Tai-Chia Lin

Undular bore theory for the Gardner equation

We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg-de Vries (KdV), equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg-de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV-type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves, and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.

New Poisson–Boltzmann type equations: one-dimensional solutions

The Poisson–Boltzmann (PB) equation is conventionally used to model the equilibrium of bulk ionic species in different media and solvents. In this paper we study a new Poisson–Boltzmann type (PB_n) equation with a small dielectric parameter 2 and non-local nonlinearity which takes into consideration the preservation of the total amount of each individual ion. This equation can be derived from the original Poisson–Nernst–Planck system. Under Robin-type boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviours of one-dimensional solutions of PB_n equations as the parameter approaches zero. In particular, we show that in case of electroneutrality, i.e. α = β, solutions of 1D PB_n equations have a similar asymptotic behaviour as those of 1D PB equations. However, as α ≠ β (non-electroneutrality), solutions of 1D PB_n equations may have blow-up behaviour which cannot be found in 1D PB equations. Such a difference between 1D PB and PB_n equations can also be verified by numerical simulations.

Symbiotic bright solitary wave solutions of coupled nonlinear Schrödinger equations

Conventionally, bright solitary wave solutions can be obtained in self-focusing nonlinear Schrodinger equations with attractive self-interaction. However, when self-interaction becomes repulsive, it seems impossible to have bright solitary wave solutions. Here we show that there exist symbiotic bright solitary wave solutions of coupled nonlinear Schrodinger equations with repulsive self-interaction but strongly attractive interspecies interaction. For such coupled nonlinear Schrodinger equations in two- and three-dimensional domains, we prove the existence of least energy solutions and study the location and configuration of symbiotic bright solitons. We use Neharis manifold to construct least energy solutions and derive their asymptotic behaviour by some techniques of singular perturbation problems.

Diffusion Limit of Kinetic Equations for Multiple Species Charged Particles

In ionic solutions, there are multi-species charged particles (ions) with different properties like mass, charge etc. Macroscopic continuum models like the Poisson–Nernst–Planck (PNP) systems have been extensively used to describe the transport and distribution of ionic species in the solvent. Starting from the kinetic theory for the ion transport, we study a Vlasov–Poisson–Fokker–Planck (VPFP) system in a bounded domain with reflection boundary conditions for charge distributions and prove that the global renormalized solutions of the VPFP system converge to the global weak solutions of the PNP system, as the small parameter related to the scaled thermal velocity and mean free path tends to zero. Our results may justify the PNP system as a macroscopic model for the transport of multi-species ions in dilute solutions.

Vortices in p-wave superconductivity

In the theory of p-wave superconductivity, the Ginzburg--Landau energy functionals with multicomponent order parameters were employed. Here we find a minimizer of a reduced form of the p-wave Ginzburg--Landau free energy with two-component order parameters. The minimizer has distinct degree-one (or minus one) vortices in each component. We also derive a system of ordinary differential equations as the motion equations of vortices in the approximated gradient flow for p-wave superconductivity.

Rigorous and generalized derivation of vortex line dynamics in superfluids and superconductors

We prove rigorously the asymptotic motion equation of a vortex line in a superconductor and a superfluid at small coherence length

Ground states of nonlinear Schrödinger systems with saturable nonlinearity in R2 for two counterpropagating beams

\epsilon

Incompressible and compressible limits of two-component Gross–Pitaevskii equations with rotating fields and trap potentials

. In superconductors, the leading order term of the motion equation of a vortex line is dominated by the curvature and the normal direction of the vortex line. In superfluids, the leading order term of the motion equation of a vortex line is determined by the curvature and the binormal direction of the vortex line. Fortunately, the motion equation of a vortex line in a superfluid has the same leading order term as the motion equation of a vortex line in an incompressible fluid at high Reynolds numbers as

Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation

\epsilon = (Reynolds number)^{-\frac{1}{2}}

Quantum crystals in a trapped Rydberg-dressed Bose-Einstein condensate

. The method of our proof is more rigorous and generalized than the formal asymptotic analysis in the dynamics of fluid dynamic vortices.