Existence and regularity for an energy maximization problem in two dimensions

Kamvissis, Spyridon; Rakhmanov, Evguenii Andreevich

doi:10.1063/1.1985069

Cited by 31 publications

(59 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In [18] we have studied the energy equilibrium problem that underlies the function g appearing in the change of variables of Chapter 4 in [17] which expresses the finite gap ansatz. We have been able to show that the equilibrium measure µ exists for a particular contour and hence that the right g exists so long as the support of µ does not touch the segment [0, iA] at more than a finite number of points.…”

Section: Application Of the Exact Wkb Results To The Focusing Nonlinementioning

confidence: 99%

See 1 more Smart Citation

Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential

Hatzizisis

Kamvissis

2020

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

In this paper we examine the semiclassical behaviour of the scattering data of a non-self-adjoint Dirac operator with analytic potential decaying at infinity. In particular, employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the reflection coefficient and the Bohr-Sommerfeld condition for the location of the eigenvalues. Our analysis has some interesting consequences concerning the focusing cubic NLS equation, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator is the basis of the solution of the NLS equation via inverse scattering theory.

show abstract

Section: Application Of the Exact Wkb Results To The Focusing Nonlinementioning

confidence: 99%

“…where the scalar functions w ± n are defined inductively by (18) w ± −1 ≡ 0, w ± 0 ≡ 1, and for n ≥ 1,…”

Section: Exact Wkb Methods For the Zakharov-shabat Systemmentioning

confidence: 99%

Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential

Hatzizisis

Kamvissis

2020

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…The obstacle with this approach from the analytical point of view is that it introduces a "free-boundary problem" which is in general hard to pin-down also because the smoothness class of the boundary is not known a priori: around these types of investigations we would like to mention the work [22] relying on the notion of S-curve (equivalent to the "Boutroux" property as defined here), and -possibly closer to the spirit of the present manuscript-the work [24].…”

Section: Introduction and Settingmentioning

confidence: 99%

Boutroux curves with external field: equilibrium measures without a variational problem

Bertola

2011

Anal.Math.Phys.

115

View full text Add to dashboard Cite

The nonlinear steepest descent method for rank-two systems relies on the notion of g-function. The applicability of the method ranges from orthogonal polynomials (and generalizations) to Painlevé transcendents, and integrable wave equations (KdV, NonLinear Schrödinger, etc.). For the case of asymptotics of generalized orthogonal polynomials with respect to varying complex weights we can recast the requirements for the Cauchy-transform of the equilibrium measure into a problem of algebraic geometry and harmonic analysis and completely solve the existence and uniqueness issue without relying on the minimization of a functional. This addresses and solves also the issue of the "free boundary problem", determining implicitly the curves where the zeroes of the orthogonal polynomials accumulate in the limit of large degrees and the support of the measure. The relevance to the quasi-linear Stokes phenomenon for Painlevé equations is indicated. A numerical algorithm to find these curves in some cases is also explained.

show abstract

“…where C(n, 0, t) is real-valued, 13) and the sign has to be chosen in accordance with a q (n, t). Here…”

Section: A Algebro-geometric Quasi-periodic Finite-gap Solutionsmentioning

confidence: 99%

“…As already noted in [4] and numerous publications in the application of the Riemann-Hilbert theory to orthogonal polynomials and random matrices (see [1] for a clear detaied exposition), when one needs to use the full power of the Deift-Zhou method an underlying Lax-Levermore problem has to be understood. In fact, a minimizer (or in some cases a maximinimizer [10,11,13]) has to be constructed! 1 In our case, the Lax-Levermore minimizer lives on a curve in a hyperelliptic surface.…”

Section: The Generalized Toda Shock Problem and The Return Of Laxlevementioning

confidence: 99%

A riemann-hilbert problem in a riemann surface

Kamvissis

2011

Acta Mathematica Scientia

Self Cite

View full text Add to dashboard Cite

A Riemann-Hilbert problem in a Riemann surface. Back in the 1990s several authors have worked on the long time asymptotics of the Toda lattice [2,7,8,19]. Initially the method used was the method of Lax and Levermore [16], reducing the asymptotic problem to the solution of a minimization problem with constraints (an "equilibrium measure" problem). Later, it was found that the asymptotic method of Deift and Zhou (analysis of the associated Riemann-Hilbert factorization problem in the complex plane) could apply to previously intractable problems and also produce more detailed information. Acta Mathematica Scientia, 31B (6).Recently, together with Gerald Teschl, we have revisited the Toda lattice; instead of solutions in a constant or steplike constant background that were considered in the 1990s we have been able to study solutions in a periodic background.Two features are worth noting here. First, the associated Riemann-Hilbert factorization problem naturally lies in a hyperelliptic Riemann surface. We thus generalize the DeiftZhou "nonlinear stationary phase method" to surfaces of nonzero genus. Second, we illustrate the important fact that very often even when applying the powerful Riemann-Hilbert method, a Lax-Levermore problem is still underlying and understanding it is crucial in the analysis and the proofs of the Deift-Zhou method!

show abstract

Existence and regularity for an energy maximization problem in two dimensions

Abstract: 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.

Cited by 31 publications

References 6 publications

Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential

Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential

Boutroux curves with external field: equilibrium measures without a variational problem

A riemann-hilbert problem in a riemann surface

Contact Info

Product

Resources

About