of stationary phase λo = ( the phase here being defined in a standard way by, say, the evolution of the Jost functions) In that region, the solution performs decaying oscillations of the same form as in the other regions, but with different parameters The order of decay is O((^) 12 )We prove our result by using the Riemann-Hilbert factorization formulation of the inverse scattering problem We recover our asymptotics by transforming our problem to one which is equivalent for large time, and which can be interpreted as the one corresponding to the genus 0 algebro-geometric solution of the equation