Marchenko Equation for the Derivative Nonlinear SchrSdinger Equation

A simple derivation of the Marchenko equation is given for the derivative nonlinear Schrodinger equation. The kernel of the Marchenko equation is demanded to satisfy the conditions given by the compatibility equations. The soliton solutions to the Marchenko equation are verified. The derivation is not concerned with the revisions of Kaup and Newell.     

On the Derivation of a High-Velocity Tail from the Boltzmann–Fokker–Planck Equation for Shear Flow

Uniform shear flow is a paradigmatic example of a nonequilibrium fluid state exhibiting non-Newtonian behavior. It is characterized by uniform density and temperature and a linear velocity profile U _x (y)=ay, where a is the constant shear rate. In the case of a rarefied gas, all the relevant physical information is represented by the one-particle velocity distribution function f(r,v)=f(V), with Vv–U(r), which satisfies the standard nonlinear integro-differential Boltzmann equation. We have studied this state for a two-dimensional gas of Maxwell molecules with a collision rate K()lim ₀ ^–2 (–), where is the scattering angle, in which case the nonlinear Boltzmann collision operator reduces to a Fokker–Planck operator. We have found analytically that for shear rates larger than a certain threshold value a _th0.3520 (where is an average collision frequency and a _th/ is the real root of the cubic equation 64x ³+16x ²+12x–9=0) the velocity distribution function exhibits an algebraic high-velocity tail of the form f(V;a)|V|^–4–(a) (;a), where tan V _y /V _x and the angular distribution function (;a) is the solution of a modified Mathieu equation. The enforcement of the periodicity condition (;a)=(+;a) allows one to obtain the exponent (a) as a function of the shear rate. It diverges when aa _th and tends to a minimum value _min1.252 in the limit a. As a consequence of this power-law decay for a>a _th, all the velocity moments of a degree equal to or larger than 2+(a) are divergent. In the high-velocity domain the velocity distribution is highly anisotropic, with the angular distribution sharply concentrated around a preferred orientation angle ~(a), which rotates from ~=–/4,3/4 when aa _th to ~=0, in the limit a.     

A Fokker–Planck Model of the Boltzmann Equation with Correct Prandtl Number for Polyatomic Gases

We propose an extension of the Fokker–Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier–Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar–Gross–Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman–Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.     

Fokker–Planck Equation for a Metastable Time Dependent Potential

We use the Fokker–Planck equation to study the diffusion process driven for a metastable potential within a temporal dependence. This potential is characterized by the existence of a barrier that increases with time and reduces the particle diffusion. Escape rate across the barrier for different values of diffusion coefficient is analyzed. The results are also associated with the diffusion process through ion channels in biological system.     

Instability for the Semiclassical Non-linear Schrödinger Equation

We adapt recent results on instability for non-linear Schrödinger equations to the semi-classical setting. Rather than work with Sobolev spaces we estimate projective instability in terms of the small constant, h, appearing in the equation. Our motivation comes from the Gross-Pitaevski equation used in the study of Bose-Einstein condensation.     

Simulation of the Weakly Nonlinear Rayleigh–Taylor Instability in Spherical Geometry

The Rayleigh-Taylor instability at the weakly nonlinear(WN) stage in spherical geometry is studied by numerical simulation.The mode coupling processes are revealed.The results are consistent with the WN model based on parameter expansion,while higher order effects are found to be non-negligible.For Legendre mode perturbation Pn(cos B),the nonlinear saturation amplitude(NS A) of the fundamental mode decreases with the mode number n.When n is large,the spherical NSA is lower than the corresponding...     

Solution to the Fokker–Planck Equation with Piecewise-Constant Drift

We study the solution to the Fokker–Planck equation with piecewise-constant drift, taking the case with two jumps in the drift as an example. The solution in Laplace space can be expressed in closed analytic form, and its inverse can be obtained conveniently using some numerical inversion methods. The results obtained by numerical inversion can be regarded as exact solutions, enabling us to demonstrate the validity of some numerical methods for solving the Fokker–Planck equation. In particular, ...     

Estimates for the KdV-Limit of the Camassa–Holm Equation

In a number of scaling limits, we prove estimates relating the solutions of the Camassa–Holm equation to the solutions of the associated KdV equation. As a consequence, suitable solutions of the water wave problem and solutions of the Camassa–Holm equation stay close together for long times.     

Unstable Manifolds of Relative Periodic Orbits in the Symmetry-Reduced State Space of the Kuramoto–Sivashinsky System

Systems such as fluid flows in channels and pipes or the complex Ginzburg–Landau system, defined over periodic domains, exhibit both continuous symmetries, translational and rotational, as well as discrete symmetries under spatial reflections or complex conjugation. The simplest, and very common symmetry of this type is the equivariance of the defining equations under the orthogonal group O(2). We formulate a novel symmetry reduction scheme for such systems by combining the method of slices with invariant polynomial methods, and show how it works by applying it to the Kuramoto–Sivashinsky system in one spatial dimension. As an example, we track a relative periodic orbit through a sequence of bifurcations to the onset of chaos. Within the symmetry-reduced state space we are able to compute and visualize the unstable manifolds of relative periodic orbits, their torus bifurcations, a transition to chaos via torus breakdown, and heteroclinic connections between various relative periodic orbits. It would be very hard to carry through such analysis in the full state space, without a symmetry reduction such as the one we present here.     

Colliding Solitons for the Nonlinear Schrödinger Equation

We study the collision of two fast solitons for the nonlinear Schrödinger equation in the presence of a slowly varying external potential. For a high initial relative speed ||v|| of the solitons, we show that, up to times of order ||v|| after the collision, the solitons preserve their shape (in L ²-norm), and the dynamics of the centers of mass of the solitons is approximately determined by the external potential, plus error terms due to radiation damping and the extended nature of the solitons. We remark on how to obtain longer time scales under stronger assumptions on the initial condition and the external potential.