Research

Differential Equations

Thursday, April 29, 2021

Abstract

The behavior of focusing media with non-zero background has received increased interest in recent years, revealing a number of interesting phenomena. This talk will present a review of recent results on this subject. Specifically:

1. First, I will characterize the nonlinear stage of modulational instability induced by localized perturbations of a uniform background on a self-focusing medium. In particular, I will show how, using the inverse scattering transform (IST) for the focusing nonlinear Schrodinger (NLS) equation, one can prove that the long-time asymptotics is universal, in the sense that a large class of initial conditions give rise to the same behavior: two outer quiescent states separated by a central, wedge-shaped region described by modulated periodic oscillations, whose features can be described analytically.
2. Next, I will show that another kind of universality also exists, in that the above behavior is not limited to the NLS equation, but is instead shared by many different continuous and discrete nonlinear system affected by modulational instability.
3. Next, I will show that the interactions between solitons and the oscillatory wedge gives rise to three different outcomes, all of which can be completely characterized analytically: soliton transmission, soliton trapping, and a mixed regime in which a soliton-generated wake is also produced.
4. I will then discuss a broad class of Riemann-problems in these kinds of systems, and I will show how some of these problems, which give rise to the formation of dispersive shock waves (DSW), are effectively described by Whitham modulation theory.
5. Finally, I will discuss the interactions between solitons and these DSW structures, and I will show how all the soliton properties (amplitude, velocity, width, internal structure) change upon passing through the DSW, but can still be correctly captured by the IST.

Thursday, April 22, 2021

Thursday, April 15, 2021

No seminar this week — Spring Break.

Thursday, April 8, 2021

Abstract

The Fokas method, also called the unified transform, is a recently developed algorithmic procedure for studying certain boundary value problems for linear PDEs (e.g., the Laplace equation over a triangle region) and for certain nonlinear integrable PDEs (e.g., the nonlinear Schr�dinger (NLS) equation in the half-line/finite interval). The method includes three steps: (1) constructing a Lax pair, (2) simultaneous spectral analysis, and (3) global relation analysis. We will mainly focus on discussing three examples: (1) two-dimensional linear PDEs in a convex polygon, (2) the NLS equation on the interval, and (3) the NLS equation on the circle.

Thursday, April 1, 2021

Abstract

We will show a brief note about the Stokes waves, the physical effect of the changes in the mean velocity and the mean height, the modulation equations, and the Stokes waves on multimedia, and lastly, we will talk about the KdV equation from the Whitham modulation point-of-view.

Thursday, March 25, 2021

Thursday, March 18, 2021

Abstract

We will discuss the modulation theory in the field of nonlinear optics. Particularly, we will consider one-dimensional modulation equations, study two different types of basic beam equations: focusing and thin beams, and analyze higher-order dispersive effects.

Thursday, March 11, 2021

Thursday, March 4, 2021

Abstract

We will talk about the modulation equations and their solutions through the average variational principle, and classify the equations and determine stability conditions based on group velocities. Higher-order dispersive effects will also be examined to explore the important differences between linear and nonlinear theories.

Thursday, February 25, 2021

Abstract

We will simply discuss Whitham�s average Lagrangian method for single-phase mode analysis of dispersive waves in the nonlinear Klein-Gordon equation. Then we will introduce Luke�s perturbation scheme for the same problem. Finally, we will talk about how Ablowitz and Benney generalized the methods to cases of multi-phase modes. All those approaches are essentially a version of the WKB method for nonlinear dispersive equations.

Thursday, February 18, 2021

Thursday, February 11, 2021

Abstract

We will discuss nonlinear dispersive wave equations and particularly, explain how periodic wavetrains can exist in a nonlinear Klein-Gordon equation, with a discussion on the variational approach to its modulation theory. Moreover, a description of the variational principle for multiple-phase wavetrains is presented in detail.

Thursday, February 4, 2021

Abstract

We will discuss how dispersive effects may be incorporated into the shallow water theory. In particular, we consider how nonlinearity affects dispersive waves. Two nonlinear wave equations, namely the Boussinesq and Korteweg-de Vries equations, will be derived. We will also discuss some special solutions such as solitary and periodic wave solutions.

Thursday, January 28, 2021

Abstract

Linear theory of waves has been a fascinating subject to study because its associated mathematical problems are familiar real-life phenomena. We will talk about the linear theory of waves by considering an inviscid incompressible fluid (water) moving under constant gravitational field. Equations of water waves will be linearized, and the dispersion relations will be obtained for waves on an interface between two fluids and under surface tension. The formulation of the initial value problems and the behavior of water waves in various depth will also be presented.

Thursday, January 21, 2021

Abstract

Wave patterns present many interesting phenomena. Those phenomena can be mathematically and physically studied by the dispersion relation of the waves. For instance, capillary waves and V-shaped waves known as the Kelvin wedge will be discussed. These waves show different behaviors on shallow water, deep water and thin sheets.

Thursday, January 14, 2021

Abstract

We will talk about linear dispersive waves and their corresponding dispersion relations. General solutions are formulated by Fourier integrals and their asymptotic behavior is determined by the method of steepest descents. Group velocity is linked to propagation of wave number, amplitude, and energy. The variational approach and asymptotic expansions are adopted for analyzing the conservation of wave action, extendable to the cases of nonuniform media and nonlinear wavetrains.