Exact travelling wave solutions for some important nonlinear physical models

The two-dimensional nonlinear physical models and coupled nonlinear systems such as Maccari equations, Higgs equations and Schrödinger–KdV equations have been widely applied in many branches of physics. So, finding exact travelling wave solutions of such equations are very helpful in the theories and numerical studies. In this paper, the Kudryashov method is used to seek exact travelling wave solutions of such physical models. Further, three-dimensional plots of some of the solutions are also given to visualize the dynamics of the equations. The results reveal that the method is a very effective and powerful tool for solving nonlinear partial differential equations arising in mathematical physics.     

Exact solutions in cosmological inflationary models

The exact solutions of the Einstein equations that correspond to a given evolution of the scale factor in the inflationary stage in the development of the universe are considered within the framework of the theory of a self-acting scalar field that interacts minimally with a gravitational field. The form of the potential that ensures a given rate of expansion and the solution for the scalar field are calculated for each type of inflation considered. Some general laws of the relation between the form of potential and the scale factor evolution are analyzed. Moscow State University Branch in Ul'yanovsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 83–88, August, 1996.     

Exact solutions of some nonlinear equations

We obtain exact solutions of three nonlinear diffusive equations and of the KdV-Burger equation by making an ansatz for the solution in each case.     

The relaxation dynamics of a confined glassy simple liquid

We use molecular-dynamics computer simulations to study the relaxation dynamics of a confined simple liquid. Two types of confining walls are considered: A rough wall and a smooth wall. The simulation is set up in such a way that the static properties of the confined system are identical to the ones of the bulk. Nevertheless, we find that upon cooling the relaxation dynamics of the confined systems differ strongly from the one of the bulk. In particular, we find that close to the rough/smooth wall this dynamics is slowed down/accelerated by orders of magnitude. Using these results we are able to extract a dynamical length scale of the system and we show that this length shows an Arrhenius dependence.Received: 1 January 2003, Published online: 14 October 2003PACS: 64.70.Pf Glass transitions - 68.15. + e Liquid thin films - 02.70.Ns Molecular dynamics and particle methods     

Exact solutions for network rewiring models

Evolving networks with a constant number of edges may be modelled using a rewiring process. These models are used to describe many real-world processes including the evolution of cultural artifacts such as family names, the evolution of gene variations, and the popularity of strategies in simple econophysics models such as the minority game. The model is closely related to Urn models used for glasses, quantum gravity and wealth distributions. The full mean field equation for the degree distribution is found and its exact solution and generating solution are given.     

Exact solutions of integrable 2D contour dynamics

A class of exact solutions of the dispersionless Toda hierarchy constrained by a string equation is obtained. These solutions represent deformations of analytic curves with a finite number of nonzero harmonic moments. The corresponding τ-functions are determined and the emergence of cusps is studied.     

Exact classical solutions of nonlinear sigma models on supermanifolds

We study two-dimensional nonlinear sigma models with target spaces being the complex super-Grassmannian manifolds, that is, coset supermanifolds G(m,p|n,q)≅U(m|n)/[U(p|q)⊗U(m−p|n−q)]$G(m,p|n,q)\cong U(m|n)/[U(p|q)\otimes U(m-p|n-q)]$ for 0?p?m$0?p?m$, 0?q?n$0?q?n$ and 1?p+q$1?p+q$. The projective superspace CPm−1|n${\mathbf{CP}}^{m-1|n}$ is a special case of p=1$p=1$, q=0$q=0$. For the two-dimensional Euclidean base space, a wide class of exact classical solutions (or harmonic maps) are constructed explicitly and elementarily in terms of Gramm–Schmidt orthonormalisation procedure starting from holomorphic bosonic and fermionic supervector input functions. The construction is a generalisation of the non-super-case published more than twenty years ago by one of the present authors.     

Exact travelling kink-like wave solutions to some nonlinear equations

New explicit and exact travelling wave solutions to some nonlinear equations are obtained. For some chosen parameters in the nonlinear equations, constants in the explicit solutions are given numerically.     

Nonequilibrium dynamics of the zeta urn model

We consider a mean-field dynamical urn model, defined by rules which give the rate at which a ball is drawn from an urn and put in another one, chosen amongst an assembly. At equilibrium, this model possesses a fluid and a condensed phase, separated by a critical line. We present an analytical study of the nonequilibrium properties of the fluctuating number of balls in a given urn, considering successively the temporal evolution of its distribution, of its two-time correlation and response functions, and of the associated fluctuation-dissipation ratio, both along the critical line and in the condensed phase. For well separated times the fluctuation-dissipation ratio admits non-trivial limit values, both at criticality and in the condensed phase, which are universal quantities depending continuously on temperature.     

Conformational dynamics of some short-chain biradicals in solutions

Intramolecular electron spin exchange, as a function of temperature and the solvent nature, has been studied by X-band electron paramagnetic resonance (EPR) spectroscopy in five short-chain flexible nitroxide biradicals. Certain thermodynamic parameters of the conformational rearrangements were calculated from the EPR spectra. The process of spin exchange in short flexible biradicals has some peculiarities in comparison with that in long-chain molecules.     

Exact soliton-like solutions in generalized dynamical models of a quasi-one-dimensional crystal

We examine the dynamics of a one-dimensional crystal on a substrate with allowance for both nonlinear interaction with the substrate and intramolecular dispersion and nonlinearity. We show that such models, with the potential chosen appropriately, have soliton-like solutions with a distinct velocity. Such a potential allows for an independent selection of the curvature of the potential well and the height of the potential barrier, and in this way “globally” models the real potential. We show that such global modeling is correct to a first approximation. Zh. éksp. Teor. Fiz. 112, 1510–1515 (October 1997)     

Nonexistence of finite-energy solutions in some gauge models

A theorem which forbids the existence of certain nontrivial static finite-energy classical solutions to Yang-Mills-Higgs equations in dependence on the spatial dimension is generalized to include fields with nonvanishing electric-like componentsF _0j .Dedicated to 30th anniversary of the Joint Institute for Nuclear Research.     

Exact and explicit solitary wave solutions to some nonlinear equations

Exact and explicit solitary wave solutions are obtained for some physically interesting nonlinear evolutions and wave equations in physics and other fields by using a special transformation. These equations include the KdV-Burgers equation, the MKdV-Burgers equation, the combined KdV-MKdV equation, the Newell-Whitehead equation, the dissipative Φ⁴-model equation, the generalized Fisher equation, and the elastic-medium wave equation.     

Exact solutions for the discrete Boltzmann models with specular reflection

We construct exact (1+1)-dimensional solutions (space x, time t), in the presence of a purely reflecting well, for both the four velocity discrete Boltzmann model and the Broadwell model. These exact solutions, sums of two similarity shock waves, are positive for x0, t0.     

Exact relaxation dynamics of a localized many-body state in the 1D Bose gas

Through an exact method, we numerically solve the time evolution of the density profile for an initially localized state in the one-dimensional bosons with repulsive short-range interactions. We show that a localized state with a density notch is constructed by superposing one-hole excitations. The initial density profile overlaps the plot of the squared amplitude of a dark soliton in the weak coupling regime. We observe the localized state collapsing into a flat profile in equilibrium for a large number of particles such as N=1000. The relaxation time increases as the coupling constant decreases, which suggests the existence of off-diagonal long-range order. We show a recurrence phenomenon for a small number of particles such as N=20.     

Electron spin-lattice relaxation of nitroxyl radicals in temperature ranges that span glassy solutions to low-viscosity liquids

Electron spin-lattice relaxation rates, 1/T1, at X-band of nitroxyl radicals (4-hydroxy-2,2,6,6-tetramethylpiperidin-1-oxyl, 4-oxo-2,2,6,6-tetramethylpiperidin-1-oxyl, 3-carbamoyl-2,2,5,5-tetramethylpyrrolidin-1-oxyl and 3-carbamoyl-2,2,5,5-tetramethylpyrrolin-1-oxyl) in glass-forming solvents (decalin, glycerol, 3-methylpentane, o-terphenyl, 1-propanol, sorbitol, sucrose octaacetate, and 1:1 water:glycerol) at temperatures between 100 and 300K were measured by long-pulse saturation recovery to investigate the relaxation processes in slow-to-fast tumbling regimes. A subset of samples was also studied at lower temperatures or at Q-band. Tumbling correlation times were calculated from continuous wave lineshapes. Temperature dependence and isotope substitution (2H and 15N) were used to distinguish the contributions of various processes. Below about 100K relaxation is dominated by the Raman process. At higher temperatures, but below the glass transition temperature, a local mode process makes significant contributions. Above the glass transition temperature, increased rates of molecular tumbling modulate nuclear hyperfine and g anisotropy. The contribution from spin rotation is very small. Relaxation rates at X-band and Q-band are similar. The dependence of 1/T1 on tumbling correlation times fits better with the Cole-Davidson spectral density function than with the Bloembergen-Purcell-Pound model.