Soliton theory and the theory of Hankel (and Toeplitz) operators have stayed essentially hermetic to each other. This paper is concerned with linking these two very active and extremely large theories. On the prototypical example of the Cauchy problem for the Kortewegde Vries (KdV) equation we demonstrate the power of the language of Hankel operators in which symbols are conveniently represented in terms of the scattering data for the Schrödinger operator associated with the initial data for the KdV equation. This approach yields short-cuts to already known results as well as to a variety of new ones (e.g., wellposedness beyond standard assumptions on the initial data) which are achieved by employing some subtle results for Hankel operators.
Preface.As is well known, many problems in soliton theory (completely integrable systems) can be formulated in terms of Riemann-Hilbert boundary problems. This has been used (explicitly or implicitly) since the early 1980s. On the other hand, it is also well known that the Riemann-Hilbert problem is closely related to the theory of Hankel and Toeplitz operators. Moreover, since the 1960s (and implicitly even earlier) the former has stimulated the latter. But, surprisingly enough, while having experienced a boom at the same time, soliton theory and the theory of Hankel and Toeplitz operators have not shown much direct interaction. It is our main goal to demonstrate that the language of Hankel/Toeplitz operators is very natural for soliton theory, providing a direct access to the wealth of deep results on Hankel and Toeplitz operators. That is why it is also our goal to capture the attention of the Hankel/Toeplitz operator community to applications of Hankel operators to integrable systems, which in turn may have a stimulating influence on the theory of Hankel and Toeplitz operators.Our new results are of an optimal nature and are based on some fine properties of Hankel operators. We do not believe they can be obtained as effectively by any other method.The purpose of reaching out to both soliton and Hankel/Toeplitz operator communities determines the style of the paper: maximally self-contained exposition with recall of basic definitions and auxiliary results.We concentrate solely on the Korteweg-de Vries (KdV) case, but it should be quite clear to anyone familiar with the area that our approach by no means is restricted to this case. Moreover, we believe that the interplay between soliton theory and Hankel *
We present an exact analytical solution of the nonlinear shallow water theory for wave run-up in inclined channels of arbitrary cross-section, which generalizes previous studies on wave run-up for a plane beach and channels of parabolic cross-section. The solution is found using a hodograph-type transform, which extends the well-known Carrier-Greenspan transform for wave run-up on a plane beach. As a result, the nonlinear shallow water equations are reduced to a single one-dimensional linear wave equation for an auxiliary function and all physical variables can be expressed in terms of this function by purely algebraic formulas. In the special case of a U-shaped channel this equation coincides with a spherically symmetric wave equation in space, whose dimension is defined by the channel cross-section and can be fractional. As an example, the run-up of a sinusoidal wave on a beach is considered for channels of several different cross-sections and the influence of the cross-section on wave run-up characteristics is studied.
ABSTRACT. We are concerned with the Cauchy problem for the KdV equation on the whole line with an initial profile V 0 which is decaying sufficiently fast at +∞ and arbitrarily enough (i.e., no decay or pattern of behavior) at −∞. We show that this system is completely integrable in a very strong sense. Namely, the solution V (x, t) admits the Hirota τ-function representationwhere M x,t is a Hankel integral operator constucted from certain scattering and spectral data suitably defined in terms of the Titchmarsh-Weyl m-functions associated with the two half-line Schrödinger operators corresponding to V 0 . We show that V (x, t) is real meromorphic with respect to x for any t > 0. We also show that under a very mild additional condition on V 0 representation (0.1) implies a strong well-posedness of the KdV equation with such V 0 's. Among others, our approach yields some relevant results due to Cohen, Kappeler, Khruslov, Kotlyarov, Venakides, Zhang and others.
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