The Spectral Problem for the Dispersionless Camassa–Holm Equation

International Mathematics Research Notices

Kostenko

2018

Section: Introductionmentioning

confidence: 96%

The Inverse Spectral Problem for Periodic Conservative Multi-peakon Solutions of the Camassa–Holm Equation

International Mathematics Research Notices

Kostenko

2018

“…And in fact, despite a large amount of articles, very little is known about the inverse problem for (1.2) in the indefinite case and almost the entire literature on this subject restricts to strictly positive and smooth ω (in which case the spectral problem can be transformed into a standard form that is known from the Kortewegde Vries equation [4,58,64]). Apart from the explicitly solvable finite dimensional case [5,33], only insufficient partial uniqueness results [7,9,10,11,32,34,37] have been obtained so far for the inverse problem in the indefinite case.…”

Section: Introductionmentioning

confidence: 99%

The Inverse Spectral Transform for the Conservative Camassa–Holm Flow with Decaying Initial Data

Arch Rational Mech Anal

2016

“…First and foremost, the coupling problem is essentially equivalent to an inverse spectral problem for second order ordinary differential equations or two-dimensional first order systems with trace class resolvents. This circumstance indicates that it is not likely for a simple elementary proof of our theorem to exist, as the uniqueness part allows one to effortlessly deduce (generalizations of) results in [3], [6], [12], [13], [17], which had to be proven in a more cumbersome way before. On the other side, the coupling problem is also of relevance for certain completely integrable nonlinear wave equations (with the Camassa-Holm equation [8], [4] and the Hunter-Saxton equation [19] being the prime examples) when the underlying isospectral problem has purely discrete spectrum.…”

mentioning

confidence: 99%

“…Hence we are able to retrieve the measure ω from the spectrum and the norming constants by means of solving a family of coupling problems. In particular, this guarantees that ω is uniquely determined by the given spectral data, a fact that usually requires considerable effort [5], [6], [11], [14], [22]. More generally, the coupling problem can also be employed to solve analogous inverse spectral problems for indefinite strings as in [15] or canonical systems with two singular endpoints.…”

mentioning

confidence: 99%

Unique Solvability of a Coupling Problem for Entire Functions

2017

Constr Approx

We establish the unique solvability of a coupling problem for entire functions that arises in inverse spectral theory for singular second-order ordinary differential equations/two-dimensional first-order systems and is also of relevance for the integration of certain nonlinear wave equations.Keywords Coupling problem for entire functions · Unique solvability · Inverse spectral theory Mathematics Subject Classification Primary 30D20, 34A55 · Secondary 34B05, 37K15 ResultsLet σ be a discrete set of nonzero real numbers such that the sumCommunicated by Arno Kuijlaars. For a given sequence η ∈R σ (referred to as coupling constants or data), where we denote byR = R∪{∞} the one-point compactification of R, we consider the following task. Coupling problem(G) Growth and positivity condition:Let us first assume that the pair ( − , + ) is a solution of the coupling problem with data η. The growth and positivity condition (G) means that the function Upon invoking the open mapping theorem, this first of all guarantees that all zeros of the functions − and + are real. It furthermore entails that the zeros of the function in the numerator of (3) and the zeros of the function in the denominator of (3) are interlacing (after possible cancelations); see [29, Theorem 27.2.1]. From this we may conclude that the functions − and + are actually of genus zero and satisfy the boundIndeed, this inequality follows essentially from roughly estimating the individual factors in the corresponding Hadamard representation, with the normalization condition (N) taken into account, and employing the interlacing property mentioned above.1 To be precise, this condition has to be read as + (λ) = 0 whenever η(λ) = ∞. We should emphasize here that this upper bound is always independent of the actual coupling constants η. On the other hand, the condition (G) also tells us that the residues of the function in (3) at all poles are negative. In conjunction with the coupling condition (C), this impliesfor all those λ ∈ σ for which the coupling constant η(λ) is finite. Unless it happens that λ is a zero of the function + , this constitutes a necessary restriction on the sign of the coupling constant η(λ) in order for a solution of the coupling problem to exist. Roughly speaking, the coupling constants are expected to have alternating signs beginning with nonnegative ones for those corresponding to the smallest (in modulus) positive and negative element of σ . Motivated by these considerations and the nature of our applications, we introduce the following terminology.Definition Coupling constants η ∈R σ are called admissible if the inequalityholds for all those λ ∈ σ for which η(λ) is finite.The main purpose of the present article is to prove that this simple condition is sufficient to guarantee unique solvability of the corresponding coupling problem. Theorem (Existence and Uniqueness) If the coupling constants η ∈R σ are admissible, then the coupling problem with data η has a unique solution.Apart from this result, we will also establish the fact ...