arXiv subjects

Giuseppe Orsatti

Publications and source records attributed to Giuseppe Orsatti.

4 recordsLinked to original sources

arXiv2024

$\bar{\partial}$-problem for focusing nonlinear Schrödinger equation and soliton shielding

We consider soliton gas solutions of the Focusing Nonlinear Schrödinger (NLS) equation, where the point spectrum of the Zakharov-Shabat linear operator condensate in a bounded domain $\mathcal{D}$ in the upper half-plane. We show that the corresponding inverse scattering problem can be formulated as a $\overline{\partial}$-problem on the domain. We prove the existence of the solution of this $\overline{\partial}$-problem by showing that the $τ$-function of the problem (a Fredholm determinant) does not vanish. We then represent the solution of the NLS equation via the $τ$ of the $\overline{\partial}$- problem. Finally we show that, when the domain $\mathcal{D}$ is an ellipse and the density of solitons is analytic, the initial datum of the Cauchy problem is asymptotically step-like oscillatory, and it is described by a periodic elliptic function as $x \to - \infty$ while it vanishes exponentially fast as $x \to +\infty$.

math-ph↗

arXiv2023

Integrable operators, $\overline{\partial}$-Problems, KP and NLS hierarchy

We develop the theory of integrable operators $\mathcal{K}$ acting on a domain of the complex plane with smooth boundary in analogy with the theory of integrable operators acting on contours of the complex plane. We show how the resolvent operator is obtained from the solution of a $\overline{\partial}$-problem in the complex plane. When such a $\overline{\partial}$-problem depends on auxiliary parameters we define its Malgrange one form in analogy with the theory of isomonodromic problems. We show that the Malgrange one form is closed and coincides with the exterior logarithmic differential of the Hilbert-Carleman determinant of the operator $\mathcal{K}$. With suitable choices of the setup we show that the Hilbert-Carleman determinant is a $τ$-function of the Kadomtsev-Petviashvili (KP) or nonlinear Schrödinger hierarchies.

math-ph↗

arXiv2023

On the role of the Integrable Toda model in one-dimensional molecular dynamics

We prove that the common Mie-Lennard-Jones (MLJ) molecular potentials, appropriately normalized via an affine transformation, converge, in the limit of hard-core repulsion, to the Toda exponential potential. Correspondingly, any Fermi-Pasta-Ulam (FPU)-like Hamiltonian, with MLJ-type interparticle potential, turns out to be $1/n$-close to the Toda integrable Hamiltonian, $n$ being the exponent ruling repulsion in the MLJ potential. This means that the dynamics of chains of particles interacting through typical molecular potentials, is close to integrable in an unexpected sense. Theoretical results are accompanied by a numerical illustration; numerics shows, in particular, that even the very standard 12--6 MLJ potential is closer to integrability than the FPU potentials which are more commonly used in the literature.

math-ph↗

arXiv2023

Soliton shielding of the focusing Nonlinear Schrödinger Equation

We first consider a deterministic gas of $N$ solitons for the Focusing Nonlinear Schrödinger (FNLS) equation in the limit $N\to\infty$ with a point spectrum chosen to interpolate a given spectral soliton density over a bounded domain of the complex spectral plane. We show that when the domain is a disk and the soliton density is an analytic function, then the corresponding deterministic soliton gas surprisingly yields the one-soliton solution with point spectrum the center of the disk. We call this effect {\it soliton shielding}. We show that this behaviour is robust and survives also for a {\it stochastic} soliton gas: indeed, when the $N$ soliton spectrum is chosen as random variables either uniformly distributed on the circle, or chosen according to the statistics of the eigenvalues of the Ginibre random matrix the phenomenon of soliton shielding persists in the limit $N\to \infty$. When the domain is an ellipse, the soliton shielding reduces the spectral data to the soliton density concentrating between the foci of the ellipse. The physical solution is asymptotically step-like oscillatory, namely, the initial profile is a periodic elliptic function in the negative $x$--direction while it vanishes exponentially fast in the opposite direction.

math-ph↗