A new model of long wave-short wave interaction generalising the Yajima-Oikawa and Newell systems: integrability and linear stability spectra

Caso Huerta, Marcos (2022) A new model of long wave-short wave interaction generalising the Yajima-Oikawa and Newell systems: integrability and linear stability spectra. Doctoral thesis, Northumbria University.

Preview

Text (Doctoral thesis) caso.huerta.marcos_phd(19022477).pdf - Submitted Version Download (5MB) | Preview

Abstract

In this thesis we consider a recent model for resonant interaction between long and short waves that we proposed unifying and generalising those first proposed by Yajima and Oikawa and by Newell, which we call Yajima-Oikawa-Newell (YON) system, which has the remarkable property of remaining integrable for any choice of the two arbitrary, non rescalable parameters it features. Long wave-short wave systems, which model the propagation of short waves that generate waves of much longer wavelength, appear in many physical settings, especially in fluid dynamics and plasma physics. Throughout the thesis, we introduce this new system through mathematical means and by recalling the physical origin of the Yajima-Oikawa system, and employ various techniques, both general and pertaining to the theory of integrable systems, to study it. In particular, we obtain several types of solutions, including bright and dark solitons, periodic solutions, breathers and rational solutions, by means of a general Ansatz approach and by using Hirota bilinearisation techniques (namely, the theory of τ -functions, which allows us to relate the system with the Kadomtsev-Petviashvili equation), with which we were able to derive the general N-soliton solution both on a zero and non-zero background, and the phase shift corresponding to the collision of two solitons (which, remarkably, only depends on the wave numbers of the two solitons).

Even though a physical derivation of the whole YON system is not available at this point of the research and remains an open problem, the fact that the Yajima-Oikawa system, in itself a subcase of the YON system, can be physically derived encourages us to try to obtain the whole system as a reduction of a physical model of resonantly interacting long and short waves. In case it can be derived in a physical context, the fact that the system features free parameters might be useful to better model and assist experiments.

Furthermore, we introduce a recent technique for the study of stability of solutions of integrable systems, proposed by Degasperis, Lombardo & Sommacal (2018), which makes use of the Lax pair associated to the system to perform a linear stability analysis of the solution by introducing a new object that we refer to as the stability spectrum, defined as an algebraic/topological structure in the complex plane. The geometric properties of this spectrum are linked to the stability or instability of the given solution, which allows us to provide a full classification of the stability behaviours in the parameter space. In the thesis we employ this technique to study the stability of the plane waves of the YON system. We also provide a few conjectures relating the topology of the stability spectrum and the existence of special kinds of functions, namely dark solitons and rational solutions. These predictions are indeed true for the YON system, as checked with the solutions derived via Hirota bilinearisation.

Actions (login required)

View Item

Downloads