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.     

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.     

The effect of magnetic field heat treatment on magnetostriction of Terfenol-D 2-2 composites

Based on the analysis of the magnetostriction for Terfenol-D composites, Terfenol-D 2-2 magnetostrictive composites have been prepared with laminations perpendicular to [1 1 2] axes. Then one of the samples was annealed in the vacuum at 423 K for 15 min at the magnetic field of 240 kA/m, which is along the direction of laminations and vertical to the [1 1 2] axes of the specimen. The static magnetostriction λ and dynamic magnetostrictive coefficient d₃₃ of samples were measured under the compressive stress of 0, 2, 4, 6 and 8 MPa. Effects of the compressive stress and the magnetic field heat treatment on the magnetostriction λ have been investigated. It is found that the magnetostriction of 2-2 composites can be improved under the compressive stress when the magnetic field is larger than 20 kA/m. The magnetostriction of 2-2 composites with the magnetic field heat treatment increases under compressive stress, and it can reach 1390×10⁻⁶ at the magnetic field of 200 kA/m and under the compressive stress of 4 MPa, much larger than the value of 860×10⁻⁶ without the magnetic field heat treatment. The highest magnetostriction of the 2-2 composite with the magnetic field heat treatment can reach 1530×10⁻⁶. The dynamic magnetostrictive coefficient d₃₃ of 2-2 composites with the magnetic field heat treatment have been improved, compared with that without magnetic field heat treatment. The maximum value of d₃₃ of the sample with magnetic field heat treatment is 71% larger than that without magnetic field heat treatment.     

线性偏振激光在相对论等离子体中的调制不稳定性

从相对论等离体中电磁波的非线性色散方程出发,利用Karpman方法获得了线性偏振波模所满足的非线性控制方程,在非线性色散方程和非线性控制方程的基础上对线性偏振激光在相对论等离体中传播的调制不稳定性进行分析,给出了调制不稳定的时间增长率与扰动态波数之间的函数关系。     

Induced Lie Algebras of a Six-Dimensional Matrix Lie Algebra

By using a six-dimensional matrix Lie algebra [Y.F. Zhang and Y. Wang, Phys. Lett. A 360 （2006） 92], three induced Lie algebras are constructed. One of them is obtained by extending Lie bracket, the others are higher-dimensional complex Lie algebras constructed by using linear transformations. The equivalent Lie algebras of the later two with multi-component forms are obtained as well. As their applications, we derive an integrable coupling and quasi-Hamiltonian structure of the modified TC hierarchy of soliton equations.     

Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-four matrices is presented, and a discrete four-by-four matrix spectral problem is introduced. A hierarchy of discrete integrable coupling systems is derived. The obtained integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discrete Hamiltonian systems.     

Travelling wave-like solutions of the Zakharov-Kuznetsov equation with variable coefficients

Travelling wave-like solutions of the Zakharov-Kuznetsov equation with variable coefficients are studied using the solutions of Raccati equation. The solitary wave-like solution, the trigonometric periodic wave solution and the rational wave solution are obtained with a constraint between coefficients. The property of the solutions is numerically investigated. It is shown that the coefficients of the equation do not change the wave amplitude, but may change the wave velocity.      

The Decay of Massive Scalar Field in Non-Static Gödel Type Universe with Viscous Fluid and Heat Flow

In this study, we have investigated the dynamics of non-static Gödel type rotating universe with massive scalar field, viscous fluid and heat flow in the presence of cosmological constant. For various cosmic matter forms, the behavior of the cosmological constant (Λ), shear (η) and bulk (ξ) viscosity coefficients and other kinematic quantities have studied in the early universe. We have showed the decay of massive scalar field in the non-static rotating Gödel type universe and we have obtained constant scalar field with and without source density. Also, we have investigated the effects of massive scalar field on the matter density and pressure. From solutions of the field equations, we have a cosmological model with non-zero expansion, shear, heat flux and rotation. Also some physical and geometrical aspects of the model discussed.     

Multi-bump, self-similar, blow-up solutions of the Ginzburg-Landau equation

For the Ginzburg-Landau equation (GL), we establish the existence and local uniqueness of two classes of multi-bump, self-similar, blow-up solutions for all dimensions 2<d<4 (under certain conditions on the coefficients in the equation). In numerical simulation and via asymptotic analysis, one class of solutions was already found; the second class of multi-bump solutions is new.In the analysis, we treat the GL as a small perturbation of the cubic nonlinear Schrödinger equation (NLS). The existence result given here is a major extension of results established previously for the NLS, since for the NLS the construction only holds for d close to the critical dimension d=2.The behaviour of the self-similar solutions is described by a nonlinear, non-autonomous ordinary differential equation (ODE). After linearisation, this ODE exhibits hyperbolic behaviour near the origin and elliptic behaviour asymptotically. We call the region where the type of behaviour changes the mid-range. All of the bumps of the solutions that we construct lie in the mid-range.For the construction, we track a manifold of solutions of the ODE that satisfy the condition at the origin forward, and a manifold of solutions that satisfy the asymptotic conditions backward, to a common point in the mid-range. Then, we show that these manifolds intersect transversely. We study the dynamics in the mid-range by using geometric singular perturbation theory, adiabatic Melnikov theory, and the Exchange Lemma.     

The semiclassical modified nonlinear Schrödinger equation I: Modulation theory and spectral analysis

We study an integrable modification of the focusing nonlinear Schrödinger equation from the point of view of semiclassical asymptotics. In particular, (i) we establish several important consequences of the mixed-type limiting quasilinear system including the existence of maps that embed the limiting forms of both the focusing and defocusing nonlinear Schrödinger equations into the framework of a single limiting system for the modified equation, (ii) we obtain bounds for the location of the discrete spectrum for the associated spectral problem that are particularly suited to the semiclassical limit and that generalize known results for the spectrum of the nonselfadjoint Zakharov-Shabat spectral problem, and (iii) we present a multiparameter family of initial data for which we solve the associated spectral problem in terms of special functions for all values of the semiclassical scaling parameter. We view our results as part of a broader project to analyze the semiclassical limit of the modified nonlinear Schrödinger equation via the noncommutative steepest descent procedure of Deift and Zhou, and we also present a selfcontained development of a Riemann-Hilbert problem of inverse scattering that differs from those given in the literature and that is well adapted to semiclassical asymptotics.