Analytical solutions to time-fractional partial differential equations in a two-dimensional multilayer annulus

In this paper, analytical solutions to time-fractional partial differential equations in a multi-layer annulus are presented. The final solutions are obtained in terms of Mittag-Leffler function by using the finite integral transform technique and Laplace transform technique. In addition, the classical diffusion equation (α=1$\alpha =1$), the Helmholtz equation (α→0$\alpha \to 0$) and the wave equation (α=2$\alpha =2$) are discussed as special cases. Finally, an illustrative example problem for the three-layer semi-circular annular region is solved and numerical results are presented graphically for various kind of order of fractional derivative.     

Exact solutions to a coupled modified KdV equations with non-uniformity terms

The bilinear form of a coupled modified KdV equations with non-uniformity terms is given and a few soliton solutions are obtained. Furthermore, the multisoliton of the coupled system is expressed by Pfaffian.     

New exact solutions to a system of coupled KdV equations

Using an improved homogeneous balance method, several kinds of exact solutions which include Wang's results are obtained for a system of coupled KdV equations.     

Some quasi-periodic solutions to the Kadometsev-Petviashvili and modified Kadometsev-Petviashvili equations

The Kadometsev-Petviashvili (KP) and modified KP (mKP) equations are retrieved from the first two soliton equations of coupled Korteweg-de Vries (cKdV) hierarchy. Based on the nonlinearization of Lax pairs, the KP and mKP equations are ultimately reduced to integrable finite-dimensional Hamiltonian systems in view of the r-matrix theory. Finally, the resulting Hamiltonian flows are linearized in Abel-Jacobi coordinates, such that some specially explicit quasi-periodic solutions to the KP and mKP equations are synchronously given in terms of theta functions through the Jacobi inversion.     

On reduction of a two-dimensional generalized Toda lattice to an ordinary differential equations system

Reduction of a two-dimensional generalized Toda lattice to an ordinary differential equations system, which defines the functional dependence of Toda functions on the corresponding separable solutions, is given. It is shown, in particular, that between all the equations of the form ²/x t=() only Liouville (L), sine-Gordon (SG) and Bullough-Dodd (BD) equations, which are associated with simple Lie algebras of a finite growth of rank 1, lead to the third Painlevé equation (P3). The last circumstance allows one, probably, to assume the existence of a deep relation between the complete integrability condition for the corresponding class of dynamical systems and the criterion of the absence of movable critical points in the solutions of an ordinary differential equation system of the second order. If this is so, it means that Painleé's equations and transcendents can be generalized for multi-component cases.Tbilisi State University     

Oscillations and concentrations in weak solutions of the incompressible fluid equations

The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations. One application of the concepts developed here is the following important result: a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of 3-D Euler defined for all positive times. The authors present several explicit examples of solution sequences for 3-D incompressible Euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations. An extensions of the concept of Young measure is developed to incorporate these complex phenomena in the measure-valued solutions constructed here.Partially supported by N.S.F. GrantPartially supported by N.S.F. Grant 84-0223 and 86-11110     

Variable separation solutions and interacting waves of a coupled system of the modified KdV and potential BLMP equations

The multilinear variable separation approach (MLVSA) is applied to a coupled modified Korteweg–de Vries and potential Boiti–Leon–Manna–Pempinelli equations, as a result, the potential fields _uy$u_y$ and _vy$v_y$ are exactly the universal quantity applicable to all multilinear variable separable systems. The generalized MLVSA is also applied, and it is found that _uy$u_y$ (_vy$v_y$) is rightly the subtraction (addition) of two universal quantities with different parameters. Then interactions between periodic waves are discussed, for instance, the elastic interaction between two semi-periodic waves and non-elastic interaction between two periodic instantons. An attractive phenomenon is observed that a dromion moves along a semi-periodic wave.     

Growth rate of transverse instabilities of solitary pulse solutions to a family of modified Zakharov-Kuznetsov equations

Using the small-k expansion method, we obtain a closed-form expression for the growth rate of long-wavelength transverse instabilities of solitary pulse solutions to a modified Zakharov-Kuznetsov equation with a nonlinearity of the form (Aup+Bu2p)ux.     

Random solutions to a system of fractional differential equations via the Hadamard fractional derivative

In this work, we use a new random fixed point theorem in vector metric spaces due to Sinacer et al. [M.L. Sinacer et al., Random Oper. Stoch. Equ. 24, 93 (2016)] to prove the existence of solutions and the compactness of solution sets of a random system of fractional differential equations via the Hadamard-type derivative. The existence, modification and stochastically continuity of an M²-solution are also proved.     

Generation of static solutions of a self-consistent system of Einstein-Klein-Gordon equations

A theorem is proved according to which a class of static solutions of a self-consistent system of Einstein-Klein-Gordon equations dependent on one arbitrary function is set in correspondence with a static solution of the Einstein equations with any given energy-momentum tensor T_ij. Two particular cases are examined as an illustration of this theorem. Methods of constructing the static solutions of a system of Einstein-Klein-Gordon equations with an ideal fluid energy-momentum tensor and a massive scalar field are indicated therein.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 11–14, June, 1989.     

两自由度带电耦合振子系统的守恒量与近似解

由于两自由度带电耦合振子系统的Lagrange函数中存在耦合项,从而导致其运动微分方程是非线性耦合的.先通过坐标变换消去Lagrange函数中的耦合项,用直接积分法求得系统的守恒量,用Adomian分解法求得系统的近似解,再通过坐标反变换求得系统在原坐标下的守恒量与近似解,并对近似解作了讨论.     

Orbital equations in a weak gravitational field

With the obtainment of the Synge equations of motion in third approximation for a test particle, we determine the relativistic force in the field generated by a massive body obtained previously in terms of its potentials. Afterwards, in order to study the motion, we obtain the integral of energy and the integral of angular momentum. By means of them, the general form for the trajectory of an equatorial orbit and for the advance of its apsidal line are obtained. As we will see, the known diverse contributions for this advance appears in the general form after strong supplementary conditions on the potentials. As application, with such assumptions these contributions are derived in a unified way.     

Exact solutions for KdV system equations hierarchy

Exact solutions for KdV system equations hierarchy are obtained by using the inverse scattering transform. Exact solutions of isospectral KdV hierarchy, nonisospectral KdV hierarchies and τ$\tau$-equations related to the KdV spectral problem are obtained by reduction. The interaction of two solitons is investigated.     

Global weak solutions of the Vlasov-Maxwell system with boundary conditions

Boundaries occur naturally in physucal systems which satisfy the Vlasov-Maxwell system. Assume perfect conductor boundary conditions for Maxwell, and either specular reflection or partial absorption for Vlasov. Then weak solutions with finite energy exist for all time.This research is supported in part by NSF Grant DMS 90-23864     

Nontrivial solutions of the Einstein-Yang-Mills equations for a system of massive monopoles

Nontrivial solutions are obtained for the Einstein-Yang-Mills equations, as applied to a system of N slowly moving massive monopoles with a linear accuracy relative to the interaction constant.Astrophysics Institute, Kazakh Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 59–62, February, 1995.     

A note on the cylindrically symmetric,static solutions of the weak field equations

The general solution of the weak field equations of the nonsymmetric unified field theory in the magnetic case is reduced to the solution of a second-order differential equation with movable critical points. Some special solutions are discussed.     

一类非线性方程的新周期解

把Jacobi椭圆函数展开法扩展到Jacobi椭圆余弦函数和第三类Jacobi椭圆函数的有限展开法,并给出了一类非线性波动方程的新周期解,并且应用这种方法得到的周期解也可以退化为冲击波解或孤波解. 关键词： Jacobi椭圆函数 非线性方程 周期解 孤波解     

Magnetoresistance of two-dimensional tight-binding electrons in a weak magnetic field

We study the Anderson model on a two-dimensional square lattice with an applied weak magnetic field B which causes the hopping matrix elements to have Peierls phase factors. The recursion method is applied and B dependent conductivity σ(B) is calculated from the Kubo formula for different system sizes N and degree of disorder W. For large W there is no appreciable change of σ(B) with B, but its system size dependence is first an increasing and then a decreasing behavior.