Crystallographic groups and homogeneous statistical solutions of navier-stokes equations

In this paper, we investigate the connection between crystallographic groups and homogeneous statistical solutions of Navier-Stokes equations. Several results of Foias and Temam are extended. Fluid flows invariant under crystallographic groups are studied. This idea may be of interest to the understanding of bifurcation and turbulence.     

On the well-posedness of the Eckhaus equation

The Eckhaus equation, which is a nonlinear Schrödinger type equation that can be linearized to the free linear Schrödinger equation, is considered. A linearized analysis of the nonlinear problem indicates that the periodic boundary value problem is ill-posed. The exact solution also demonstrates the ill-posedness, even though the L² norm of the solution is a constant of the motion. The ill-posedness disappears on the infinite line with square integrable data, but the solitonic solutions, which are not square integrable, are seriously unstable.     

On the derivation of thermohydrodynamic equations from the Boltzmann equations

The Boltzmann equation deals with a distributionf(x, ), wherex denotes the space variable and is the momentum. The hydrodynamic equations deal with-moments of the distribution. The paper deals with the derivation of the hydrodynamic equations in the case that the collision kernel is Maxwellian, i.e., independent of the velocity. For such a kernel, a computational tool, based on the theory of representations of the orthogonal group, is developed. With this tool it is possible to derive systems of equations for any number of moments. The construction of closed systems is based on asymptotic estimates for solutions of Boltzmann equations. These show that, in some definite sense, an approximating system involving moments of high order is more accurate than a system of lower order.     

On the global well-posedness of the Euler-Boussinesq system with fractional dissipation

We study the global well-posedness of the Euler-Boussinesq system with the term dissipation on the temperature equation. We prove that for α>1 the coupled system has a global unique solution for initial data with critical regularities.     

On a class of solutions of nonlinear Boltzmann equations

In a recent paper by Krook and Wu, the nonlinear Boltzmann equation for an infinite, spatially homogeneous, isotropic monoatomic gas of constant density and kinetic energy and with an elastic differential cross section that varies inversely as relative speed has been reduced to an infinite sequence of moment equations. The present note observes that the moment equations are successively integrable and shows that as time goes to infinity, the distribution tends to be Maxwellian.     

On the periodic solutions for both nonlinear differential and difference equations: A unified approach

A direct and unifying scheme for disclosure of periodic wave solutions of both nonlinear differential and difference equations is presented. The scheme is based on Hirota's bilinear form and certain Riemann theta function formulae. The relations between the periodic wave solutions and soliton solutions are rigorously established.     

Some new results for the multidimensional Landau-Lifshitz equations

In this Letter, we study the multidimensional Landau-Lifshitz equations for two cases of external magnetic fields depending on both time and position, and obtain some new interesting results. For the strong degenerate Landau-Lifshitz equation, we obtain a new blow up solution.     

Numerical solutions to the time-dependent Bloch equations revisited

The purpose of this study was to demonstrate a simple and fast method for solving the time-dependent Bloch equations. First, the time-dependent Bloch equations were reduced to a homogeneous linear differential equation, and then a simple equation was derived to solve it using a matrix operation. The validity of this method was investigated by comparing with the analytical solutions in the case of constant radiofrequency irradiation. There was a good agreement between them, indicating the validity of this method. As a further example, this method was applied to the time-dependent Bloch equations in the two-pool exchange model for chemical exchange saturation transfer (CEST) or amide proton transfer (APT) magnetic resonance imaging (MRI), and the Z-spectra and asymmetry spectra were calculated from their solutions. They were also calculated using the fourth/fifth-order Runge-Kutta-Fehlberg (RKF) method for comparison. There was also a good agreement between them, and this method was much faster than the RKF method. In conclusion, this method will be useful for analyzing the complex CEST or APT contrast mechanism and/or investigating the optimal conditions for CEST or APT MRI.     

Bifurcation methods of dynamical systems for handling nonlinear wave equations

By using the bifurcation theory and methods of dynamical systems to construct the exact travelling wave solutions for nonlinear wave equations, some new soliton solutions, kink (anti-kink) solutions and periodic solutions with double period are obtained.      

Integrable Geodesic Flows on the (Super)extension of the Bott–Virasoro Group

In this Letter, we present an answer to the question posed by Marcel, Ovsienko and Roger in their paper (Lett. Math. Phys. 40 (1997), 31–39). The Itô equation, modified dispersive water wave equation and modified dispersionless long wave equation are shown to be the geodesic flows with respect to an L ² metric on the semidirect product space Diff ^s C (S ¹), where Diff ^s (S ¹) is the group of orientation-preserving Sobolev H ^s diffeomorphisms of the circle. We also study the geodesic flows with respect to H ¹ metric. The geodesic flows in this case yield different integrable systems admitting nonlinear dispersion terms. These systems exhibit more general wave phenomena than usual integrable systems. Finally, we study an integrable geodesic flow on the extended Neveu–Schwarz space.     

Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Exp-function method

A new generalized solitary solution of the Jaulent-Miodek equations is obtained using the Exp-function method. By a transformation, the solitary solution can be easily converted into a generalized compacton-like solution. The free parameters in the obtained generalized solutions might imply some meaningful results in physical process.     

On the Bethe-Lamb equations

A derivation is given of an effective Hamiltonian to be used in treating the decay of atomic states perturbed by a static applied field. This Hamiltonian is an improvement on that usually used in writing Bethe-Lamb equations. It includes the virtues of a Stark-perturbed golden rule method which has recently proved successful in treating decays of metastable states in helium, where the usual Bethe-Lamb approach fails.     

Reduction operators and exact solutions of generalized Burgers equations

Reduction operators of generalized Burgers equations are studied. A connection between these equations and potential fast diffusion equations with power nonlinearity of degree −1 via reduction operators is established. Exact solutions of generalized Burgers equations are constructed using this connection and known solutions of the constant-coefficient potential fast diffusion equation.     

几种辅助方程与非线性发展方程的无穷序列精确解

本文为了获得非线性发展方程新的无穷序列精确解,给出了几种辅助方程的Böcklund变换和解的非线性叠加公式,并构造了一些非线性发展方程新的无穷序列精确解,其中包括无穷序列Jacobi椭圆函数解、无穷序列双曲函数解和无穷序列三角函数解.该方法在构造非线性发展方程无穷序列精确解方面具有普遍意义. 关键词： 辅助方程法 解的非线性叠加公式 无穷序列解 非线性发展方程     

On Nth-order rogue wave solution to nonlinear coupled dispersionless evolution equations

The generalized Darboux transformation is applied to obtain Nth-order rogue wave solution of nonlinear coupled dispersionless evolution equations (NLCDEE). In particular, the interesting structures for these solutions from first to third order rogue wave are shown.     

三Riccati 方程的新展开法及其在非线性数学物理方程中的应用

通过对tanh函数法和射影Riccati方程法的探讨, 提出一种新的算法来求解非线性发展方程. 并且通过求解高阶 schrodinger 方程和 mKdV 方程来说明该算法.得到了这些方程的新形式解,包括新的孤立波解, 周期解等, 并图示一些新形式解.     

On the discretization of Laine equations

We consider the discretization of Darboux integrable equations. For each of the integrals of a Laine equation we constructed either a semi-discrete equation which has that integral as an n-integral, or we proved that such an equation does not exist. It is also shown that all constructed semi-discrete equations are Darboux integrable.     

On the universality of Einstein equations

It is proved that a Lagrangian field theory based on a linear connection in space-time is equivalent to Einstein's general relativity interacting with additional matter fields.     

Narrow bounds for numerical integration of differential equations

Through a detailed analysis of the properties of a system of differential equations, bounds are given for the error affecting the final result of a numerical integration. These bounds appear to be narrower than those obtained with other methods. The key procedure is to consider carefully the linear part of the system and to bound it taking account of all possible errors. No very significant restriction is made on the system.This work was partially supported by the Ministero della Pubblica Istruzione.