Symmetry Reduction and Cauchy Problems for a Class of Fourth-Order Evolution Equations

We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations to Cauchy problems for systems of ordinary differential equations (ODEs). We classify a class of fourth-order evolution equations which admit certain higher-order generalized conditional symmetries (GCSs) and give some examples to show the main reduction procedure. These reductions cannot be derived within the framework of the standard Lie approach, which hints that the technique presented here is something essential for the dimensional reduction of evolution equations.     

Spinless Duffin-Kemmer-Petiau Oscillator in a Galilean Non-commutative Phase Space

We examine Galilei-invariant linear wave equations in a non-commutative phase space. Specifically, we establish and solve the Galilean covariant Duffin-Kemmer-Petiau equation for spin-0 fields in a harmonic oscillator potential. We obtain these wave equations with a Galilean covariant approach, based on a (4+1)-dimensional manifold with light-cone coordinates followed by a reduction to a (3+1)-dimensional spacetime. We find the exact wave functions and their energy levels, and we examine the effects of non-commutativity.     

Riemann Invariants and Rank-k Solutions of Hyperbolic Systems

Abstract

In this paper we employ a “direct method” to construct rank-k solutions, expressible in Riemann invariants, to hyperbolic system of first order quasilinear di!erential equations in many dimensions. The most important feature of our approach is the analysis of group invariance properties of these solutions and applying the conditional symmetry reduction technique to the initial equations. We discuss in detail the necessary and su"cient conditions for existence of these type of solutions. We demonstrate our approach through several examples of hydrodynamic type systems; new classes of solutions are obtained in a closed form.     

INTEGRABILITY OF LIE SYSTEMS THROUGH RICCATI EQUATIONS

Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyze a geometric method to construct integrability conditions for Riccati equations following these approaches. Our procedure provides us with a unified geometrical viewpoint that allows us to analyze some previous works on the topic and explain new properties. Moreover, this new approach can be straightforwardly generalized to describe integrability conditions for any Lie system. Finally we show the usefulness of our treatment in order to study the problem of the linearizability of Riccati equations.     

Maxwell-Schrödinger equations for a dilute gas Bose-Einstein condensate coupled to an electromagnetic field

We give a general formulation of the semiclassical approach to solving the problem of interaction between a Bose-Einstein condensate of dilute gas and electromagnetic radiation without using the commonly applied mean-field approximation. We suggest variants of the systems of Maxwell-Schrödinger equations whose solution describes such effects as superradiant light scattering, light beam amplification, atomic wave (atomic laser) amplification, induced transparency, and reduction in the group velocity of light.     

Similarity Reductions and Painlev6 Properties for the Kupershmidt Equations

Five types of similarity reductions of the Kupershmidt equations which admit a tri-Hamiltonian structure are found by a direct method. Two types of reduction equations which are Painlevé Ⅱ and IV types are coincident with those obtained by classical Lie approach. Both algebraic and logarithmic branch points for time t can be entered into the solutions of Kupershmidt equations. The integrability of the Kupershmidt equations is re-examined by the singularity analysis using the Weiss-Kruskal approach and the A blowitz-Ramani-Segur algorithm.     

Non-linear models of geophysical hydrodynamics and the problem of forecasting stream patterns I

The dynamic stochastic approach to the study of mathematical models of thermohydrodynamic large-scale fields is developed in which the mathematical image of stochasticity is the strange attractor of the real atmosphere. This approach is based on methods of analysing non-linear equations of atmosphere. The approximation of these equations is mostly effected using Galerkin's procedures. This reduction is based on the theorem of invariant manifold, especially on the theorem of central manifold for semi-flow of the Navier-Stokes equation. According to the theorem of the central manifold, all significant phenomena related to dynamic systems of thermohydrodynamic equations of atmosphere occur in a particular finite-dimensional formulation. General circulation could be modelled with the aid of relatively single dynamic systems. We shall continue deal with the finite-dimensional approximations of the dissipative, non-divergent, barotropic flow and the bifurcation analysis of the spectral models with a small number of spectral modes and an external force acting only on fundamental modes.     

Tau Functions of q-Painlevé III and IV Equations

We present a geometric approach to τ-functions of the q-Painlevé III and IV equations via rational surfaces with affine Weyl group symmetry of type (A ₂+A ₁)⁽¹⁾. We also study a similarity reduction of the q-KP hierarchy to the equations. Mathematics Subject Classification (2000). 34M55, 37K10, 39A13     

Fourier–Mukai and Nahm transforms for holomorphic triples on elliptic curves

We define a Fourier–Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases, the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic equations on a triple—vortices—and explore some of its basic properties. Our approach combines direct methods with dimensional reduction techniques, relating triples over a curve with vector bundles over the product of the curve with the complex projective line.     

Fourier–Mukai and Nahm transforms for holomorphic triples on elliptic curves

We define a Fourier–Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases, the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic equations on a triple—vortices—and explore some of its basic properties. Our approach combines direct methods with dimensional reduction techniques, relating triples over a curve with vector bundles over the product of the curve with the complex projective line.     

Lie Point Symmetries and Exact Solutions of Couple KdV Equations

The simple Lie point symmetry reduction procedure is used to obtain infinitely many symmetries to a new integrable system of coupled KdV equations. Using some symmetry subalgebra of the equations, five types of the significant similarity reductions are obtained by virtue of the Lie group approach, and obtain abundant solutions of the coupled KdV equations, such as the solitary wave solution, exponential solution, rational solution, polynomial solution, etc.     

Symmetries and conservation laws of lattice Boussinesq equations

Sequences of canonical conservation laws and generalized symmetries for the lattice Boussinesq and the lattice modified Boussinesq systems are successively derived. The interpretation of these symmetries as differential-difference equations leads to corresponding hierarchies of such equations for which conservation laws and Lax pairs are constructed. Finally, using the continuous symmetry reduction approach, an integrable, multidimensionally consistent system of partial differential equations is derived in relation with the lattice modified Boussinesq system.     

Special Conditional Similarity Reduction Solutions for Two Nonlinear Partial Differential Equations

We present a method of special conditional similarity reduction solutions for nonlinear partial differential equations, As concrete examples of its application, we apply this method to the （2＋1）-dimensional modified Broer- Kaup equations and the variable coefficient KdV-mKdV equation, which have extensive physics backgrounds, and obtain abundant exact solutions derived from some reduction equations.     

Special Conditional Similarity Reduction Solutions for Two Nonlinear Partial Differential Equations

We present a method of special conditional similarity reduction solutions for nonlinear partial differential equations. As concrete examples of its application, we apply this method to the (2 1)-dimensional modified Broer-Kaup equations and the variable coefficient KdV-mKdV equation, which have extensive physics backgrounds, and obtain abundant exact solutions derived from some reduction equations.     

Symplectic reduction of holonomic open-chain multi-body systems with constant momentum

This paper presents a two-step symplectic geometric approach to the reduction of Hamilton’s equation for open-chain, multi-body systems with multi-degree-of-freedom holonomic joints and constant momentum. First, symplectic reduction theorem is revisited for Hamiltonian systems on cotangent bundles. Then, we recall the notion of displacement subgroups, which is the class of multi-degree-of-freedom joints considered in this paper. We briefly study the kinematics of open-chain multi-body systems consisting of such joints. And, we show that the relative configuration manifold corresponding to the first joint is indeed a symmetry group for an open-chain multi-body system with multi-degree-of-freedom holonomic joints. Subsequently using symplectic reduction theorem at a non-zero momentum, we express Hamilton’s equation of such a system in the symplectic reduced manifold, which is identified by the cotangent bundle of a quotient manifold. The kinetic energy metric of multi-body systems is further studied, and some sufficient conditions are introduced, under which the kinetic energy metric is invariant under the action of a subgroup of the configuration manifold. As a result, the symplectic reduction procedure for open-chain, multi-body systems is extended to a two-step reduction process for the dynamical equations of such systems. Finally, we explicitly derive the reduced dynamical equations in the local coordinates for an example of a six-degree-of-freedom manipulator mounted on a spacecraft, to demonstrate the results of this paper.     

Whitham-Broer-Kaup浅水波方程的对称性约化

本文利用群论方法和直接法给出Whitham-Broer-Kaup浅水波方程的5种类型的对称性约化。群论方法得到的Painlevé Ⅱ型约化仅仅是直接法约化的一种特殊情况。在直接法的约化结果中包含有关于时间变量t的3种类型的奇点:极点,代数支点和对数支点。 关键词：     

Superspace approach to quantum gauge theories

We present an approach to quantum gauge theories formulated entirely on a superspace. We show that at the classical level the field equations are the same as in the usual Minkowski-space approach. In particular the a-flatness conditions, which represent the BRS and anti-BRS covariance in the usual approach, appear as field equations. We show that the theory is renormalizable and the a-flatness conditions are stable under renormalization. We speculate about the relevance of this approach to the confinement problem.     

联立薛定谔方程的不传播光孤子和传播光孤子

映射法是一种非常经典、有效而且非常成熟的一种求解非线性演化方程的方法,其最大的特点是可以有无穷多个不同形式的设解,使得最终求得的解丰富多彩。传统的方法是在行波约化的前提下,即在常微分方程下进行映射。将这种方法进行扩展,推广成变系数的非行波约化下的映射,取得了成功,并利用改进的里卡蒂(Riccati)方程映射法,得到了联立薛定谔方程(负KdV方程)新的精确解。根据所得到的解模拟出了联立薛定谔方程的不传播光孤子(时间光孤子和亮-暗脉冲光孤子)和传播光孤子,以及光孤子的中和现象。     

Reduction by λ –symmetries and σ –symmetries: a Frobenius approach

Different kinds of reduction for ordinary differential equations, such as λ –symmetry and σ –symmetry reductions, are recovered as particular cases of Frobenius reduction theorem for distribution of vector fields. This general approach provides some hints to tackle the reconstruction problem and to solve it under suitable assumptions on the distribution involved in the reduction process.     

Compact and noncompact structures for nonlinear fractional evolution equations

This Letter deals with compact and noncompact solutions for nonlinear evolution equations with time-fractional derivatives. We present a reliable approach of the homotopy perturbation method to handle nonlinear fractional evolution equations. The validity of the approach is verified through illustrative examples. New exact solitary wave and compacton solutions are developed. The proposed technique could lead to a promising approach for a wide class of nonlinear fractional evolution equations.