arXiv subjects

Matteo Sommacal

Publications and source records attributed to Matteo Sommacal.

5 recordsLinked to original sources

arXiv2022

Periodic and solitary wave solutions of the long wave-short wave Yajima-Oikawa-Newell model

Models describing long wave-short wave resonant interactions have many physical applications from fluid dynamics to plasma physics. We consider here the Yajima-Oikawa-Newell (YON) model, which has been recently introduced combining the interaction terms of two long wave-short wave, integrable models, one proposed by Yajima-Oikawa, and the other one by Newell. The new YON model contains two arbitrary coupling constants and it is still integrable - in the sense of possessing a Lax pair - for any values of these coupling constants. It reduces to the Yajima-Oikawa or the Newell systems for special choices of these two parameters. We construct families of periodic and solitary wave solutions, which display the generation of very long waves. We also compute the explicit expression of a number of conservation laws.

nlin.SI↗

arXiv2021

A new integrable model of long wave-short wave interaction and linear stability spectra

We consider the propagation of short waves which generate waves of much longer (infinite) wave-length. Model equations of such long wave-short wave resonant interaction, including integrable ones, are well-known and have received much attention because of their appearance in various physical contexts, particularly fluid dynamics and plasma physics. Here we introduce a new long wave-short wave integrable model which generalises those first proposed by Yajima-Oikawa and by Newell. By means of its associated Lax pair, we carry out the linear stability analysis of its continuous wave solutions by introducing the stability spectrum as an algebraic curve in the complex plane. This is done starting from the construction of the eigenfunctions of the linearised long wave-short wave model equations. The geometrical features of this spectrum are related to the stability/instability properties of the solution under scrutiny. Stability spectra for the plane wave solutions are fully classified in the parameter space together with types of modulational instabilities.

nlin.SI↗

arXiv2018

Integrability and linear stability of nonlinear waves

It is well known that the linear stability of solutions of partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation by using only their associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general $N \times N$ matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schroedinger system and the multi-wave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for $N = 3$ for the particular system of two coupled nonlinear Schroedinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants.

nlin.SI↗

arXiv2011

Understanding complex dynamics by means of an associated Riemann surface

We provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated Riemann surface. Thanks to this geometric description an explicit formula for the period of the orbits can be derived, which is shown to depend on the initial data and the continued fraction expansion of a simple ratio of the coupling constants of the problem. For rational values of this ratio and generic values of the initial data, all orbits are periodic and the system is isochronous. For irrational values of the ratio, there exist periodic and quasi-periodic orbits for different initial data. Moreover, the dependence of the period on the initial data shows a rich behavior and initial data can always be found such the period is arbitrarily high.

nlin.CD↗

arXiv2007

Solvable Nonlinear Evolution PDEs in Multidimensional Space

A class of solvable (systems of) nonlinear evolution PDEs in multidimensional space is discussed. We focus on a rotation-invariant system of PDEs of Schrödinger type and on a relativistically-invariant system of PDEs of Klein-Gordon type. Isochronous variants of these evolution PDEs are also considered.

nlin.SI↗