Juan Carlos Muñoz Grajales

Stiff Microscale Forcing and Solitary Wave Refocusing

This study is focused on wave propagation in heterogeneous media and on capturing the cumulative effect due to small-scale orographic features. More specifically, we focus on designing a numerical method capable of capturing these effects over long propagation distances (the macroscale) when a weakly dispersive, weakly nonlinear solitary wave (the mesoscale) is forced by a disordered microscale}. An accurate and stable numerical scheme is presented for solving a variable coefficient nonlinear Boussinesq system, in the presence of highly oscillatory solutions, together with a new result on its linear stability analysis. The stability analysis focuses on describing the stiffness promoted by the topographys microscale and how it can be removed through an appropriate change of coordinates. The new mathematical formulation acts as a preconditioning at the differential equation level, and the spectrums range becomes insensitive to the microscale. The other goal of this paper is to present the time-reversal an...

EFFECTIVE BEHAVIOR OF SOLITARY WAVES OVER RANDOM TOPOGRAPHY

The deformation of a nonlinear pulse traveling in a dispersive random medium can be studied with asymptotic analysis based on separation of scales when the propagation distance is large compared to the correlation length of the random medium. We consider shallow water waves with a spatially random depth. We use a formulation in terms of a terrain-following Boussinesq system. We compute the effective evolution equation for the front pulse which can be written as a dissipative Kortweg-de Vries equation. We study the soliton dynamics driven by this system. We show, both theoretically and numerically, that a solitary wave is more robust than a linear wave in the early steps of the propagation. However, it eventually decays much faster after a critical distance corresponding to the loss of about half of its initial amplitude. We also perform an asymptotic analysis for a class of random bottom topographies. A universal behavior is captured through the asymptotic analysis of the metric term for the corresponding change to terrain-following coordinates. Within this class we characterize the effective height for highly disordered topographies. The probabilistic asymptotic results are illustrated by performing Monte Carlo simulations with a Schwarz-Christoffel Toolbox.

Existence and Numerical Approximation of Solutions of an Improved Internal Wave Model

AbstractIn this paper we establish local existence of solutions for a new model to describe the propagation of an internal wave propagating at the interface of two immiscible fluids with constant densities, contained at rest in a long channel with a horizontal rigid top and bottom. We also introduce a spectral-type numerical scheme to approximate the solutions of the corresponding Cauchy problem and perform a complete error analysis of the semidiscrete scheme.

Modulation instability in nonlinear propagation of pulses in optical fibers

We consider a generalized system consisting of two coupled Schrodinger-type equations which models nonlinear pulse propagation in a linearly birefringent Kerr optical fiber. An analysis of stability of periodic solutions of the system is performed by using a second-order accurate spectral numerical solver. The main novelty in our study is that we do not assume high or weak birefringence in the fiber and thus the model considered incorporates nonlinear terms which were neglected in the stability analysis performed in previous works. In the case of anomalous dispersion, our numerical simulations indicate that a periodic solution develops a type of modulation instability against small disturbances, given that the perturbation frequency remains within certain range (computed analytically by means of a linear analysis) which depends on the models parameters and the amplitude of the perturbing signals. We also find that the gain spectrum of modulation instability is altered by the additional nonlinear terms included in the above-mentioned system.

Analysis of a Galerkin approach applied to a system of coupled Schrödinger equations

Abstract We study error and convergence rate of a fully discrete Fourier–Galerkin scheme to approximate the solutions of a system consisting of two coupled cubic Schrodinger equations with cross modulation, which is a model for the propagation of a nonlinear pulse in a linearly birefringent Kerr optical fiber. We also establish a result on existence and uniqueness of a solution of the corresponding initial value problem.

Error estimates for a Galerkin numerical scheme applied to a generalized BBM equation

Herein we obtain error estimates for a new Galerkin spectral scheme to approximate the solutions of a dispersive-type equation which is a model to describe the propagation of a wave on the surface of a channel with a flat bottom.Herein we obtain error estimates for a new Galerkin spectral scheme to approximate the solutions of a dispersive-type equation which is a model to describe the propagation of a wave on the surface of a channel with a flat bottom.

Decay of solutions of a Boussinesq-type system with variable coefficients

We introduce a modified Chebyshev rational approximation applied to a weakly nonlinear, weakly dispersive Boussinesq formulation considered on the whole line, which is a model for waves propagating on the surface of a channel with variable bottom. We also establish rigorous convergence and error estimates in L 2 of the semidiscrete and fully discrete numerical schemes and validate the numerical results using the theory derived in a previous paper on the decay of a linear pulse propagating along the surface of a channel with a random bottom.

Error analysis of a Fourier–Galerkin method applied to the Schrödinger equation

We study error and convergence of a fully discrete Fourier–Galerkin scheme to approximate the solutions of the time-dependent cubic Schrödinger equation. The evolution is carried out combining a second-order accurate scheme and a Fourier-spectral discretization in space. The illustrative cases include simulations with analytic periodic standing waves of the Schrödinger equation. The numerical results are in perfect agreement with our analytical results.