Charles Bu

On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation

We present analytical methods to investigate the Cauchy problem for the complex Ginzburg-Landau equation u 1 = ( v + i α)Δ u − (κ + iβ) | u | 2 q u + γ u in 2 spatial dimensions (here all parameters are real). We first obtain the local existence for v > 0, κ ≥ 0. Global existence is established in the critical case q = 1. In addition, we prove the global existence when .

Nonlinear Schrödinger Equation with Inhomogeneous Dirichlet Boundary Data

In this article we study the following nonlinear Schrodinger equation iut=Δu−g∣u∣p−1u in a domain Ω⊂Rn with initial condition u(x,0)=ϕ(x) and the Dirichlet boundary condition u(x,t)=Q(x,t) on ∂Ω, where ϕ, Q are given smooth functions. The nonlinear term contributes a negative term to the energy (i.e., g<0). We present the existence theorem for a global solution of finite energy when p⩽1+2∕n.

An initial-boundary value problem for the Ginzburg-Landau equation

Abstract This paper establishes existence and uniqueness of the weak solution to the Ginzburg-Landau equation posed in a finite domain Ω = [0, L] for t ⩾ 0, with certain initial-boundary data.

A Dirichlet boundary value problem for a generalized Ginzburg-Landau equation

Abstract We study the following generalized 1D Ginzburg-Landau equation on Ω = (0,∞) × (0, ∞): u t = (1 + iμ)u xx + (a 1 + ib 1 )|u| 2 u x + (a 2 + ib 2 )u 2 u x − (1 + iν)|u| 4 u with initial and Dirichlet boundary conditions u(x,0) = h(x),u(0,t) = Q(t). Under suitable conditions, we prove that there is a unique H1 solution that exists for all time.

Almost periodic solution for a model of tumor growth

A mathematical model of tumor growth governed by diffusion equation is studied where the source of mitotic inhibitor is almost periodic and time-dependent within the tissue. Existence and uniqueness of an almost periodic solution for this model have been proved.

Forced cubic Schrödinger equation with Robin boundary data: continuous dependency result

For the cubic Schrodinger equation iu t = u xx + k | u | 2 u , 0 ≤ x, t u(x , 0) = u 0 ( x ) ∈ H 2 [0, ∞), and Robin boundary data u x (0, t ) + α u (0, t ) = R(t) ∈ C 2 [0, ∞) (where α is real), we show that the solution u depends continuously on u 0 and R .

Modified Korteweg–de Vries equation with generalized functions as initial values

In this article the existence of generalized solutions to the modified Korteweg–de Vries equation ut−6σu2ux+uxxx=0 is studied. The solutions are found in certain algebras of new generalized functions containing spaces of distributions.

The generalized Ginzburg-Landau equation: posed in a quarter plane

This paper is concerned with a generalized 1D Ginzburg-Landau equation involving a fifth order term and two nonlinear terms containing spatial derivatives. The equation is posed in a quarter plane 0=

Numerical simulation of blow-up of a 2D generalized Ginzburg-Landau equation

Abstract We study the Cauchy problem for the following generalized Ginzburg-Landau equation u t = ( ν + iα ) Δu − ( κ + iβ )| u | 2 q u + γu in two spatial dimensions for q > 1 (here α, β, γ are real parameters and ν , κ > 0). A blow-up of solutions is found via numerical simulation in several cases for q > 1.