Yuxi Zheng

On a completely integrable nonlinear hyperbolic variational equation

Abstract We show that the nonlinear partial differential equation, ( u t +uu x ) xx =1/2( u x 2 ) x , is a completely integrable, bi-variational, bi-Hamiltonian system. The corresponding equation for w=u xx belongs to the Harry Dym hierarchy. This equation arises in two different physical contexts in two nonequivalent variational forms. It describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field, and it is the high-frequency limit of the Camassa-Holm equation, which is an integrable model equation for shallow water waves. Using the bi-Hamiltonian structure, we derive a recursion operator, a Lax pair, and an infinite family commuting Hamiltonian flows, together with the associated conservation laws. We also give the transformation to action-angle coordinates. Smooth solutions of the partial differential equation break down because their derivative blows up in finite time. Nevertheless, the Hamiltonian structure and complete integrability appear to remain valid globally in time, even after smooth solutions break down. We show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions. We compute the bi-Hamiltonian structure on this invariant manifold which is obtained by restricting the bi-Hamiltonian structure of the partial differential equation.

On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions

We study the nonlinear hyperbolic partial differential equation, (ut+uux)x=1/2ux2. This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal.Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up.There are no a priori estimates on the second derivatives in any Lp space, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation.We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave.

Effect of solutes on dislocation motion ¿a phase-field simulation

Abstract Based on recent advances in phase-field models for integrating phase and defect microstructures as well as dislocation dynamics, the interactions between diffusional solutes and moving dislocations under applied stresses are studied in three dimensions. A new functional form for describing the eigenstrains of dislocations is proposed, eliminating the dependence of the magnitude of the dislocation Burgers vector on the applied stress and providing correct stress fields of dislocations. A relationship between the velocity of the dislocation and the applied stress is obtained by theoretical analysis and numerical simulations. The operation of Frank–Read sources in the presence of diffusional solutes, the effect of chemical interactions in solid solution on the equilibrium distribution of Cottrell atmosphere, and the drag effect of Cottrell atmosphere on dislocation motion are examined. The results demonstrate that the phase-field model correctly describes the long-range elastic interactions and short-range chemical interactions that control dislocation motion.

Conservative Solutions to a Nonlinear Variational Wave Equation

We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation utt − c(u)(c(u)ux)x=0, for initial data of finite energy. Here c(·) is any smooth function with uniformly positive bounded values.

Concentrations in the one-dimensional Vlasov-Poisson equations, I.: temporal development and non-unique weak solutions in the single component case

Abstract Weak solutions of the one-component Vlasov-Poisson equation in a single space dimension are proposed and studied here as a simpler analogue problem for the behavior of weak solutions of the two-dimensional incompressible Euler equations with non-negative vorticity. The physical, structural, and functional analytic analogies between these two problems are developed in detail here. With this background, explicit solutions for electron sheet initial data, the analogue of vortex sheet initial data, are presented, which display the phenomena of singularity formation in finite time as well as the explicit temporal development of charge concentrations. Other rigorous explicit examples with charge concentration are developed where there are non-unique weak solutions with the same initial data. In one of these non-unique weak solutions, an electron sheet completely collapses to a point charge in finite time. The detailed limiting behavior of regularizations such as the diffusive Fokker-Planck equation are developed through a very efficient numerical method which yields extremely high resolution for these simpler analogue problems. A striking consequences of the numerical results reported here is the fact that there is not a selection principle for a unique weak solution in some situations where there are several weak solutions with charge concentration for the same initial data. In particular, two explicit weak solutions with the same initial data are constructed here where it is demonstrated that the zero smoothing limit of time reversible particle methods converges to one of these solutions while the zero diffusion limit of the Fokker-Planck equation converges to the other weak solution.

Transonic Shock Formation in a Rarefaction Riemann Problem for the 2D Compressible Euler Equations

It is perhaps surprising for a shock wave to exist in the solution of a rarefaction Riemann problem for the compressible Euler equations in two space dimensions. We present numerical evidence and generalized characteristic analysis to establish the existence of a shock wave in such a 2D Riemann problem, defined by the interaction of four rarefaction waves. We consider both the customary configuration of waves at the right angle and also an oblique configuration for the rarefaction waves. Two distinct mechanisms for the formation of a shock wave are discovered as the angle between the waves is varied.

TRANSIENT ANALYSIS OF IMMIGRATION BIRTH–DEATH PROCESSES WITH TOTAL CATASTROPHES

Very few stochastic systems are known to have closed-form transient solutions. In this article we consider an immigration birth and death population process with total catastrophes and study its transient as well as equilibrium behavior. We obtain closed-form solutions for the equilibrium distribution as well as the closed-form transient probability distribution at any time t ≥ 0. Our approach involves solving ordinary and partial differential equations, and the method of characteristics is used in solving partial differential equations.

RAREFACTIVE SOLUTIONS TO A NONLINEAR VARIATIONAL WAVE EQUATION OF LIQUID CRYSTALS

We study a nonlinear wave equation derived from a simplified liquid crystal model, in which the wave speed is a given function of the wave amplitude. We formulate a viscous approximation of the equation and establish the global existence of smooth solutions for the viscously perturbed equation. For a monotone wave speed function in the equation, we find an invariant region in the phase space in which we discover: (a) smooth data evolve smoothly forever; (b) both the viscous regularization and the smooth solutions obtained through data smoothing for rough initial data yield weak solutions to the Cauchy problem of the nonlinear variational wave equation. The main tool is the Young measure theory and related techniques.

On a nonlinear hyperbolic variational equation: II. The zero-viscosity and dispersion limits

We consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation (ut+uux)x=1/2ux2 with the simplest initial data such that ux blows up in finite time. We prove that the zero-viscosity limit selects a unique global weak solution of the partial differential equation without viscosity. We also present numerical experiments which indicate that the zero-dispersion limit selects a different global weak solution of the same initial-value problem.

Singularities and Oscillations in a Nonlinear Variational Wave Equation

This paper analyzes a nonlinear variational wave equation in which the wave speed is a function of the dependent variable. The wave equation arises is a number of different physical contexts and is the simplest example of an interesting class of nonlinear hyperbolic partial differential equations. We describe a blow-up result for the one-dimensional wave equation which shows that smooth solutions break down in finite time. We illustrate this result with some numerical solutions. We also derive a closed system of equations which describe the interaction between the mean field of a solution and oscillations in its spatial derivative.