Peakon Solutions to Supersymmetric Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper,the supersymmetric Camassa-Holm equation and Degasperis-Procesi equation are derived from a general superfield equations by choosing different parameters.Their peakon-type solutions are shown in weak sense.At the same time,the dynamic behaviors are analyzed particularly when the two peakons collide elastically,and some results are compared with each other between the two equations.     

Solutions of CDG Equation and Modified CDG Equation

This article puts forward a new way to find solutions of CDG equation. The main results are:(i) According to the Lax pair of CDG equation, we introduce the modified CDG equation. (ii) An invariance depending on two parameters of M-CDG equation is found. (iii) Some solutions for CDG equation are obtained by using the invariance.     

Propagation-Dispersion Equation

A propagation-dispersion equation is derived for the first passage distribution function of a particle moving on a substrate with time delays. The equation is obtained as the hydrodynamic limit of the first visit equation, an exact microscopic finite difference equation describing the motion of a particle on a lattice whose sites operate as time-delayers. The propagation-dispersion equation should be contrasted with the advection-diffusion equation (or the classical Fokker–Planck equation) as it describes a dispersion process in time (instead of diffusion in space) with a drift expressed by a propagation speed with non-zero bounded values. The temporal dispersion coefficient is shown to exhibit a form analogous to Taylor's dispersivity. Physical systems where the propagation-dispersion equation applies are discussed.     

与量子输运方程相一致的夸克经典输运方程

用Sμν作为描述夸克自旋的动力学变量建立了它的经典输运方程，并讨论了该方程与量子输运方程的一致性，通过一致性的比较探讨了经典分布函数在色空间和自旋空间的一些特性.     

On a Relation Between the Bach Equation and the Equation of Geometrodynamics

The Bach equation and the equation of geometrodynamics are based on two quite different physical motivations, but in both approaches, the conformal properties of gravitation play the key role. In this paper we present an analysis of the relation between these two equations and show that the solutions of the equation of geometrodynamics are of a more general nature. We show the following non-trivial result: there exists a conformally invariant Lagrangian, whose field equation generalizes the Bach equation and has as solutions those Ricci tensors which are solutions to the equation of break geometrodynamics.     

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.     

A Search on Dirac Equation

The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger-like one. Earlier results are discussed in a unified framework, and some solutions of a large class of potentials are given.     

相对论类空方程

根据Dirac类空自洽性条件的思想,通过引入类空因子定义了类空波函数.它的物理部分与Bethe-Salpeter波函数相一致.利用普适的相互作用核的重排技术,导出了对于束缚态和散射态的相对论类空方程,并且将它们推广到多粒子的情形.也得到了束缚态类空波函数的归一化条件和散射态类空方程的非齐次项的解.因此,建立起了相对论类空方程体系.     

Relativistic BUU Equation

Based on the Waleck's models QHD-Ⅰ and QHD-Ⅱ describing the nucleon-nucleon interaction,the Boltzmann-Uehling-Uhlenbeck (BUU) equation,which is the time evolution of the nucleon distribution function including the Hartree and Fock self-energy terms as well as the Born collision term and its exchange term,has been derived by using the closed-time path Green's function technique and assuming that the Green's functions and the self-energy terms are slowly varying functions of the centre-of-mass coordinates.Our result shows that the BUU equation for proton and that for neutron are simultaneous each other.     

Generalized Variable-Coefficient KP Equation

The variable-coefficient generalizations of thecelebrated KP equation (GvcKPs) are realistic models forvarious physical and engineering situations. In thisnote, the application of symbolic computation and the truncated Painleve expansion leads toan auto-Backlund transformation and soliton-typedsolutions to a type of the GvcKPs.     

The stochastic Burgers Equation

We study Burgers Equation perturbed by a white noise in space and time. We prove the existence of solutions by showing that the Cole-Hopf transformation is meaningful also in the stochastic case. The problem is thus reduced to the anaylsis of a linear equation with multiplicativehalf white noise. An explicit solution of the latter is constructed through a generalized Feynman-Kac formula. Typical properties of the trajectories are then discussed. A technical result, concerning the regularizing effect of the convolution with the heat kernel, is proved for stochastic integrals.     

Darboux Transformation for Drinfel'd-Sokolov-Wilson Equation

A coupled system known as tie Drinfel'd-Sokolov-Wilson equation is reexamined.With the help of a Lax operator of fourth order,its proper Darboux transformation is constructed.Also,a nonlinear superposition formula is worked out for the associated Backlund transformation and some solutions are calculated.     

Darboux Transformation for Drinfel'd-Sokolov-Wilson Equation

A coupled system known as the Drinfel'd-Sokolov-Wilson equation is reexamined. With the help of a Lax operator of fourth order, its proper Darboux transformation is constructed. Also, a nonlinear superposition formula is worked out for the associated Bäcklund transformation and some solutions are calculated.     

General form of DMPK Equation

The Dorokhov–Mello–Pereyra–Kumar (DMPK) equation, using in the analysis of quasi-onedimensional systems and describing evolution of diagonal elements of the many-channel transfer-matrix, is derived under minimal assumptions on the properties of channels. The general equation is of the diffusion type with a tensor character of the diffusion coefficient and finite values of non-diagonal components. We suggest three different forms of the diagonal approximation, one of which reproduces the usual DMPK equation and its generalization suggested by Muttalib and co-workers. Two other variants lead to equations of the same structure, but with different definitions of their parameters. They contain additional terms, which are absent in the first variant. The coefficients of additional terms are shown to be finite beyond the metallic phase by calculation of the Lyapunov exponents and their comparison with numerical experiments. Consequences of the obtained equations for the problem of the conductance distribution and the status of the nonlinear sigma-models are discussed.     

Mathieu Equation and Elliptic Curve

We present a relation between the Mathieu equation and a particular elliptic curve. We find that the Floquet exponent of the Mathieu equation, for both q<<1 and q>>1, can be obtained from the integral of a differential one form along the two homology cycles of the elliptic curve. Certain higher order differential operators are needed to generate the WKB expansion. We make a few conjectures about the general structure of these differential operators.     

Chain Solutions of the BKP Equation

We obtain a new class of exact analytic solutions of the BKP equation which are called chain solutions because they are analogous to the, soliton solutions of the KP equation. The rational solutions called lumps are shown to include in this class as a Special limiting situation.     

New Exact Solutions for Konopelchenko-Dubrovsky Equation Using an Extended Riccati Equation Rational Expansion Method

Taking the Konopelchenko-Dubrovsky system as a simple example, some families of rational formal hyperbolic function solutions, rational formal triangular periodic solutions, and rational solutions are constructed by using the extended Riccati equation rational expansion method presented by us. The method can also be applied to solve more nonlinear partial differential equation or equations.     

Novel Solutions of KdV Equation

Some novel solutions of the KdV equation are obtained through the modified bilinear B\"{a}cklund transformation.     

Energy Level and Riccati Equation

By an integral transformation, the energy eigenvalue problem of quantum bound state is converted to a nonlinear Riccati equation which is easily solved with the help of node theorem. Some typical examples are used to show the conciseness of this new approach.