Structure equation and Mei conserved quantity for Mei symmetry of Appell equation

This paper investigates structure equation and Mei conserved quantity of Mei symmetry of Appell equations for non-Chetaev nonholonomic systems. Appell equations and differential equations of motion for non-Chetaev nonholonomic mechanical systems are established. A new expression of the total derivative of the function with respect to time t along the trajectory of a curve of the system is obtained, the definition and the criterion of Mei symmetry of Appell equations under the infinitesimal transformations of groups are also given. The expressions of the structure equation and the Mei conserved quantity of Mei symmetry in the Appell function are obtained. An example is given to illustrate the application of the results.     

Non relativistic limit of the Landau–Lifshitz equation: A new equation

It is shown that Ford equation is not adequate in general to describe the motion of a charged particle including the reaction force in the non relativistic limit. As in General Relativity where a post-Newtonian method is developed in order to describe the gravitational effects at low velocities and small energies, an extra term inherited from Special Relativity must be added to the Ford equation. This is due to that the new term is greater than the reaction force in many physical situations. The Coulombic case is analyzed showing the necessity of including the new term. Comparison with General Relativity results is analyzed. The Vlasov equation to first order in 1/c²$1/c^2$ is proposed for the constant electric and magnetic fields.     

Bianchi transformation between the real hyperbolic Monge-Ampère equation and the Born-Infeld equation

By casting the Born-Infeld equation and the real hyperbolic Monge-Ampère equation into the form of equations of hydrodynamic type, we find that there exists an explicit transformation between them. This is Bianchi transformation.     

Roy equation analysis of ππ scattering

We analyse the Roy equations for the lowest partial waves of elastic ππ scattering. In the first part of the paper, we review the mathematical properties of these equations as well as their phenomenological applications. In particular, the experimental situation concerning the contributions from intermediate energies and the evaluation of the driving terms are discussed in detail. We then demonstrate that the two S-wave scattering lengths a₀⁰ and a₀² are the essential parameters in the low energy region: Once these are known, the available experimental information determines the behaviour near threshold to within remarkably small uncertainties. An explicit numerical representation for the energy dependence of the S- and P-waves is given and it is shown that the threshold parameters of the D- and F-waves are also fixed very sharply in terms of a₀⁰ and a²₀. In agreement with earlier work, which is reviewed in some detail, we find that the Roy equations admit physically acceptable solutions only within a band of the (a₀⁰,a₀²) plane. We show that the data on the reactions e⁺e⁻→ππ and τ→ππν reduce the width of this band quite significantly. Furthermore, we discuss the relevance of the decay K→ππeν in restricting the allowed range of a₀⁰, preparing the grounds for an analysis of the forthcoming precision data on this decay and on pionic atoms. We expect these to reduce the uncertainties in the two basic low energy parameters very substantially, so that a meaningful test of the chiral perturbation theory predictions will become possible.     

Breather solutions of modified Benjamin-Bona-Mahony equation

New two-component vector breather solution of the modified Benjamin-Bona-Mahony(MBBM)equation is considered.Using the generalized perturbation reduction method,the MBBM equation is reduced to the coupled nonlinear Schr¨odinger equations for auxiliary functions.Explicit analytical expressions for the profile and parameters of the vector breather oscillating with the sum and difference of the frequencies and wavenumbers are presented.The two-component vector breather and single-component scalar breather of the MBBM equation is compared.     

A new supersymmetric classical Boussinesq equation

In this paper, we obtain a supersymmetric generalization for the classical Boussinesq equation. We show that the supersymmetric equation system passes the Painlev\'{e test and we also calculate its one- and two-soliton solutions.     

The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq–Burgers equation

This paper studies the coupled Burgers equation and the high-order Boussinesq–Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink (soliton) solutions and multiple-singular-kink (soliton) solutions are derived for the two equations.     

Some new exact solutions of Jacobian elliptic function about the generalized Boussinesq equation and Boussinesq-Burgers equation

By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.     

Geometrical equivalence of a deformed heisenberg spin equation and the generalized nonlinear schrödinger equation

It is shown that the general SO(3) invariant deformed Heisenberg spin chain discussed by Mikhailov and Shabat is geometrically equivalent to a generalized nonlinear Schrödinger equation through a moving space curve formalism. They are also mutually gauge equivalent in the sense noted by Kundu in a different context.     

Gauge and Bäcklund transformations for the generalized sine-Gordon equation and its η-dependent modified equation

We study the generalized sine-Gordon hierarchy and its associated-dependent modified sine-Gordon hierarchy. Two Bäcklund transformations for these two families are constructed. One of them is a generalization of the Bäcklund transformations of Wadatiet al. and the other one is new. Gauge transformations of a relevant AKNS system are employed to reduce the integration of these equations via the Bäcklund transformations to quadratures. Three generations of explicit solutions of the sine-Gordon equation are presented.     

Soliton and rogue wave solutions of two-component nonlinear Schr?dinger equation coupled to the Boussinesq equation

The nonlinear Schr?dinger(NLS) equation and Boussinesq equation are two very important integrable equations.They have widely physical applications. In this paper, we investigate a nonlinear system, which is the two-component NLS equation coupled to the Boussinesq equation. We obtain the bright–bright, bright–dark, and dark–dark soliton solutions to the nonlinear system. We discuss the collision between two solitons. We observe that the collision of bright–bright soliton is inelastic and two solitons oscillating periodically can happen in the two parallel-traveling bright–bright or bright–dark soliton solution. The general breather and rogue wave solutions are also given. Our results show again that there are more abundant dynamical properties for multi-component nonlinear systems.     

物理学咬文嚼字之三十九 Secular,equation

Secular equation,既是别称characteristic equation的久期方程,又是指行星运动的长期均差.西文中本来浑然一体的概念,在汉语语境中却似乎不搭边.     

Approximate analytic solutions for a generalized Hirota—Satsuma coupled KdV equation and a coupled mKdV equation

<正>This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries(KdV) equation and a coupled modified Korteweg-de Vries(mKdV) equation. This method provides a sequence of functions which converges to the exact solution of the problem and is based on the use of the Lagrange multiplier for the identification of optimal values of parameters in a functional.Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.     

Exact solution of the Rijke tube equation

In the nrevlous rticle,“A slmnle theorr of RIJke tube oscillation”,（Chin.J.Acoustics,2002; 21（1）:1-8）,Ohly ppr。mate SOllltion、presellted to sh。the pfopertleS Of the pketube oscillation,and a constant factor was thus lost.The exact solution Is presented now. Based on Euler’s fluid dyn。cs eqllatlons;the Rike tube、eqllation（Eq.（4）inprevious、rk,and the s。e ordering of equations are。ed In present、rk）was derl、d as OZDI OZD OU’ d-id一c凸。‘（”-“＊（‘）where。’Is t…     

Can the Bogoliubov–de Gennes equation be interpreted as a ‘one-particle’ wave equation?

The solutions of the Bogoliubov–de Gennes (BdG) equation are usually interpreted as the excitations from the superconducting ground state. This viewpoint is not easily applied to a strongly coupled heterojunction since the ground state changes across the interface and it is not clear how the ground state should be connected across the heterointerface. In this paper, we present a different viewpoint that does not suffer from this conceptual drawback. We show that the BdG equation can be viewed as a ‘one-particle’ wave equation whose eigenstates (including the negative energy states) can be filled up systematically to describe the superconducting state, in much the same way that we fill the eigenstates of the Schrödinger equation to describe normal conductors. The only difference is that we need to start from a special vacuum | V〉, consisting of a full band of down-spin electrons, instead of the usual vacuum devoid of all particles. Any quantity of interest, A (such as the charge density or the current density), can be interpreted as the sum of a ‘vacuum contribution’A_{V AC}due to the vacuum | V〉 and a one-particle contribution A_BdGdue to the filled eigenstates of the BdG equation. This picture is easily applied even to strongly coupled heterojunctions since the vacuum | V〉 is the same on both sides of a heterointerface. As such, we believe it puts the scattering theory of transport for superconductors on a firmer conceptual basis.     

Pfaffianization of the variable-coefficient Kadomtsev-Petviashvili equation＊

This paper constructs more general exact solutions than $N$-soliton solution and Wronskian solution for variable-coefficient Kadomtsev--Petviashvili (KP) equation. By using the Hirota method and Pfaffian technique, it finds the Grammian determinant-type solution for the variable-coefficient KP equation (VCKP), the Wronski-type Pfaffian solution and the Gram-type Pfaffian solutions for the Pfaffianized VCKP equation.     

Evolution property of soliton solutions for the Whitham－Broer－Kaup equation and variant Boussinesq equation

Using the standard Painlevé analysis approach, the (1+1)-dimensional Whitham－Broer－Kaup (WBK) and variant Boussinesq equations are solved. Some significant and exact solutions are given. We investigate the behaviour of the interactions between the multi-soliton-kink-type solution for the WBK equation and the multi-solitonic solutions and find the interactions are not elastic. The fission of solutions for the WBK equation and the fusions of those for the variant Boussinesq equation may occur after their interactions.