Spectral inverse problems for compact Hankel operators

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.

Effective integrable dynamics for some nonlinear wave equation

We consider the following degenerate half wave equation on the one dimensional torus $$quad ipartial_t u-|D|u=|u|^2u, ; u(0,cdot)=u_0. $$ We show that, on a large time interval, the solution may be approximated by the solution of a completely integrable system-- the cubic Szego equation. As a consequence, we prove an instability result for large $H^s$ norms of solutions of this wave equation.