Minimum dissipative relaxed states in toroidal plasmas

Relaxation of toroidal discharges is described by the principle of minimum energy dissipation together with the constraint of conserved global helicity. The resulting Euler-Lagrange equation is solved in toroidal coordinates for an axisymmetric torus by expressing the solutions in terms of Chandrasekhar-Kendall (C-K) eigenfunctions analytically continued in the complex domain. The C-K eigenfunctions are obtained as hypergeometric functions that are solutions of scalar Helmholtz equation in toroidal coordinates in the large aspect-ratio approximation. Equilibria are constructed by assuming the current to vanish at the edge of plasma. For the m=0, n=0 (m and n are the poloidal and toroidal mode numbers respectively) relaxed states, the magnetic field, current, q (safety factor) and pressure profiles are calculated for a given value of aspect-ratio of the torus and for different values of the eigenvalue λ _{r 0}. The new feature of the present model is that solutions allow for both tokamak as well as RFP-like behaviour with increase in the values of λ _{r 0}, which is related directly to volt-sec in the experiment.     

Some extensions on the soliton solutions for the Novikov equation with cubic nonlinearity

In this paper, by using the bifurcation method of dynamical systems, we derive the traveling wave solutions of the nonlinear equation UU_τyy ? U_yU_τy + U²U_τ + 3U_y = 0. Based on the relationship of the solutions between the Novikov equation and the nonlinear equation, we present the parametric representations of the smooth and nonsmooth soliton solutions for the Novikov equation with cubic nonlinearity. These solutions contain peaked soliton, smooth soliton, W-shaped soliton and periodic solutions. Our work extends some previous results.     

Charged sphere of shear-free fluids with $$\frac{\partial}{\rho} \partial r \leq 0$$

The Einstein-Maxwell equations for non-static charged shear-free spherically symmetric perfect fluid distribution reduce to a second-order non-linear differential equation in the radial parameter. Several solutions of this equation have been obtained in earlier work without considering the general requirement for physical relevance of the solutions. Generally physically acceptable relativistic fluid models demand that the solutions satisfy the reality conditions ρ ≥ 0, p ≥ 0, ρ _r ≤ 0, etc. throughout the fluid model. In this article the expression for density gradient ρ _x (or ρ _r ) has been utilized to produce charged shear-free relativistic fluid models with non-positive density gradient (NDG)ρ _r ≤ 0. Eventually, we have found that none of the Riccati solutions have NDG including Vaidya metric. Also, the solutions with NDG neither possess Lie-symmetries nor Painlevé property. Further, it is observed that the solutions with NDG have no uncharged analogue.     

Properties of Perturbative Solutions of Unilateral Matrix Equations

A left-unilateral matrix equation is an algebraic equation of the form a ₀+a ₁ x+a ₂ x ²+·+a _n x ⁿ =0 where the coefficients a _r and the unknown x are square matrices of the same order and all coefficients are on the left (similarly for a right-unilateral equation). Recently certain perturbative solutions of unilateral equations and their properties have been discussed. We present a unified approach based on the generalized Bezout theorem for matrix polynomials. Two equations discussed in the literature, their perturbative solutions and the relation between them are described. More abstractly, the coefficients and the unknown can be taken as elements of an associative, but possibly noncommutative, algebra.     

On the oscillons in the signum-Gordon model

New periodic solutions of signum-Gordon equation are presented. We first find solutions φ₀(x, t) defined for (x, t) ∈ ? × [0, T ] and satisfying the condition φ₀(x, 0) = φ₀(x, T ) = 0. Then these solutions are extended to the whole spacetime by using (2.4).     

A simple-looking relative of the Novikov,Hirota-Satsuma and Sawada-Kotera equations

We study the simple-looking scalar integrable equation f_xxt 3( f_x f_t 1) = 0, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a Bäcklund transformation, soliton and merging soliton solutions (some exhibiting instabilities), two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries, a Painlevé series, a scaling reduction to a third order ODE and its Painlevé series, and the Hirota form (giving further multisoliton solutions).     

Trial Equation Method Based on Symmetry and Applications to Nonlinear Equations Arising in Mathematical Physics

To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a ₀+a ₁tan ξ+a ₂tan ² ξ, which indicates that some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method.     

On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, II: Universality of Critical Behaviour

Hamiltonian perturbations of the simplest hyperbolic equation u _t + a(u) u _x = 0 are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution. Moreover, this behaviour is described by a special solution to an integrable fourth order ODE.     

The solution space of the unitary matrix model string equation and the Sato Grassmannian

The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on pointsV ₁ andV ₂ in the big cell Gr⁽⁰⁾ of the Sato Grassmannian Gr. This is a consequence of a well-defined continuum limit in which the string equation has the simple form matrices of differential operators. These conditions onV ₁ andV ₂ yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraintsL _n (n0), whereL _n annihilate the two modified-KdV -functions whose product gives the partition function of the Unitary Matrix Model.     

The Rigged Hilbert Space of the Free Hamiltonian

We explicitly construct the Rigged Hilbert Space (RHS) of the free Hamiltonian H ₀. The construction of the RHS of H ₀ provides yet another opportunity to see that when continuous spectrum is present, the solutions of the Schrödinger equation lie in a RHS rather than just in a Hilbert space.     

Stability for Quasi-Periodically Perturbed Hill's Equations

We consider a perturbed Hill's equation of the form +(p₀(t)+ɛp₁(t))ϕ=0, where p₀ is real analytic and periodic, p₁ is real analytic and quasi-periodic and ɛ ∈ℝ is ``small'. Assuming Diophantine conditions on the frequencies of the decoupled system, i.e. the frequencies of the external potentials p₀ and p₁ and the proper frequency of the unperturbed (ɛ=0) Hill's equation, but without making any assumptions on the perturbing potential p₁ other than analyticity, we prove that quasi-periodic solutions of the unperturbed equation can be continued into quasi-periodic solutions if ɛ lies in a Cantor set of relatively large measure in where ɛ₀ is small enough. Our method is based on a resummation procedure of a formal Lindstedt series obtained as a solution of a generalized Riccati equation associated to Hill's problem.     

Solitary waves in the Madelung's fluid: Connection between the nonlinear Schrödinger equation and the Korteweg-de Vries equation

An investigation to deepen the connection between the family of nonlinear Schr?dinger equations and the one of Korteweg-de Vries equations is carried out within the context of the Madelung's fluid picture. In particular, under suitable hypothesis for the current velocity, it is proven that the cubic nonlinear Schr?dinger equation, whose solution is a complex wave function, can be put in correspondence with the standard Korteweg-de Vries equation, is such a way that the soliton solutions of the latter are the squared modulus of the envelope soliton solution of the former. Under suitable physical hypothesis for the current velocity, this correspondence allows us to find envelope soliton solutions of the cubic nonlinear Schr?dinger equation, starting from the soliton solutions of the associated Korteweg-de Vries equation. In particular, in the case of constant current velocities, the solitary waves have the amplitude independent of the envelope velocity (which coincides with the constant current velocity). They are bright or dark envelope solitons and have a phase linearly depending both on space and on time coordinates. In the case of an arbitrarily large stationary-profile perturbation of the current velocity, envelope solitons are grey or dark and they relate the velocity u₀ with the amplitude; in fact, they exist for a limited range of velocities and have a phase nonlinearly depending on the combined variable x-u_{0 s} (s being a time-like variable). This novel method in solving the nonlinear Schr?dinger equation starting from the Korteweg-de Vries equation give new insights and represents an alternative key of reading of the dark/grey envelope solitons based on the fluid language. Moreover, a comparison between the solutions found in the present paper and the ones already known in literature is also presented. Received 20 February 2002 and Received in final form 22 April 2002 Published online 6 June 2002     

Long-Time Asymptotics for Strong Solutions¶of the Thin Film Equation

In this paper we investigate the large-time behavior of strong solutions to the one-dimensional fourth order degenerate parabolic equation u _t =−(u u _xxx ) _x , modeling the evolution of the interface of a spreading droplet. For nonnegative initial values u ₀(x)∈H ¹(ℝ), both compactly supported or of finite second moment, we prove explicit and universal algebraic decay in the L ¹-norm of the strong solution u(x,t) towards the unique (among source type solutions) strong source type solution of the equation with the same mass. The method we use is based on the study of the time decay of the entropy introduced in [13] for the porous medium equation, and uses analogies between the thin film equation and the porous medium equation. Received: 2 February 2001 / Accepted: 7 October 2001     

Madelung Representation for Complex Nonlinear D’Alembert Equations in n-Dimensional Minkowski Space

Abstract

The Madelung representation ψ = u exp(iv) is considered for the d’Alembert equation □ _n ψ?F (|ψ|)ψ = 0 to develop a technique for finding exact solutions. We classify the nonlinear function F for which the amplitude and phase of the d’Alembert equation are related to the solutions of the compatible d’Alembert–Hamiltonian system.

The equations are studied in n-dimensional Minkowski space.     

On Integration of the Nonlinear d’Alembert-Eikonal System and Conditional Symmetry of Nonlinear Wave Equations

Abstract

We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear d’Alembert equation □u = F (u) and nonlinear eikonal equation u _xµ u _x µ = G(u) in the complex Minkowski space R(1, 3). A method suggested makes it possible to establish necessary and sufficient compatibility conditions and construct a general solution of the d’Alembert-eikonal system for all cases when it is compatible. The results obtained can be applied, in particular, to construct principally new (non-Lie, non-similarity) solutions of the non-linear d’Alembert, Dirac, and Yang-Mills equations. Solutions found in this way are shown to correspond to conditional symmetry of the equations enumerated above. Using the said approach, we study in detail conditional symmetry of the nonlinear wave equation □w = F ₀(w) in the four-dimensional Minkowski space. A number of new (non-Lie) reductions of the above equation are obtained giving rise to its new exact solutions which contain arbitrary functions.     

Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base

The Knizhnik-Zamolodchikov equation associated withsl ₂ is considered. The transition functions between asymptotic solutions to the Knizhnik-Zamolodchikov equation are described. A connection between asymptotic solutions and the crystal base in the tensor product of modules over the quantum groupU _q sl ₂ is established, in particular, a correspondence between the Bethe vectors of the Gaudin model of an inhomogeneous magnetic chain and the Q-basis of the crystal base.Dedicated to the memory of Ansgar SchnizerThe author was supported by NSF Grant DMS-9203929     

On the problem of one-dimensional disordered chain

Conclusion Some properties of a one-dimensional disordered homogeneous chain were studied in this paper. Using standard techniques of probability theory, expressions for the frequency distribution function (2) and the localization length (6) were derived. Having considered only pair correlations between atoms, both these expressions contained only one unknown function — the joint probability distribution of the massm _n and the ratiot _n ^± = –ku _n±1/u _n which could be found as a solution of the integral equation (5). Our approach to the problem was applied on the ideal lattice and the lattice with low concentration of impurities. In these cases the solutions of the integral equation (5) reduced to the functional form (7) were found analytically. Using these solutions, old well-known results for ( ²) and the local vibration of impurities were derived by this method.Derivation of all equations in this paper is straightforward from the equations of motion. The quantities we deal with have a clear physical meaning, which facilitated, for instance to find the solutions of functional equation (7) in the special case of the ideal crystal. This is what we consider to be the advantages of our approach.     

Cuspons and Smooth Solitons of the Degasperis–Procesi Equation Under Inhomogeneous Boundary Condition

This paper is contributed to explore all possible single peakon solutions for the Degasperis–Procesi (DP) equation m _t  + m _x u + 3mu _x  = 0, m = u − u _xx . Our procedure shows that the DP equation either has cusp soliton and smooth soliton solutions only under the inhomogeneous boundary condition lim_|x|→ ∞ u =A ≠0, or possesses the regular peakon solutions ce ^{− |x − ct|} ∈ H ¹ (c is the wave speed) only when lim_|x|→ ∞ u = 0 (see Theorem 4.1). In particular, we first time obtain the stationary cuspon solution of the DP equation. Moreover we present new cusp solitons (in the space of ) and smooth soliton solutions in an explicit form. Asymptotic analysis and numerical simulations are provided for smooth solitons and cusp solitons of the DP equation.      

幂函数叠加势的径向薛定谔方程的解析解

研究多种正幂势函数与逆幂势函数紧密耦合条件下薛定谔径向方程解析解的求解方法.对势函数为V（r）=α₁r⁸＋α₂r³+α₃r²+β₃r^-1＋β₂r^-3＋β₁r^-4的径向薛定谔方程存在解析解的条件以及精确的解析解进行了研究. 根据量子系统波函数必须满足单值、有界和连续的标准条件,首先求出径向坐标r→∞以及r→0时的渐近解,然后采用非正则奇点邻域附近的波函数级数解法与求得的渐近解相结合,通过幂级数系数比较法得到径向薛定谔方程在势函数系数紧密耦合条件下的一系列定态波函数解析解以及相应的能级结构,并作适当讨论与结论. 关键词： 级数解法 幂势函数 径向波函数 渐近解     

A strictly geometric interpretation of gravitation in general relativity

A geometric interpretation of gravitation is given using general relativity. The law of gravitation is taken in the formR ₄₄=0, whereR ₄₄is the component of the contracted Riemann-Christoffel (Ricci) tensor representing the curvature of time. The remaining curvature components of the contracted Riemann-Christoffel tensor may or may not vanish. All that is required in addition toR ₄₄=0 is that the Gaussian curvatureR be nowhere infinite. The conditionR ₄₄=0 yields a nonlinear wave equation. One of the static degenerate solutions represents the gravitational field surrounding a static gravitational point singularity. It is found that for this solution, the three famous predictions of general relativity are obtained in the weak-field approximation. In addition, it is found that there is a correction to the Kepler period of revolution for an orbit.