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Soliton theory and the theory of Hankel (and Toeplitz) operators have stayed essentially hermetic to each other. This paper is concerned with linking together these two very active and extremely large theories. On the prototypical example of the Cauchy problem for the Korteweg-de 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 Schrodinger 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.
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In this note we show a comparison principle for nonlinear heat Rockland operators on graded groups. We give a simple proof for it using purely algebraic relations. As an application of the established comparison principle we prove the global in $t$-boundedness of solutions for a class of nonlinear equations for the heat $p$-sub-Laplacian on stratified groups.
We use the theory of functions of noncommuting operators (noncommutative analysis) to solve an asymptotic problem for a partial differential equation and show how, starting from general constructions and operator formulas that seem to be rather abstract from the viewpoint of differential equations, one can end up with very specific, easy-to-evaluate expressions for the solution, useful, e.g., in the tsunami wave problem.
Given two arbitrary sequences $(lambda_j)_{jge 1}$ and $(mu_j)_{jge 1}$ of real numbers satisfying $$|lambda_1|>|mu_1|>|lambda_2|>|mu_2|>...>| lambda_j| >| mu_j| to 0 ,$$ we prove that there exists a unique sequence $c=(c_n)_{ninZ_+}$, real valued, such that the Hankel operators $Gamma_c$ and $Gamma_{tilde c}$ of symbols $c=(c_{n})_{nge 0}$ and $tilde c=(c_{n+1})_{nge 0}$ respectively, are selfadjoint compact operators on $ell^2(Z_+)$ and have the sequences $(lambda_j)_{jge 1}$ and $(mu_j)_{jge 1}$ respectively as non zero eigenvalues. Moreover, we give an explicit formula for $c$ and we describe the kernel of $Gamma_c$ and of $Gamma_{tilde c}$ in terms of the sequences $(lambda_j)_{jge 1}$ and $(mu_j)_{jge 1}$. More generally, given two arbitrary sequences $(rho_j)_{jge 1}$ and $(sigma_j)_{jge 1}$ of positive numbers satisfying $$rho_1>sigma_1>rho_2>sigma_2>...> rho_j> sigma_j to 0 ,$$ we describe the set of sequences $c=(c_n)_{ninZ_+}$ of complex numbers such that the Hankel operators $Gamma_c$ and $Gamma_{tilde c}$ are compact on $ell ^2(Z_+)$ and have sequences $(rho_j)_{jge 1}$ and $(sigma_j)_{jge 1}$ respectively as non zero singular values.
In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces $A^2_{omega _varphi}$, where $omega _varphi= e^{-varphi}$ and $varphi$ is a subharmonic function. We consider compact Hankel operators $H_{overline {phi}}$, with anti-analytic symbols ${overline {phi}}$, and give estimates of the trace of $h(|H_{overline phi}|)$ for any convex function $h$. This allows us to give asymptotic estimates of the singular values $(s_n(H_{overline {phi}}))_n$ in terms of decreasing rearrangement of $|phi |/sqrt{Delta varphi}$. For the radial weights, we first prove that the critical decay of $(s_n(H_{overline {phi}}))_n$ is achieved by $(s_n (H_{overline{z}}))_n$. Namely, we establish that if $s_n(H_{overline {phi}})= o (s_n(H_{overline {z}}))$, then $H_{overline {phi}} = 0$. Then, we show that if $Delta varphi (z) asymp frac{1}{(1-|z|^2)^{2+beta}}$ with $beta geq 0$, then $s_n(H_{overline {phi}}) = O(s_n(H_{overline {z}}))$ if and only if $phi $ belongs to the Hardy space $H^p$, where $p= frac{2(1+beta)}{2+beta}$. Finally, we compute the asymptotics of $s_n(H_{overline {phi}})$ whenever $ phi in H^{p }$.
Formulas relating Poincare-Steklov operators for Schroedinger equations related by Darboux-Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary.
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