New exact solutions for two nonlinear equations

In this Letter, we investigate two nonlinear equations given by ut−uxxt+3u²ux=2uxuxx+uuxxx and ut−uxxt+4u²ux=3uxuxx+uuxxx. Through some special phase orbits we obtain four new exact solutions for each equation above. Some previous results are extended.     

Expanding the class of general exact solutions for interacting two field kinks

In this work we extend the range of applicability of a method recently introduced where coupled first-order nonlinear equations can be put into a linear form, and consequently be solved completely. Some general consequences of the present extension are then commented.     

Integrals of motion in quantum mechanics

It is shown within the framework of the local approach that the dimensionality of the commutative subalgebras of observables, leading to a separation of variables, is equal to the number of independent variables. A number of questions relating to the degeneracy of the spectrum are also considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 13–17, November, 1976.     

Painlevé analysis and exact solutions of two dimensional Korteweg-de Vries-Burgers equation

Two dimensional Korteweg-de Vries-Burgers equation is shown to be non-integrable using Painlevé analysis. Exact travelling wave solutions are obtained using an algorithmic approach of truncating the Painlevé series expansions.     

Nonergodicity of the motion in three-dimensional steep repelling dispersing potentials

It is demonstrated numerically that smooth three degrees of freedom Hamiltonian systems that are arbitrarily close to three-dimensional strictly dispersing billiards (Sinai billiards) have islands of effective stability, and hence are nonergodic. The mechanism for creating the islands is corners of the billiards domain.     

Integrals of the motion for a nonlinear quantum oscillator

We construct and discusss explicitly time dependent integrals of the motion of non-autonomous quantum systems. Such integrals may exist even when the classical limit of the dynamics is non-integrable.     

Reproduction of solutions in the plane problem on motion of a free-boundary fluid

This study is devoted to finding exact solutions of the plane unsteady problem on the motion of an ideal incompressible free-boundary fluid. A certain procedure of reproduction making it possible to obtain a two-parametrical family of new exact solutions from one known solution is proposed.     

New exact solutions of the Dirac equation

The search for new exact solutions to the Dirac and Klein-Gordon equations initiated in [1] is continued. New solutions are found for axisymmetric fields and one type of nonstationary field of special configuration. The basic notation and system of units of [1] are retained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 10–16, April, 1980.     

The exact equation of motion for a two level system. Zubarev like approach

A simple model, one spinS=1/2 interacting with an alternating transverse field and with a single mode of a boson field, treated as an open system, is rigorously investigated. An approach based on the Zubarev method is applied. An integro-differential equation for a mean value of the Zeeman operator is derived. Kernels of this equation are defined as solutions of some functional (integral) equations. A particular case, known as the Jaynes-Cummings model in quantum optics, is considered. The Markovian limit of the integro-differential equation for the Jaynes-Cummings model leads to a simple relaxation equation.Supported in part by the Polish Academy of Sciences.The author would like to thank Prof. A. Pawlikowski for his continual interest in the present work and useful comments and discussions. The author is also indebted to Dr. E. Malec and Dr. J. Aksamit for their help during the preparation of this paper.     

On exact solutions of the Bogoyavlenskii equation

Exact solutions for the Bogoyavlenskii equation are studied by the travelling wave method and the singular manifold method. It is found that the linear superposition of the shock wave solution and the complex solitary wave solution for the physical field is still a solution of the equation of interest, except for a phase-shift. The dromion-like structures with elastic and nonelastic interactions are found.     

Integrals of motion for some equations onq-minkowski space

The conservation laws for second order linear equation with constant coefficients on braided linear space are derived. As an example we study conserved currents connected with symmetry operators for scalar and spinor wave equations onq-Minkowski space. Then we formulate set of conditions for integral over spatial dimensions and use the postulated integral in construction of constant of motion for arbitrary conserved current onq-Minkowski space. Presented at the 6th International Colloquium on Quantum Groups “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

修正的BBM方程新的精确解

应用进一步修正的简单方程法对修正的 Benjamin -Bona -Mahoney (mBBM )方程进行求解,给出了mBBM方程新的精确类孤波解,取定某些参数值,便可得到精确孤波解.这种方法也可用于寻找其它常系数以及变系数非线性发展方程(组)的精确解,具有一定的普适性.     

New exact solutions of the Yang-Mills equation

New exact classical solutions of the Euclidean Yang-Mills equation are constructed. The exact solutions presented here are characterized by infinite values of topological charges and therefore they are in striking contrast to the well-known instanton and meron solutions which have finite topological charges.     

Parametrically driven pendulum and exact solutions

The parametrically driven pendulumx +f ₁(t) x +f ₂(t) sinx = 0 cannot be solved in closed form for arbitrary functionvsf ₁,f ₂. We apply the Painlevé test to obtain the constraint on the functionsf ₁, andf ₂ for which the equation passes the test. The constraint onf ₁, andf ₂, a differential equation whichf ₁ andf ₂ obey, is discussed and solutions are given. The third Painlevé transcendent plays a central role.     

Nonlocal gravitational models and exact solutions

A nonlocal gravity model with a function f (□^?1 R), where □ is the d’Alembert operator, is considered. The algorithm, allowing to reconstruct f(□^?1 R), corresponding to the given Hubble parameter and the state parameter of the matter, is proposed. Using this algorithm, we find the functions f(□^?1 R), corresponding to de Sitter solutions.     

Nonlinear model-Boltzmann equations and exact solutions

The present paper is devoted to the subject of photonuclear reactions of the type (γ, p), (γ, n), (γ, np), (γ, nn), (γ, pp) as well as nucle on capture reactions as (p, γ) at intermediate photon energies. The theoretical part of this paper reviews and extends earlier calculations of the authors up t gamma energies of E_γ ≈ 400 MeV. The paper is not intended to give a review on different theoretical approaches which have been underta ken in the past. Instead, emphasis is put on the dynamical aspects of the photonuclear reactions in order to see which elementary processes govern the reactions at different photon energies. In view of the success of the present treatment up to E_γ ≈ 120 MeV photon energy, we tempt the predictive power of the model up to photon energies of E_γ ≈ 400 MeV for the reactions in ⁴He and ¹⁶O.     

Invariance transformations and exact solutions of the Rabi model

The invariance transformations of the Rabi model describing the interaction of a two-level system with anl-mode electromagnetic field are constructed. On this basis explicit expressions for the coherent states and Green function of the problem are obtained.     

The construction of exact solutions of the model problem on rotating fluid in domains with angular points

A new method has been developed for the construction of a two-dimensional model problem on the oscillations of rotating ideal fluid in some domains containing angular points. It is proved that these solutions correspond to the absolutely continuous component of the spectrum of the linear operator connected with the problem.     

New explicit and exact solutions of the Benney--Kawahara--Lin equation

In this paper, we present a combination method of constructing the explicit and exact solutions of nonlinear partial differential equations. And as an illustrative example, we apply the method to the Benney-Kawahara-Lin equation and derive its many explicit and exact solutions which are all new solutions.