Program

Thursday Sept 5

Rodrigo Cienfuegos

Rodrigo Cienfuegos

Nonlinear energy transfers in very shallow coastal waters: rogue waves forced by infragravity waves penetrating in the laguna Cáhuil

Long infragravity waves with periods between 30 seconds and 10 minutes are generated by nonlinear surfzone processes experienced by high energy swells reaching the coast of Chile. During a field campaign conducted in August 2023, we measured the free penetration of these long-waves into the sallow bar-built estuary of laguna Cáhuil located in the O’Higgins Region. We show that they enter into the laguna with very high Ursell numbers and fission in a dispersive shock wave regime. Solitonic wave trains are thus generated, refracting within the lagoon and reflecting at its borders. The continuos input of infragravity energy from the incoming swells and the crossing of solitonic wave trains, result in a soliton gas like state within the laguna, where strongly non-gaussian wave fields and rogue waves are observed. We show that numerical simulations using the fully nonlinear and weakly dispersive Serre-Green-Naghdi equations are able to reproduce these wave processes reaching similar statistical and spectral characteristics. These field and numerical experiments support previous theoretical studies where intense solitonic wave-wave interactions were shown to explain rough wave appearance in shallow waters.

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Olivier Goubet

Olivier Goubet

Propagation of long waves in shallow water with varying bathymetry.

We discuss here some asymptotical models for long waves in shallow water when the bathymetry fluctuates.
How a sudden modification of the bathymetry may provide tsunami waves?
May we slow down rough waves by controling the bathymetry?

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Sebastian Tapia

Sebastián Tapia

Fractal Dynamics in Quintic NLS Soliton Collisions Review

We will explore the multi bouncing and fractal phenomena observed in the velocity-phase relationship during soliton collisions in the quintic nonlinear Schrödinger equation. We will present a classical finite difference scheme adapted for simulating these collisions, followed by the derivation and simulation of the governing system of ODEs for soliton trajectories and velocities. Additionally, we will discuss potential invariants that may help clarify the origins of the fractal patterns, both within the original PDE and the reduced ODE system.

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Friday Sept 6

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María Eugenia Martínez

María Eugenia Martínez

The Soliton Problem in Water Wave Models with a Varying Medium

We explore the behavior of solitary waves in two deep water wave models: the Whitham equation and the Zakharov water wave system. Our study focuses on how these waves respond to environmental changes, such as the introduction of a potential in the power cases of the Whitham equation or alterations in the fluid domain’s bottom in the Zakharov water wave system.
During these interactions, the solitary wave experiences changes in speed (and amplitude), governed by an ODE that describes the interaction dynamics. We propose that the essence of understanding these interactions lies in analyzing the dynamical system that dictates the speed of the solitary wave throughout the interaction regime.

 

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Clémentine Courtès

Clémentine Courtès

Numerical study for some water waves models

One of the aims of the PANDA project is to model and better understand waves propagation on the ocean surface. This talk will focus on two numerical aspects of the PANDA project :
– the numerical modeling of the solitary waves for the Korteweg-de Vries equation or the Whitham equation with a non-flat bottom
– the numerical analysis of a finite-difference scheme for the abcd system.
I will present the very first results in progress in each of these two directions, as well as the future steps we plan to take.

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Felipe Poblete

Felipe Poblete

The scattering problem for Hamiltonian ABCD
Boussinesq systems in the energy space

We will explore the Boussinesq abcd system originally derived by Bona, Chen, and Saut. We will focus particularly on the regime where b, d > 0 and a, c < 0, noting that if b = d, the system becomes Hamiltonian. During this presentation, we will analyze the problem of decay and scattering in this regime. We will show that for suciently dispersive abcd systems, all small solutions must decay to zero in the energy space within a proper subset of the light cone. This will be shown using virial functionals that enable global-in-time control of local terms in H^{1} × H^{1}, without requiring additional assumptions on parity or decay.

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