Abstract : Recent advances in the theory of nonlinear waves have allowed a better understanding of the underlying mechanisms leading to the formation of space-time localised extreme waves, often referred in the literature as rogue waves, in systems modelled by nonlinear PDEs of integrable and non-integrable type. Many theoretical questions remain open as for a qualitative and quantitative description of the evolution of a localised or periodic perturbation on a given background. The aim of this minisymposium is to gather world-leading experts in the field to discuss the most recent results about the onset and recurrence of rogue waves in nonlinear media.
Organizer(s) : Prof Sara Lombardo (Heriot-Watt University, UK), Dr Matteo Sommacal (Northumbria University, UK)
[03779] Maximal Amplitudes of N-Phase Solutions of a Modified NLS Equation
Format : Online Talk on Zoom
Author(s) :
Otis Wright (Cedarville University)
Abstract : An effective method for finding the maximal amplitudes of N-phase solutions of a modified nonlinear Schrödinger equation is discussed.
[05214] The effects of damping on rogue wave formation and permanent downshifting
Format : Online Talk on Zoom
Author(s) :
Cosntance Schober (University of Central Florida)
Annalisa Maria Calini (College of Charleston )
Abstract : The effects of damping on the B-F instability, rogue wave formation, and permanent downshifting are discussed
in the framework of the viscous damped
higher order nonlinear Schrodinger (v-HONLS) equation. The linear stability analysis of the damped Stokes wave
solution is presented.
Numerical simulations of the v-HONLS with unstable Stokes wave initial data indicate the
inclusion of viscosity enables permanent downshifting and rogue waves typically
do not develop after the time of permanent downshift.
[04450] Rogue-wave formation scenarios for focusing NLS with parabolic initial data
Format : Talk at Waseda University
Author(s) :
Francesco Demontis (University of Cagliari)
Giovanni Ortenzi (University of Torino)
Giacomo Roberti (Northumbria University)
Matteo Sommacal (Northumbria University)
Abstract : We study focussing NLS for compactly-supported parabolic initial data with constant chirp. In the absence of dispersion, we provide a criterion for blow-up, generalising a result by Talanov et al. In the presence of dispersion, the same criterion determines, even beyond the semi-classical regime, the formation of rogue-waves, whose onset time is predicted by the corresponding dispersionless catastrophe time. Numerics suggest that the chirp controls the prevailing scenario among two competing mechanisms for rogue-wave formation.
[04494] Stability of plane waves for the Yajima-Oikawa-Newell equation
Format : Talk at Waseda University
Author(s) :
Marcos Caso-Huerta (Northumbria University)
Abstract : A new, integrable long wave-short wave model is proposed, encompassing Yajima-Oikawa and Newell systems as particular choices of the coefficients. The stability of its plane waves is studied in an algebraic-geometric approach making use of its Lax pair. The stability spectra are explicitly computed, leading to identifying a relation between the topology of the spectra and the gain of the system. This allows one to predict regions of existence for rogue wave type solutions.
Paolo Maria Santini (Dept. of Physics, University "La Sapienza")
Petr Grinevich (Steklov Math. Institute, Moscow)
Francesco Coppini (Dept. of Physics, University "La Sapienza")
Abstract : We summarize recent results on the theory of rogue waves. 1) Relevant exact rogue wave (RW) solutions of integrable continuous, discrete, and relativistic NLS type field theories in 1+1 and/or 2+1 dimensions. 2) Analytic description of the recurrence of RWs. 3) Stability properties of exact RW solutions. 4) The effect of perturbations of the model on the RW dynamics.
[04499] Finite-gap approach to the Davey-Stewardson-2 rogue waves
Format : Online Talk on Zoom
Author(s) :
Petr G. Grinevich (Steklov Mathematical Institute, RAS)
Paolo Maria Santini (University Roma-1 "La Sapienza", INFT)
Abstract : In a recent series of paper we showed that for the 1+1 dimensional soliton systems the finite-gap formulas for solutions describing the generation of rogue waves can be essentially simplified in the leading order. The origin of this simplification is that the Cauchy problem for rogue waves naturally contains a small parameter therefore the spectral curve is a small perturbation of a rational one.
We show that this approach can be naturally extended to the focusing Davey-Stewardson 2 equation, which is 2+1 integrable model admitting rogue waves type solutions. Again, in the leading order we obtain elementary formulas for the rogue waves Cauchy problem.
[04876] Non-commutative soliton equations: some solutions of matrix mKdV equation
Abstract : Solutions of matrix mKdV equation are presented. They can be termed soliton solutions since they exhibit
the typical behaviour of solitons. The asymptotics of 2-soliton solutions of the d x d-matrix modified
Korteweg de-Vries equation is given under the assumption that the involved spectral matrices are
invertible. This work is motivated by explicit solutions by the authors, and bases on an explicit solution
formula for the N-soliton solutions previously obtained.
[05198] Spectral approaches to wave instability
Format : Talk at Waseda University
Author(s) :
Sara Lombardo (Heriot-Watt University)
Abstract : Recent spectral techniques to study the stability of nonlinear waves will be reviewed in connection with more established results, with a particular focus on the case of nonlinear evolution equations of integrable type.
