Particular solutions of the linearized Boltzmann equation for a binary mixture of rigid spheres

Particular solutions that correspond to inhomogeneous driving terms in the linearized Boltzmann equation for the case of a binary mixture of rigid spheres are reported. For flow problems (in a plane channel) driven by pressure, temperature, and density gradients, inhomogeneous terms appear in the Boltzmann equation, and it is for these inhomogeneous terms that the particular solutions are developed. The required solutions for temperature and density driven problems are expressed in terms of previously reported generalized (vector-valued) Chapman–Enskog functions. However, for the pressure-driven problem (Poiseuille flow) the required particular solution is expressed in terms of two generalized Burnett functions defined by linear integral equations in which the driving terms are given in terms of the Chapman–Enskog functions. To complete this work, expansions in terms of Hermite cubic splines and a collocation scheme are used to establish numerical solutions for the generalized (vector-valued) Burnett functions.     

Particular solutions of the linearized Boltzmann equation for a binary mixture of rigid spheres

Particular solutions that correspond to inhomogeneous driving terms in the linearized Boltzmann equation for the case of a binary mixture of rigid spheres are reported. For flow problems (in a plane channel) driven by pressure, temperature, and density gradients, inhomogeneous terms appear in the Boltzmann equation, and it is for these inhomogeneous terms that the particular solutions are developed. The required solutions for temperature and density driven problems are expressed in terms of previously reported generalized (vector-valued) Chapman–Enskog functions. However, for the pressure-driven problem (Poiseuille flow) the required particular solution is expressed in terms of two generalized Burnett functions defined by linear integral equations in which the driving terms are given in terms of the Chapman–Enskog functions. To complete this work, expansions in terms of Hermite cubic splines and a collocation scheme are used to establish numerical solutions for the generalized (vector-valued) Burnett functions.     

Diffusion limit for the linear boltzmann equation of the neutron transport theory

In this paper we present the asymptotic analysis of the linear Boltzmann equation for neutrons with a small positive parameter ? related to the mean free path, based upon the Chapman–Enskog procedure of the kinetic theory. We prove that if proper initial conditions derived by considering initial layer solutions are used, the diffusion equation gives the uniform approximation to the neutron density function with the O(?²) accuracy.     

Microscopic solutions of kinetic equations and the irreversibility problem

As established by N.N. Bogolyubov, the Boltzmann-Enskog kinetic equation admits the so-called microscopic solutions. These solutions are generalized functions (have the form of sums of delta functions); they correspond to the trajectories of a system of a finite number of balls. However, the existence of these solutions has been established at the “physical” level of rigor. In the present paper, these solutions are assigned a rigorous meaning. It is shown that some other kinetic equations (the Enskog and Vlasov-Enskog equations) also have microscopic solutions. In this sense, one can speak of consistency of these solutions with microscopic dynamics. In addition, new kinetic equations for a gas of elastic balls are obtained through the analysis of a special limit case of the Vlasov equation.     

Group-invariant solutions of the Fokker-Planck equation

Group-invariance under infinitesimal transformations is used to generate a wide class of solutions of some Fokker-Planck equations. The partial differential equation in two variables is reduced to an ordinary differential equation; reduction of the latter to standard forms is noted in most cases. Some of the known existing solutions are obtained as particular cases. Only self-similar types of solutions are discussed. The appearance of a free parameter that can be treated as an eigenvalue (or transform variable) offers flexibility in constructing new solutions. Some solutions of this parabolic equation have wave-like features. The general results can also be used to solve some types of moving-boundary problems.     

Doubly periodic solutions of the modified Kawahara equation

Some doubly periodic (Jacobi elliptic function) solutions of the modified Kawahara equation are presented in closed form. Our approach is to introduce a new auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly periodic solutions of the modified Kawahara equation. When the module m → 1, these solutions degenerate to the exact solitary wave solutions of the equation. Then we reveal the relation of some exact solutions for the modified Kawahara equation obtained by other authors.     

Explicit solutions of the Bogoyavlensky-Konoplechenko equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the Bogoyavlensky-Konoplechenko equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the BK equation. Then we get the reductions using the symmetry and give some exact solutions of the BK equation.     

Infinite energy solutions of the surface quasi-geostrophic equation

We study the formation of singularities of a 1D non-linear and non-local equation. We show that this equation provides solutions of the surface quasi-geostrophic equation with infinite energy. The existence of self-similar solutions and the blow-up for classical solutions are shown.     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

Decreasing solutions and convex solutions of the polynomial-like iterative equation

In this paper we prove the existence and uniqueness of decreasing solutions for the polynomial-like iterative equation so as to answer Problem 2 in [J. Zhang, L. Yang, W. Zhang, Some advances on functional equations, Adv. Math. (China) 24 (1995) 385-405] (or Problem 3 in [W. Zhang, J.A. Baker, Continuous solutions of a polynomial-like iterative equation with variable coefficients, Ann. Polon. Math. 73 (2000) 29-36]). Furthermore, we completely investigate increasing convex (or concave) solutions and decreasing convex (or concave) solutions of this equation so that the results obtained in [W. Zhang, K. Nikodem, B. Xu, Convex solutions of polynomial-like iterative equations, J. Math. Anal. Appl. 315 (2006) 29-40] are improved.     

Novel exact solutions to the short pulse equation

In this paper, the bridge connecting the short pulse equation (SPE for short) with the sine-Gordon equation is applied to construct the novel solutions to the short pulse equation. It is shown that the solutions of the sine-Gordon equation can be used to obtain many different kinds of solutions to the short pulse equation with the aid of symbolic computation and plot representation of Maple.     

New travelling wave solutions to the Ostrovsky equation

In this work, new travelling wave solutions to the Ostrovsky equation are studied by employing the improved tanh function method. With this method, the Ostrovsky equation is reduced to the nonlinear ordinary differential equation and then the different types of exact solutions are derived based on the solutions of the Riccati equation. We will compare our solutions with those gained by the other methods.     

Exact solutions of Kolmogorov-Petrovskii-Piskunov equation using the modified simple equation method

The modified simple equation method is employed to find the exact solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov （KPP） equation. When certain parameters of the equations are chosen to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the modified simple equation method provides an effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.     

Finite-gap elliptic solutions of the KdV equation

A method is proposed for constructing finite-gap elliptic inx or/and int solutions of the Korteweg-de Vries equation. Dynamics of poles for two-gap elliptic solutions of the KdV equation are considered. Numerous examples of new elliptic solutions of the KdV equation are given.Dedicated to the memory of J.-L. Verdier     

Uniqueness of solutions for the extended Fisher-Kolmogorov equation

We consider stationary solutions of the Extended Fisher-Kolmogorov (EFK) equation, a fourth-order model equation for bi-stable systems. We show that as long as the stable equilibrium points are real saddles, the paths in the (u, u')-plane of two bounded solutions do not cross. As a consequence we derive that the bounded solutions of the EFK equation correspond exactly to those of the classical Fisher-Kolmogorov equation.     

求Nizhnik-Novikov-Veselov方程新精确解

引入改进的F-广义方法,并将其应用于（2＋1）维Nizhnik-Novikov-Veselov（NNV）方程.在符号计算软件的帮助下,可以得到NNV方程的许多新解.该方法用于获取包括雅可比椭圆函数解的一系列解,在数学物理中可应用于其他的非线性偏微分方程.     

Exact solutions of the Kudryashov-Sinelshchikov equation

The Kudryashov-Sinelshchikov equation for describing the pressure waves in liquid with gas bubbles is studied. New exact solutions of this equation are found. Modification of truncated expansion method is used for obtaining exact solution of this equation.     

New double Wronskian solutions of the AKNS equation

Soliton solutions, rational solutions, Matveev solutions, complexitons and interaction solutions of the AKNS equation are derived through a matrix method for constructing double Wronskian entries. The latter three solutions are novel. Moreover, rational solutions of the nonlinear Schrodinger equation are obtained by reduction.     

Similarity inner solutions for the Pulsar equation

Lie symmetries are applied to classify the source of the magnetic field for the Pulsar equation near to the surface of the neutron star. We find that there are six possible different admitted Lie algebras. We apply the corresponding Lie invariants to reduce the Pulsar equation close to the surface to an ordinary differential equation. This equation is solved either with the use of Lie symmetries or the application of the ARS algorithm for singularity analysis to write the analytic solution as a Laurent expansion. These solutions are called inner solutions.