Three types of generalized Kadomtsev－Petviashvili equations arising from baroclinic potential vorticity equation

By means of the reductive perturbation method, three types of generalized (2+1)-dimensional Kadomtsev--Petviashvili (KP) equations are derived from the baroclinic potential vorticity (BPV) equation, including the modified KP (mKP) equation, standard KP equation and cylindrical KP (cKP) equation. Then some solutions of generalized cKP and KP equations with certain conditions are given directly and a relationship between the generalized mKP equation and the mKP equation is established by the symmetry group direct method proposed by Lou et al. From the relationship and the solutions of the mKP equation, some solutions of the generalized mKP equation can be obtained. Furthermore, some approximate solutions of the baroclinic potential vorticity equation are derived from three types of generalized KP equations.     

An extension of the modified Sawada—Kotera equation and conservation laws

Based on the modified Sawada-Kotera equation, we introduce a 3 × 3 matrix spectral problem with two potentials and derive a hierarchy of new nonlinear evolution equations. The second member in the hierarchy is a generalization of the modified Sawada-Kotera equation, by which a Lax pair of the modified Sawada-Kotera equation is obtained. With the help of the Miura transformation, explicit solutions of the Sawada-Kotera equation, the Kaup-Kupershmidt equation, and the modified Sawada-Kotera equation are given. Moreover, infinite sequences of conserved quantities of the first two nonlinear evolution equations in the hierarchy and the modified Sawada-Kotera equation are constructed with the aid of their Lax pairs.     

Symmetries and conserved quantities of discrete wave equation associated with the Ablowitz-Ladik-Lattice system

In this paper, we present a new method to obtain the Lie symmetries and conserved quantities of the discrete wave equation with the Ablowitz-Ladik-Lattice equations. Firstly, the wave equation is transformed into a simple difference equation with the Ablowitz-Ladik-Lattice method. Secondly, according to the invariance of the discrete wave equation and the Ablowitz-Ladik-Lattice equations under infinitesimal transformation of dependent and independent variables, we derive the discrete determining equation and the discrete restricted equations. Thirdly, a series of the discrete analogs of conserved quantities, the discrete analogs of Lie groups, and the characteristic equations are obtained for the wave equation. Finally, we study a model of a biological macromolecule chain of mechanical behaviors, the Lie symmetry theory of discrete wave equation with the Ablowitz-Ladik-Lattice method is verified.     

Wronskian Form Solutions for a Variable Coefficient Kadomtsev–Petviashvili Equation

Starting from a simple transformation, and with the aid of symbolic computation, we establish the relation- ship between the solution of a generalized variable coefficient Kadomtsev-Petviashvili （vKP） equation and the solution of a system of linear partial differential equations. According to this relation, we obtain Wronskian form solutions of the vKP equation, and further present N-soliton-like solutions for some degenerated forms of the vKP equation. Moreover, we also discuss the influences of arbitrary constants on the soliton and N-soliton solutions of the KPII equation.     

Lie symmetries and conserved quantities for a two-dimentional nonlinear diffusion equation of concentration

The Lie symmetries and conserved quantities of a two-dimensional nonlinear diffusion equation of concentration are considered. Based on the invariance of the two-dimensional nonlinear diffusion equation of concentration under the infinitesimal transformation with respect to the generalized coordinates and time, the determining equations of Lie symmetries are presented. The Lie groups of transformation and infinitesimal generators of this equation are obtained. The conserved quantities associated with the nonlinear diffusion equation of concentration are derived by integrating the characteristic equations. Also, the solutions of the two-dimensional nonlinear diffusion equation of concentration can be obtained.     

An Approach for Solving Short-Wave Models for Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper, to construct exact solution of nonlinear partial differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived. We investigate the short wave model for the Camassa-Holm equation and the Degasperis-Procesi equation respectively. One-cusp soliton solution of the Camassa-Flolm equation is obtained. One-loop soliton solution of the Degasperis- Procesi equation is also obtained, the approximation of which in a closed form can be obtained firstly by the Adomian decomposition method. The obtained results in a parametric form coincide perfectly with those given in the present reference. This illustrates the efficiency and reliability of our approach.     

Lie symmetries and conserved quantities for a two-dimensional nonlinear diffusion equation of concentration

The Lie symmetries and conserved quantities of a two-dimensional nonlinear diffusion equation of concentration are considered. Based on the invariance of the two-dimensional nonlinear diffusion equation of concentration under the infinitesimal transformation with respect to the generalized coordinates and time, the determining equations of Lie symmetries are presented. The Lie groups of transformation and infinitesimal generators of this equation are obtained. The conserved quantities associated with the nonlinear diffusion equation of concentration are derived by integrating the characteristic equations. Also, the solutions of the two-dimensional nonlinear diffusion equation of concentration can be obtained.     

On the fundamental equation of nonequilibrium statistical physics—Nonequilibrium entropy evolution equation and the formula for entropy production rate

In this paper the author presents an overview on his own research works. More than ten years ago, we proposed a new fundamental equation of nonequilibrium statistical physics in place of the present Liouville equation. That is the stochastic velocity type’s Langevin equation in 6N dimensional phase space or its equivalent Liouville diffusion equation. This equation is time-reversed asymmetrical. It shows that the form of motion of particles in statistical thermodynamic systems has the drift-diffusion duality, and the law of motion of statistical thermodynamics is expressed by a superposition of both the law of dynamics and the stochastic velocity and possesses both determinism and probability. Hence it is different from the law of motion of particles in dynamical systems. The stochastic diffusion motion of the particles is the microscopic origin of macroscopic irreversibility. Starting from this fundamental equation the BBGKY diffusion equation hierarchy, the Boltzmann collision diffusion equation, the hydrodynamic equations such as the mass drift-diffusion equation, the Navier-Stokes equation and the thermal conductivity equation have been derived and presented here. What is more important, we first constructed a nonlinear evolution equation of nonequilibrium entropy density in 6N, 6 and 3 dimensional phase space, predicted the existence of entropy diffusion. This entropy evolution equation plays a leading role in nonequilibrium entropy theory, it reveals that the time rate of change of nonequilibrium entropy density originates together from its drift, diffusion and production in space. From this evolution equation, we presented a formula for entropy production rate (i.e. the law of entropy increase) in 6N and 6 dimensional phase space, proved that internal attractive force in nonequilibrium system can result in entropy decrease while internal repulsive force leads to another entropy increase, and derived a common expression for this entropy decrease rate or another entropy increase rate, obtained a theoretical expression for unifying thermodynamic degradation and self-organizing evolution, and revealed that the entropy diffusion mechanism caused the system to approach to equilibrium. As application, we used these entropy formulas in calculating and discussing some actual physical topics in the nonequilibrium and stationary states. All these derivations and results are unified and rigorous from the new fundamental equation without adding any extra new assumption.     

Solitary Wave Solutions of KP equation,Cylindrical KP Equation and Spherical KP Equation

Three(2+1)-dimensional equations–KP equation, cylindrical KP equation and spherical KP equation, have been reduced to the same Kd V equation by different transformation of variables respectively. Since the single solitary wave solution and 2-solitary wave solution of the Kd V equation have been known already, substituting the solutions of the Kd V equation into the corresponding transformation of variables respectively, the single and 2-solitary wave solutions of the three(2+1)-dimensional equations can be obtained successfully.     

Dynamic characteristics of resonant gyroscopes study based on the Mathieu equation approximate solution

Dynamic characteristics of the resonant gyroscope are studied based on the Mathieu equation approximate solution in this paper.The Mathieu equation is used to analyze the parametric resonant characteristics and the approximate output of the resonant gyroscope.The method of small parameter perturbation is used to analyze the approximate solution of the Mathieu equation.The theoretical analysis and the numerical simulations show that the approximate solution of the Mathieu equation is close to the dynamic output characteristics of the resonant gyroscope.The experimental analysis shows that the theoretical curve and the experimental data processing results coincide perfectly,which means that the approximate solution of the Mathieu equation can present the dynamic output characteristic of the resonant gyroscope.The theoretical approach and the experimental results of the Mathieu equation approximate solution are obtained,which provides a reference for the robust design of the resonant gyroscope.     

Jacobi Elliptic Function Solutions of Some Nonlinear Evolution Equations

In this letter, the modified Jacob/elliptic function expansion method is extended to solve M-coupled KdV equation, M-coupled Ito equation, vKdV equation, and AKNS equation. Some new Jacob/elliptic function solutions are obtained by using Mathematica. When the modulus m→1, those periodic solutions degenerate as the corresponding soliton solutions.     

An exact solution of Fisher equation and its stability

In this paper, the Fisher equation is analysed. One of its travelling wave solution is obtained by comparing it with KdV--Burgers (KdVB) equation. Its amplitude, width and speed are investigated. The instability for the higher order disturbances to the solution of the Fisher equation is also studied.     

New exact solitary wave solutions to generalized mKdV equation and generalized Zakharov--Kuzentsov equation

In this paper, based on hyperbolic tanh-function method and homogeneous balance method, and auxiliary equation method, some new exact solitary solutions to the generalized mKdV equation and generalized Zakharov--Kuzentsov equation are constructed by the method of auxiliary equation with function transformation with aid of symbolic computation system Mathematica. The method is of important significance in seeking new exact solutions to the evolution equation with arbitrary nonlinear term.     

Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations

In this paper we derive generalized forms of the Camassa-Holm (CH) equation from a Boussinesq-type equation using a two-parameter asymptotic expansion based on two small parameters characterizing nonlinear and dispersive effects and strictly following the arguments in the asymptotic derivation of the classical CH equation. The resulting equations generalize the CH equation in two different ways. The first generalization replaces the quadratic nonlinearity of the CH equation with a general power-type nonlinearity while the second one replaces the dispersive terms of the CH equation with fractional-type dispersive terms. In the absence of both higher-order nonlinearities and fractional-type dispersive effects, the generalized equations derived reduce to the classical CH equation that describes unidirectional propagation of shallow water waves. The generalized equations obtained are compared to similar equations available in the literature, and this leads to the observation that the present equations have not appeared in the literature.     

Exact solutions to a class of nonlinear Schrödinger-type equations

A class of nonlinear Schrödinger-type equations, including the Rangwala-Rao equation, the Gerdjikov-Ivanov equation, the Chen-Lee-Lin equation and the Ablowitz-Ramani-Segur equation are investigated, and the exact solutions are derived with the aid of the homogeneous balance principle, and a set of subsidiary higher order ordinary differential equations (sub-ODEs for short).     

Peaked Traveling Wave Solutions to a Generalized Novikov Equation with Cubic and Quadratic Nonlinearities

The Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation are the three typical integrable evolution equations admitting peaked solitons. In this paper, a generalized Novikov equation with cubic and quadratic nonlinearities is studied, which is regarded as a generalization of these three well-known studied equations. It is shown that this equation admits single peaked traveling wave solutions, periodic peaked traveling wave solutions, and multi-peaked traveling wave solutions.     

Trial Equation Method to Nonlinear Evolution Equations with Rank Inhomogeneous:Mathematical Discussions and Its Applications

A trial equation method to nonlinear evolution equation with rank inhomogeneous is given. As applications, the exact traveling wave solutions to some higher-order nonlinear equations such as generalized Boussinesq equation, generalized Pochhammer-Chree equation, KdV-Burgers equation, and KS equation and so on, are obtained. Among these, some results are new. The proposed method is based on the idea of reduction of the order of ODE. Some mathematical details of the proposed method are discussed.     

The stochastic Boltzmann equation and hydrodynamic fluctuations

Based on the assumption of a kinetic equation in space, a stochastic differential equation of the one-particle distribution is derived without the use of the linear approximation. It is just the Boltzmann equation with a Langevin-fluctuating force term. The result is the general form of the linearized Boltzmann equation with fluctuations found by Bixon and Zwanzig and by Fox and Uhlenbeck. It reduces to the general Landau-Lifshitz equations of fluid dynamics in the presence of fluctuations in a similar hydrodynamic approximation to that used by Chapman and Enskog with respect to the Boltzmann equation.This work received financial support from the Alexander von Humboldt Foundation.     

Exact solutions to the generalized lienard equation and its applications

Some new exact solutions of the generalized Lienard equation are obtained, and the solutions of the equation are applied to solve nonlinear wave equations with nonlinear terms of any order directly. The generalized one-dimensional Klein-Gordon equation, the generalized Ablowitz (A) equation and the generalized Gerdjikov-Ivanov (GI) equation are investigated and abundant new exact travelling wave solutions are obtained that include solitary wave solutions and triangular periodic wave solutions.      

Exact Solutions of a Fractional-Type Differential-Difference Equation Related to Discrete MKdV Equation

The extended simplest equation method is used to solve exactly a new differential-difference equation of fractional-type, proposed by Narita [J. Math. Anal. Appl. 381 （2011） 963] quite recently, related to the discrete MKdV equation. It is shown that the model supports three types of exact solutions with arbitrary parameters： hyperbolic, trigonometric and rational, which have not been reported before.