Conditional Invariance and Exact Solutions of a Nonlinear System

Abstract

The Lie and Q-conditional invariance of one nonlinear system of PDEs of the third-order is searched. The ansatze have been built which reduce the PDEs system to ODEs. The classes of exact solutions of the given system are obtained. The relation between the Korteweg-de Vries equation and Harry-Dym equation has been established.     

Transformations RS42(3) of the Ranks ≤ 4¶and Algebraic Solutions of the Sixth Painlevé Equation

Compositions of rational transformations of independent variables of linear matrix ordinary differential equations (ODEs) with the Schlesinger transformations (RS-transformations) are used to construct algebraic solutions of the sixth Painlevé equation. RS-Transformations of the ranks 3 and 4 of 2 × 2 matrix Fuchsian ODEs with 3 singular points into analogous ODE with 4 singular points are classified. Received: 17 August 2001 / Accepted: 14 February 2002     

Analysis of the static spherically symmetricSU(n)-Einstein-Yang-Mills equations

The singular boundary value problem that arises for the static spherically symmetricSU(n)-Einstein-Yang-Mills equations in the so-called magnetic case is analyzed. Among the possible actions ofSU(2) on aSU(n)-principal bundles over space-time there is one which appears to be the most natural. If one assumes that no electrostatic type component is present in the Yang-Mills fields and the gauge is suitably fixed a set ofn-1 second order and two first order differential equations is obtained forn-1 gauge potentials and two metric components as functions of the radial distance. This system generalizes the one for the casen=2 that leads to the discrete series of the Bartnick-Mckinnon and the corresponding black hole solutions. It is highly nonlinear and singular atr= and atr=0 or at the black hole horizon but it is known to admit at least one series of discrete solutions which are scaled versions of then=2 case. In this paper local existence and uniqueness of solutions near these singular points is established which turns out to be a nontrivial problem for generaln. Moreover, a number of new numerical soliton (i.e. globally regular) numerical solutions of theSU(3)-EYM equations are found that are not scaledn=2 solutions.     

Lie-Backlund vector fields for the nonlinear system,Q t=AQxx+F(Qx,Q)

We have analyzed the class of nonlinear second-order equations written asQ _t=AQ_xx +F(Qx, Q) withQ =( _v ^u ) andA, F are, respectively, matrix and vector functions depending onQ, Q _x, from the point of view of Lie-Backlund vector fields. When the vector functionF does not depend onQ _x, these equation set reduces to the coupled diffusion equations discussed by Steeb. But our generalized system encompasses a large class of physically meaning full nonlinear equations, such as (i) dispersive water waves and (ii) a completely anisotropic Heisenberg spin chain. We also exhibit a new nonlinear coupled system which do have nontrivial Lie-Backlund vector fields. Also our approach yields more information about the symmetry generators for a wider class of nonlinear equations than the function space approach of Fuchsteiner in a much simpler way.     

Topological invariants and the dynamics of an axial vector torsion field

A generalized theory of gravitation is discussed which is based on a Riemann-Cartan space-time,U ₄, with an axial vector torsion field. Besides Einstein's equations determining the metric of theU ₄, a system of nonlinear field equations is established coupling an axial vector source current to the axial vector torsion field. The properties of the solutions of these equations are discussed assuming a London-type condition relating the axial current and torsion field. To characterize the solutions use is made of the Euler and Pontrjagin forms and the associated quadratic curvature invariants for theU ₄ space-time. It is found that there exists for a Riemann-Cartan space-time a relation between the zeros of the axial vector torsion field and the singularities of the Pontrjagin invariant, which is analogous to the well-known Hopf relation between the zeros of vector fields and the Euler characteristic.     

Riccati-coupled similarity shock wave solutions for multispeed discrete Boltzmann models

We study nonstandard shock wave similarity solutions for three multispeed discrete Boltzmann models: (1) the square 8_i, model with speeds 1 and 2 with thex axis along one median, (2) the Cabannes cubic 14_i model with speeds 1 and 3 and thex axis perpendicular to one face, and (3) another 14_i, model with speeds 1 and 2. These models have five independent densities and two nonlinear Riccati-coupled equations. The standard similarity shock waves, solutions of scalar Riccati equations, are monotonic and the same behavior holds for the conservative macroscopic quantities. First, we determine exact similarity shock-wave solutions of coupled Riccati equations and we observe nonmonotonic behavior for one density and a smaller effect for one conservative macroscopic quantity when we allow a violation of the microreversibility. Second, we obtain new results on the Whitham weak shock wave propagation. Third, we solve numerically the corresponding dynamical system, with microreversibility satisfied or not, and we also observe the analogous nonmonotonic behavior.     

Rarefied Flow Computations Using Nonlinear Model Boltzmann Equations

High resolution finite difference schemes for solving the nonlinear model Boltzmann equations are presented for the computations of rarefied gas flows. The discrete ordinate method is first applied to remove the velocity space dependency of the distribution function which renders the model Boltzmann equation in phase space to a set of hyperbolic conservation laws with source terms in physical space. Then a high order essentially nonoscillatory method due to Harten et al. (J. Comput. Phys. 71, 231, 1987) is adapted and extended to solve them. Explicit methods using operator splitting and implicit methods using the lower-upper factorization are described to treat multidimensional problems. The methods are tested for both steady and unsteady rarefied gas flows to illustrate its potential use. The computed results using model Boltzmann equations are found to compare well both with those using the direct simulation Monte Carlo results in the transitional regime flows and those with the continuum Navier-Stokes calculations in near continuum regime flows.     

Exponential Convergence to the Maxwell Distribution for Some Class of Boltzmann Equations

We consider a class of nonlinear Boltzmann equations describing return to thermal equilibrium in a gas of colliding particles suspended in a thermal medium. We study solutions in the space where is the one-particle phase space and is the Liouville measure on Γ⁽¹⁾. Special solutions of these equations, called “Maxwellians,” are spatially homogenous static Maxwell velocity distributions at the temperature of the medium. We prove that, for dilute gases, the solutions corresponding to smooth initial conditions in a weighted L ¹-space converge to a Maxwellian in , exponentially fast in time.     

Untersuchungen zum Existenzbereich einer BOLTZMANN-Verteilung im Argonbogenplasma

Using as plasma source a wall stabilized argon arc working within a restricted parameter range (inner tube diameter = 7 mm, pressure = 30–120 Torr, current = 5-20 A)) the existence of a Boltzmann equilibrium between spectral energy levels is checked by comparing measured occupation number densities of higher excited levels (N_{m, exp}) with the corresponding number densities calculated under the assumption of Boltzmann equilibrium (N_{m, calc}). The methods for determination of the quantities N_{m, exp'} T_g (2300–5405°K), T_e (7170–9950°K) and N_e (0.33 – 2.4 × 10¹⁵ cm^?3) needed for this comparison are described. It can be shown within the limit of experimental error that a Boltzmann equilibrium exists at least for electron densities of N_e > 3 · 10¹⁴ cm^?3. The problem of energy balance of that type of arcs used in these experiments is discussed.     

On Diffusive Equilibria in Generalized Kinetic Theory

We investigate the solvability of equations Q(f,f)+ ² f=0 in term of nonnegative integrable densities fL ¹ ₊(R ³). Here, Q(f, f) is a generalized collision operator. If Q is the Boltzmann operator, the only solution is 0. In contrast, we show that if Q is the pseudo-Maxwellian collision operator for granular flow, then there are non -trivial weak solutions of ().     

Exact solutions to the time-dependent Lorentz gas Boltzmann Equation: The approach to hydrodynamics

New exact solutions to the time-dependent Lorentz gas Boltzmann equation are presented for two classes of nonequilibrium initial value problems: thedecay of localized disturbances and theresponse to applied electric fields. These exact results are used to gain some insight into the crossover of the nonequilibrium state from the early-timekinetic regime to the late-timehydrodynamic regime.     

New Discrete Model Boltzmann Equations for Arbitrary Partitions of the Velocity Space

Modified discrete Boltzmann equations for arbitrary partitions of the velocity space are established. The new equations can be derived from the continuous Boltzmann equation and are a generalization of previous discrete-velocity models. They preserve mass, momentum, and energy, and an H-theorem holds. The new model equations are tested by comparing their solutions with the analytical ones of the continuous Boltzmann equation for the Krook–Wu and the very hard particle models.     

Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation

We first consider the Boltzmann equation with a collision kernel such that all kinematically possible collisions are run at equal rates. This is the simplest Boltzmann equation having the compressible Euler equations as a scaling limit. For it we prove a stability result for theH-theorem which says that when the entropy production is small, the solution of the spatially homogeneous Boltzmann equation is necessarily close to equilibrium in the entropie sense, and therefore strongL ¹ sense. We use this to prove that solutions to the spatially homogeneous Boltzmann equation converge to equilibrium in the entropie sense with a rate of convergence which is uniform in the initial condition for all initial conditions belonging to certain natural regularity classes. Every initial condition with finite entropy andp ^th velocity moment for some p>2 belongs to such a class. We then extend these results by a simple monotonicity argument to the case where the collision rate is uniformly bounded below, which covers a wide class of slightly modified physical collision kernels. These results are the basis of a study of the relation between scaling limits of solutions of the Boltzmann equation and hydrodynamics which will be developed in subsequent papers; the program is described here.On leave from School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332.On leave from C.F.M.C. and Departamento de Matemática da Faculdade de Ciencias de Lisboa, 1700 Lisboa codex, Portugal.     

A multidimensional radiation-filled universe

We consider a radiation-filled universe which possesses the product symmetry: (N-dimensional space of constant curvature) × (n sphere). The solutions of all the types, within this class, to the classical field equations are given. In the case of theN-dimensional space of zero or negative curvature constant, the solutions exhibit a tendency to approach asymptotically the Kasner-like state in which theN-dimensional subspace expands while then sphere shrinks to the final singularity. Our conclusions based on the phase-diagram method are in agreement with the results concerning the ^N × S ⁿ universe calculated by Sahdev with the help of numerical methods.     

Dynamics and stability of volume-scavenging drop arrays: Coarsening by capillarity

N drops, pinned by circular contact lines, are arranged in an array and coupled by a network of conduits. Inertialess exchange of volume among drops is driven by capillarity through the minimization of total surface energy. Drops scavenge volume from one another based on pressure differences, proportional to the surface tension, and arising from curvature differences. The system coarsens in the sense that, with time, volume is increasingly localized and ends up in a single ‘winner’ drop. Numerical simulations show that the identity of the winner can depend discontinuously on the initial condition and connectivity network. This motivates a study of the corresponding N-dimensional nonlinear dynamical system. All fixed points and their linear stabilities, obtained analytically, are found to be independent of connectivity. To determine which of the stable fixed points will be the winner, manifolds separating the attracting regions are found using a method which combines local information (eigenvectors at fixed points) with global information (invariant manifolds due to symmetry). This method is demonstrated for three N=3 systems with various connectivity networks, and is used to explain the numerical observations.     

Statistical properties of the laser emission in a low-Q cavity

We investigate theoretically the stationary statistical properties of the laser radiation in a low-Q cavity with field, polarization, and population fluctuations. Eliminating adiabatically the electric field from the Maxwell-Bloch equations, coupled Langevin equations with bothadditive andmultiplicative noises are derived and are transformed into the multivariable Fokker-Planck equation of a probability density of the light intensity and the population difference. It is solved by the expansion into orthonormal sets, and a vector recurrence equation of motion of the expansion coefficients is given whose stationary solutions are analytically obtained in theMatrix continued-fraction. The stationary distribution function of the radiation intensity are calculated with several values of control parameters. We discuss the variance of the intensity distribution, the photon-counting coefficient, and the cross-correlation between intensity and population as a function of the pump parameter, and reveal the novel and characteristic features of the bad-cavity laser system. The comparison with the good-cavity (high-Q cavity) case is also made.     

Quantum Integrable Models and Discrete Classical Hirota Equations

The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A _k-1 -type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T−Q-relation are derived. Received: 15 May 1996 / Accepted: 25 November 1996     

Uniqueness and a priori bounds for certain homoclinic orbits of a Boussinesq system modelling solitary water waves

This paper establishes surprisingly precise a priori bounds on theL -norm of certain singular solutions of a system of two nonlinear Sturm-Liouville equations which model solitary water waves.These solutions can be interpreted as homoclinic orbits for a system of four first order ordinary differential equations. The uniqueness of these homoclinic orbits is established for certain choices of a parameterc, the phase speed of the waves. These observations do not result from perturbation of linear theory, but are global.     

Three-dimensional stationary cyclic symmetric Einstein–Maxwell solutions; black holes

From a general metric for stationary cyclic symmetric gravitational fields coupled to Maxwell electromagnetic fields within the (2 + 1)-dimensional gravity the uniqueness of wide families of exact solutions is established. Among them, all uniform electromagnetic solutions possessing electromagnetic fields with vanishing covariant derivatives, all fields having constant electromagnetic invariants F_μνF^μν and T_μνT^μν, the whole classes of hybrid electromagnetic solutions, and also wide classes of stationary solutions are derived for a third-order nonlinear key equation. Certain of these families can be thought of as black hole solutions. For the most general set of Einstein–Maxwell equations, reducible to three nonlinear equations for the three unknown functions, two new classes of solutions – having anti-de Sitter spinning metric limit – are derived. The relationship of various families with those reported by different authors’ solutions has been established. Among the classes of solutions with cosmological constant a relevant place is occupied by the electrostatic and magnetostatic Peldan solutions, the stationary uniform and spinning Clement classes, the constant electromagnetic invariant branches with the particular Kamata–Koikawa solution, the hybrid cyclic symmetric stationary black hole fields, and the non-less important solutions generated via SL(2,R)-transformations where the Clement spinning charged solution, the Martinez–Teitelboim–Zanelli black hole solution, and Dias–Lemos metric merit mention.