Special Class Of Solutions Research Articles

THE COMPACTIFIED JACOBIAN OF A REDUCIBLE HYPERELLIPTIC CURVE

There is a one-to-one correspondence between nonsingular pencils of quadrics in an odd-dimensional projective space and nonsingular hyperelliptic curves. There is also a close relation between the moduli spaces of vector bundles of ranks 1, 2 on the hyperelliptic curve and spaces associated to the intersection of quadrics of the corresponding pencil. This has been studied extensively by many authors [1, 5, 8, 9, 11]. An interesting application of this was in the proof of the Verlinde formulae by Szenes [12]. This relationship was generalised to a relation between spaces related to singular pencils of quadrics associated to irreducible nodal curves and moduli of torsion-free sheaves on the nodal curves [2, 4]. The projective models of the moduli of vector bundles on smooth curves (given by this relationship) led to explicit representations of integrable systems of [6, 7]. Similar results on singular curves would provide important classes of special solutions. The pencils of quadrics considered in previous works had associated Segre symbols of type [1 … 1] and [2 … 2 1 … 1]. Kleiman (and Emma Previato) had raised the question: What happens in the case of Segre symbol [2 2 2]? In this case the associated curve Y has two rational components intersecting transversely at three points. Unlike in the previous cases, the compactified Jacobian has more than one component. Moreover the stability of line bundles and the compactification of the Jacobian depends on the choice of a polarisation on Y, making the problem more interesting and difficult.

Geometrical aspects of the complex eikonal equations associated to the generalized Weierstrass representation

Certain geometrical aspects of a Laplace-Beltrami equation subjected to a complex eikonal equation are studied. It is shown that all principal curvatures of a solution surface are constant for this system. To illustrate the theoretical considerations, we look for a class of solutions of an overdetermined system composed of the sigma model and a system of complex eikonal equations. The connection between the sigma model and the generalized Weierstrass system for inducing constant mean curvature surfaces allows us to construct special classes of solutions for classical configurations of strings.

Characterization of special classes of solutions for some functional equations on orthogonal vectors

Taking into account some recent results concerning the conditional d'Alembert equation on orthogonal vectors ¶¶ $ f(x + y) + f(x - y) = 2 f(x) f(y) $ for all $ x,y \in X $ with (x,y) = 0 (1)1¶in the class of the functionals f from a real inner product space X into $ {\Bbb R} $ , we deal here with two other conditional functional equations related to (1)1, namely ¶¶ $ g : X \to {\Bbb R}, \quad g(x + y) g(x - y) = g^2(x) + g^2(y) $ for $ x,y \in X $ with (x,y) = 0(2)1¶¶ and¶¶ $ h : X \to {\Bbb R}, \quad h(x + y) h(x - y) = h^2(x) + h^2(y) - 1 $ for $ x,y \in X $ with (x,y) = 0,(3)1¶where we denote by $ g^2(\cdot) $ and $ h^2(\cdot) $ the pointwise product of functions.¶Our main purpose in this note is to find and characterize some special classes of solutions of the above equations (2)1 and (3)1.¶We remark that some results are obtained in the class of the functionals defined on a real inner product space X with $ \dim X \ge 2 $ , while other theorems are proved if $ \dim X \ge 3 $ .

Variational Approach to Uniform Rigid Rotation of Ginzburg—Landau Vortices

Time periodic solutions for the hyperbolic gauged Ginzburg–Landau system, with spatial domain the unit disc, are shown to exist. Time periodic solutions representing bound states of vortices rotating about one another have been previously obtained in the near self-dual limit, using perturbative techniques. In contrast, we here take a variational approach, the solutions being obtained as critical points of an indefinite functional. We consider a special class of solutions which map out, uniformly in time, an orbit of the rotation group SO(2). It is shown that in the limit of large coupling constant the solutions have nontrivial time dependence or, as is shown to be equivalent, are not radially symmetric in any gauge.

Application of weak equivalence transformations to a group analysis of a drift-diffusion model

A group analysis of a class of drift-diffusion systems is performed. In account of the presence of arbitrary constitutive functions, we look for Lie symmetries starting from the weak equivalence transformations. Applications to the transport of charges in semiconductors are presented and a special class of solutions is given for particular doping profiles.

Trilinear resonant interactions of semilinear hyperbolic waves

An important question in the study of nonlinear equations is the understanding of the asymptotic behavior of families of solutions, or approximate solutions. One of the difficulty is known to be the nonlinear interaction of oscillations. This phenomenon has been studied for special classes of solutions, such as oscillatory functions with phases and amplitudes. This is the domain of geometric optics. In some cases, general families of bounded solutions have also been considered. The present paper enters the later category, but is highly inspired by the results of nonlinear geometric optics. Geometric optics provides a precise description of the asymptotic structure of solutions to both linear (e.g. [Wh]) and nonlinear ([J], [HK], [MR], [HMR], [JMR 1], [JMR 2], [S], . . .) hyperbolic PDEs whose initial data u(0, x) = u0(x) + eu1(x, ~ φ0(x) e ) have rapid oscillations with prescribed phases ~ φ0 and a single prescribed scale 1e . Such solutions are important in both theory and applications ([L], [KK]). In the lowest approximation, oscillations having different spatial scales interact only when m equals zero, since otherwise the amplitudes of higher-frequency oscillations become negligible relative to those of lower frequencies. Taking m = 0 requires L∞ estimates. Systems that are well-posed in L∞ occur mostly in one spatial dimension. For such systems, arbitrary uniformly bounded sequences of initial data may be considered, but comparatively little is known about the asymptotic structure of the resulting solutions. Since the oscillations in such solutions may be extremely complex, a natural problem is to determine their non-oscillatory part, which may be identified with the weak limit of the sequence of solutions. The highly oscillatory nature of the solutions manifests itself in the absence of strong convergence in general. Of course, some information about those oscillations will generally be needed in order to determine the weak limit. The asymptotic structure of uniformly bounded solutions depends strongly on the nature of the nonlinearity: For entropy solutions of a genuinely nonlinear conservation law ([T 1]) or of a pair of such equations in one spatial dimension ([DP]), oscillations of amplitude O(1) do not persist for positive times, so a sequence of solutions converges strongly to the solution whose initial data is the weak limit of the initial data. Although oscillations do persist in linear systems, the weak limit of a sequence of solutions to such

Viscosity and Strain Gradient Effects on the Kinetics of Propagating Phase Boundaries in Solids.

The theory of thermoelastic materials undergoing solid-solid phase transformations requires constitutive information that governs the evolution of a phase boundary. This is known as a kinetic relation which relates a driving traction to the speed of propagation of a phase boundary. The kinetic relation is prescribed in the theory from the onset. Here, though, a special kinetic relation is derived from an augmented theory that includes viscous, strain gradient and heat conduction effects. Based on a special class of solutions, namely travelling waves, the kinetic relation is inherited from the augmented theory as the viscosity, strain gradient and heat conductivity are removed by a suitable limit process.

Uniform asymptotic formulas for curved solitons of the Kadomtsev-Petviashvili equations

A special class of solutions to the Kadomtsev-Petviashvili equations is investigated in the limit t → ∞. It is proved that these solutions split into an infinite series of curved solitons in the neighborhood of the wave front. Parameters of these solitons depend on the variable Y=y/t. Asymptotic formulas uniform in Y are obtained.

Inertial Constraints in a Curved Spacetime

A precise kinematical definition of the deformation of an arbitrary space or spacetime is considered. This approach introduces which, combined with the vacuum field equations of general relativity, lead to a unique class of solutions involving plane gravitational waves. A general expression which relates accelerations in a smoothly deformed back­ ground has been proposed recently.l) This particular formula has also been used for the definition of the inertial backgrounds (i.e., the global inertial frames of reference) which can be associated, in principle, with an arbitrary curved space or spacetime. An approach of this particular kind is useful for the reexamination of the various covariant formulations of mechanics in n-dimensions (n~4) in relation or not to the general theory of relativity.2) However, the physical interpretation of the terms involved in the aforementioned expression is rather obscure and in a certain sense misleading. In particular, the analysis is restricted in the subcase of affine motions of the background, the general case of a deformation not being considered. In this sense it is worth reconsidering the whole subject from a different point of view. It turns out that a precise kinematical definition of the deformation of an arbitrary space is always possible. In particular, we may use the deformation of a test particle's acceleration in order to establish criteria which give us infomation about the structure of the underlying space. Clearly, there is a hierarchy of such possibilities. How­ ever, even the simplest physically interesting one, where the test particle's accelera­ tion does not feel the deformation of the background, gives non-trivial results. In fact, a combination of the aforementioned case with the vacuum field equations of general relativity prescribes uniquely, a special class of solutions which involve plane gravitational waves. In an arbitrary space M, where space means any connected paracompact mani­ fold, the acceleration vector (in contradistinction to the velocity one) cannot be defined without an explicit reference to the geometry of the space under consideration. This means that a (symmetric) affine connection or a Finslerian one is at least required so that the acceleration vector is properly and consistently defined in some local neighborhood of the point mEM. If the space is continuouslydeformedl) then we have

LOCAL WORLD SHEET WEYL-INVARIANCE IN THE PRESENCE OF BACKGROUND CHARGES

We derive the consistency conditions that follow from imposing local Weyl-invariance on the Polyakov path integral in a general background. In contrast to previous attempts our approach allows for the inclusion of curvature-dependent couplings like the dilaton and in particular of background charges. The resulting equation displays explicitly the corresponding modifications of the trace anomaly and the scale dimensions of vertex operators. A special class of solutions is discussed which, in particular, allows one to interpret the screening operators of Fateev and Dotsenko as an interaction Lagrangian.

�ber die Minimall�sung der Poincar�-Perronschen Differenzengleichung

This paper deals with a special class of solutions of the higher order linear difference equation. It is shown that the minimal solution of Poincare-Perron type equations can be expressed in terms of generalized continued fractions.

Analysis of a Diffusion-Convection System Modeling a Contamination Problem

A system of coupled diffusion-convection equations which model a contamination problem are analyzed. The equations are reformu-lated as an abstract problem which is used to obtain existence, uniqueness and posit ivity results for the solutions. A minimum principle is also proved and a special class of solutions which have bearing on the model are derived

A certain class of solutions of the nonlinear wave equation

In this paper are investigated some differential geometry methods in the theory of the nonlinear wave equation ∇2u=Φ(u,(∇u‖∇u)). A special class of solutions is discussed for which (∇u‖∇u) is constant on each level of the function u. It is proved that levels of such solutions form in the space of independent variable’s hypersurfaces with all principal curvatures constant. The general form of such hypersurfaces is given. Then it is proved that via the method of characteristics it is possible to construct (in principle) all the solutions of the discussed class. They may be obtained by integration of an ODE of second order using a special class of the polynomial functions. Some new solutions are given for equations⧠v=4Av3+3Bv2+2cv+D, ⧠v=μ exp v, ⧠v=sin v, ⧠v=cosh v, and ⧠v=sinh v.

Reachable solution set of a fuzzy relation equation

The largest element of the solution set of a fuzzy relation equation has been found by E. Sanchez ( Inform. and Control 30 (1976) , 38–48) but the smallest element does not exist. It is difficult to expose the solution of the fuzzy relation equation. In the case of the determinate relation equations, complete consequences have been found by Wang Peizhuang and Yuan Meng (“Relation Equation and Relation Inequalities,” Selected papers on fuzzy subsets, Beijing Normal University, March 1980). In the case of the fuzzy relation equations, Wang and Yuan have given a class of special solutions which probably possesses some minimality characterizations. In this paper, the reachable solution set of the fuzzy relation equation is given. For the fuzzy relation equation on the finite sets, a neat and efficient method for solving it is given.

Two-dimensional, steady-state diffusion of a passive scalar in a flow field

The two-dimensional, steady-state equation governing the diffusion of a passive scalar field Ψ, ∇⋅(uΨ)=η∇2Ψ, is shown to yield a set of analytic solutions for η-independent flows u. For the case of vorticity flow, this equation can be reduced to an equation solved by Darboux. For the case of potential flow, a special class of solutions is found, of which that demonstrated by Matthaeus and Montgomery is a special case.

Generalized field theories and solutions

By beginning with a special class of solutions of field equations, new field equations are derived.

Geometrical mass of charged particle in Brans-Dicke theory

We develop a special class of solutions of the Brans-Dicke equations from vacuum solutions of Einstein's equations by applying a method due to Teixeira, Wolk, and Som. It is shown that the electric charge contributes positively to the geometric mass while the Brans-Dicke scalar field diminishes the geometric mass effectively.

One Dimensionally Alligned Decay Mode Solutions of the Two Dimensional Nonlinear Schrödinger Equation

It is shown that the arbitrary solutions of the 1-dimensional nonlinear Schrodinger equation can be transformed to the special class of solutions (1-dimensionally aligned and decaying in time) of the 2-dimensional nonlinear Schrodinger equation.

A semi-analytical approach to time-dependent coronal expansion

In this paper we point out the existence of a special class of solutions to the nonlinear hydrodynamic equations describing the time-dependent solar wind, namely that for which the velocity profile is time-invariant but the density at each point of the corona changes exponentially with time. Theoretical velocity curves are calculated for the case of isothermal expansion and compared with the Parker model for steady-state expansion. These solutions can be used to obtain quantitative estimates for the degree of departure from the latter of a real corona undergoing evolution on a finite time scale.

Normal-dominated singularities in static space-times

We define “normal-dominated” singularities of static solutions of the Einstein equations and show that a uniquely and invariantly defined structure can be assigned to these singularities. We find for the general solution that the dominant term of the Riemann tensor near the singularity is of Petrov Type N. Except for one special class of solutions, it seems that in general the shear of the null geodesics blows up at the same rate as their convergence near the singularity, in contradistinction to the “elementary singularity” of Newman and Posadas. We compute the structure for a variety of known static solutions as well as the stationary Kerr-Newman metrics.