The Painlevé property and a partial differential equation with an essential singularity

It is shown that the equation u²_t = 2uu²_x - (1 + u²)u_xx possesses the Painlevé property for partial differential equations as defined by Weiss, Tabor and Carnevale, yet does not satisfy the necessary conditions of the Painlevé conjecture to be completely integrable since it is reducible, via a similarity reduction, to an ordinary differential equation which has a movable essential singularity. It is further shown that in a more general sense, the equation does not possess the Painlevé property for partial differential equations.     

Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent

The quantal system of Bose particles described by the non-linear Schrödinger equation i?φ/?t = -$\text{1}\text{2}$?²φ/?x² + cφ1φ², with c= cxf∞ and via the ground state with finite particle density, is the 1- dimensional gas of impenetrable bosons studied by M. Girardeau, T.D. Schultz, A. Lenard, H.G. Vaidya and C.A. Tracy. We show that the 2-point (resp. 2n-point) function, or the 1-particle (resp. n-particle) reduced density matrix, of this system satisfies a non-linear differential equation (resp. a system of non-linear partial differential equations) of Painlevé type. Derivation of these equations is based on the link between field operators in a Clifford group and monodromy preserving deformation theory, which was previously established and applied to the 2-dimensional Ising model and other problems. Several related topics are also discussed.     

Operator products in 2-dimensional critical theories

A noteworthy feature of certain conformally invariant 2-dimensional theories, such as the Ising and 3-state Potts models at the critical point, is the existence of “degenerate primary fields” associated with nullvectors of the Virasoro algebra. Such fields are endowed with a remarkably simple multiplication table under the operator product expansion, known as the fusion rules. In addition, correlation functions made up of these fields satisfy a system of linear homogeneous partial differential equations. We show here that these two properties are intimately related: for any n-point function, the number of conformally invariant solutions to the system of equations equals the number of times that the identity operator appears in the fusion of all n fields in the correlator. This theorem permits the calculation of some apparently intractable correlation functions. Finally, we generalize these ideas to the Neveu-Schwarz sector of superconformal theories.     

Dynamics of conservative peakons in a system of Popowicz

We consider a two-component Hamiltonian system of partial differential equations with quadratic nonlinearities introduced by Popowicz, which has the form of a coupling between the Camassa–Holm and Degasperis–Procesi equations. Despite having reductions to these two integrable partial differential equations, the Popowicz system itself is not integrable. Nevertheless, as one of the authors showed with Irle, it admits distributional solutions of peaked soliton (peakon) type, with the dynamics of N peakons being determined by a Hamiltonian system on a phase space of dimension 3N. As well as the trivial case of a single peakon ($N=1$), the case $N=2$ is Liouville integrable. We present the explicit solution for the two-peakon dynamics, and describe some of the novel features of the interaction of peakons in the Popowicz system.     

On the integration of some classes of (<Emphasis Type="Italic">n</Emphasis>+1)-dimensional nonlinear equations of mathematical physics

This paper presents a method for construction of the families of particular solutions to some new classes of (n+1)-dimensional nonlinear partial differential equations (PDE). The method is based on the specific link between algebraic matrix equations and PDEs. Admittable solutions depend on arbitrary functions of n variables. Examples of deformed Burgers-type equations are given.     

Necessary and sufficient conditions for the existence of a Lagrangian in field theory II. Direct analytic representations of tensorial field equations

By using a characterization of the concept of analytic representation and a variational approach to self-adjointness introduced in a preceding paper, we prove a theorem, according to which a necessary and sufficient condition for a class $\text{C}$², regular, tensorial, quasi-linear system of field equations to admit an ordered direct analytic representation in terms of the Lagrange equations in a region R of its variables is that the system is self-adjoint in R. We point out as a first corollary that if the ordering requirement is removed from the definition of analytic representation, then the condition of self-adjointness of the field equations is only sufficient for the existence of a Lagrangian density. We then provide as a second corollary a methodology for the computation of the Lagrangian density for the representation of self-adjoint quasi-linear tensorial field equations. This methodology is also particularized for ordinary semilinear systems of tensorial field equations through a third corollary. The above results are interpreted from the viewpoint of interactions. We first recover, through a fourth corollary, the conventional structure of the total Lagrangian density L_Tot = Σ_{1 a}ⁿL_Free^(a) + L_Int for the semilinear form of the field equations, and then introduce through a fifth corollary a generalized structure of the type L_Tot = Σ_{1 a}ⁿL_{Int, I}^(a)L_Free^(a) + L_Int.II for t representations of the field equations in the quasi-linear form. Therefore, our analysis seems to indicate that a general form of representing interacting fields is characterized by (n+1)-interaction terms in the Lagrangian: n multiplicative terms and one additive term to the Lagrangian for the free fields.     

Systems of ordinary differential equations with nonlinear superposition principles

This work is concerned with the derivation of superposition rules which express the general solution of ordinary differential equations.

$\text{x}\text{?}\text{= η(x,t).}\text{(x, η ?}\text{R}{}^{\mathrm{n}}\text{, t ?}\text{R}\text{)}$

. in terms of a finite number of particular solutions. The point of departure is Lie's criterion according to which such a rule exists if and only if the vector fields η(x,t). ? generate a finite dimensional Lie algebra. We provide three different constructive methods for deriving superposition rules and apply them to systems of coupled Riccati equations of the projective and conformal types based, respectively, on the Lie algebra sl(n + 1, R) and o(p + 1, n ? p + 1).     

A new derivation of the Picard-Fuchs equations for effective N = 2 super Yang-Mills theories

A new method to obtain the Picard-Fuchs equations of effective, N = 2 supersymmetric gauge theories in 4 dimensions is developed. It includes both pure super Yang-Mills and supersymmetric gauge theories with massless matter hypermultiplets. It applies to all classical gauge groups, and directly produces a decoupled set of second-order, partial differential equations satisfied by the period integrals of the Seiberg-Witten differential along the 1-cycles of the algebraic curves describing the vacuum structure of the corresponding N = 2 theory.     

The classical limit of the relativistic Vlasov-Maxwell system

Solutions of the relativistic Vlasov-Maxwell system of partial differential equations are considered in three space dimensions. The speed of light,c, appears as a parameter in this system. For smooth Cauchy data, classical solutions are shown to exist on a time interval that is independent ofc. Then, using an integral representation for the electric and magnetic fields due to Glassey and Strauss [6], conditions are given under which solutions of the relativistic Vlasov-Maxwell system converge in pointwise sense to solutions of the non-relativistic Vlasov-Poisson system at the asymptotic rate of 1/c, asc tends to infinity.     

Representations and inequalities for Ising model Ursell functions

We describe and investigate representations for the Ursell functionu _n of a family ofn random variables {σ _i}. The representations involve independent but identically distributed copies of the family. We apply one of these representations in the case that the random variables are spins of a finite ferromagnetic Ising model with quadratic Hamiltonian to show that (?1) ^n/2+1 u _n(σ ₁, ...,σ _n) ≧ 0 forn=2, 4, and 6 by proving the stronger statement \(( - 1 )^{\frac{n}{2} + 1} \frac{{\partial ^m }}{{\partial J_{i1j1} \cdots \partial J_{imjm} }}Z^{\frac{n}{2}} u_n \left| {_{J = 0} } \right. \geqq {}^\backprime 0\) forn=2, 4, and 6, theJ _ij being coupling constants in the Hamiltonian andZ the partition function. For generaln we combine this result with various reductions to show that sufficiently simple derivatives of (?1) ^n/2+1 Z ^n/2u_n, evaluated at zero coupling, are nonnegative. In particular, we conclude that (?1) ^n/2+1 u _n ≧ 0 if all couplings are nonzero and the inverse temperature β is sufficiently small or sufficiently large, though this result is not uniform in the ordern or the system size. In an appendix we give a simple proof of recent inequalities which boundn-spin expectations by sums of products of simpler expectations.     

A Hamilton-Jacobi formalism for thermodynamics

We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it is 2n+1-dimensional. The equations of state of a substance pick out an n-dimensional submanifold. A family of substances whose equations of state depend on n parameters define a hypersurface of co-dimension one. This can be described by the vanishing of a function which plays the role of a Hamiltonian. The ordinary differential equations (characteristic equations) defined by this function describe a dynamical system on the hypersurface. Its orbits can be used to reconstruct the equations of state. The ‘time’ variable associated to this dynamics is related to, but is not identical to, entropy. After developing this formalism on well-grounded systems such as the van der Waals gases and the Curie-Weiss magnets, we derive a Hamilton-Jacobi equation for black hole thermodynamics in General Relativity. The cosmological constant appears as a constant of integration in this picture.     

Self-duality and octonionic analyticity of S7-valued antisymmetric fields in eight dimensions

We exhibit two octonionic extensions of the Kalb-Ramond type fields with rank two and four in eigth dimensions. By analogy to the d = 4 (anti-) self-duality of the SU(2) ≈ S³ quaternionic gauge field we consider the respective d = 8 (anti-) self-duality equations for these nonlinear, S⁷-valued antisymmetric fields. By way of an octonionic 't Hooft ansatz these equations reduce to the same generalized Fueter-Cauchy-Riemann equations over S⁸. Explicit (9n + 8) parameter S⁷ → S⁷ mapping solutions, n being a winding number, are found in terms of holomorphic functions of the spacetime octonion. An infinite number of local continuity equations results.     

Response to V. A. Marichev's reply [Surf. Sci. 605(2011)2097] to the comments written by E. M. Gutman [Surf. Sci. 605(2011)1923]

For the first time, it is shown that the first and second Gokhshtein equations were originally derived by him from false fundamental equations violating criteria for applications of Maxwell relations and Legendre transforms. These equations are erroneous. For example, the direct connection between the electrode potential and charge density is actually independent of the tangential elastic strain of flat ideally polarizable electrode. The sequence of equations of Shuttleworth–Herring–Eriksson–Couchman–Jesser–Everet–Davidson–Rusanov has a common mathematical defect which consists of illegally equating the total differential to the partial differential.     

A gravitating SO(3, 1) gauge field

In this article, we postulate SO(3, 1) as a local symmetry of any relativistic theory. This is equivalent to assuming the existence of a gauge field associated with this noncompact group. This SO(3, 1) gauge field is the spinorial affinity which usually appears when we deal with weighting spinors, which, as is well known, cannot be coupled to the metric tensor field. Furthermore, according to the integral approach to gauge fields proposed by Yang, it is also recognized that in order to obtain models of gravity we have to introduce ordinary affinities as the gauge field associated with GL(4) (the local symmetry determined by the parallel transport). Thus if we assume both L(4) and SO(3, 1) as local independent symmetries we are led to analyze the dynamical gauge system constituted by the Einstein field interacting with the SO(3, 1) Weyl-Yang gauge field. We think this system is a possible model of strong gravity. Once we give the first-order action for this Einstein-Weyl-Yang system we study whether the SO(3, 1) gauge field could have a tetrad associated with it. It is also shown that both fields propagate along a unique characteristic cone. Algebraic and differential constraints are solved when the system evolves along a null coordinate. The unconstrained expression for the action of the system is found working in the Bondi gauge. That allows us to exhibit an explicit expression of the dynamical generator of the system. Its signature turns out to be nondefinite, due to the nondefinite contribution of the Weyl-Yang field, which has the typical spinorial behavior. A conjecture is made that such an unpleasant feature could be overcome in the quantized version of this model.     

Hydrodynamics of an n-component phonon system

The dynamic properties of an n-component phonon system in d dimensions, which serves as a model for a structural phase transition of second order, are investigated. The symmetry group of the hamiltonian is the group of orthogonal transformations $\text{O}$(n). For n ≥ 2 a continuous symmetry is broken for T<T_c, where T_c is the transition temperature. We derive the hydrodynamic equations for the generators of this group, the $\text{1}\text{2}\text{n (n ? 1)}$ angular-momentum variables. Besides the usual hydrodynamics of a phonon system, there are $\text{1}\text{2}\text{n (n ? 1)}$ additional independent diffusive modes for T > T_c. In the ordered phase we find 2 (n ? 1) propagating modes with linear dispersion and quadratic damping. Formally the hydrodynamics is similar in the isotropic Heisenberg ferromagnet (n = 2) or the isotropic antiferromagnet (n ≥ 3). The relaxing modes for T < T_c require special care. We study the critical dynamics by means of the dynamical scaling hypothesis and by a mode-coupling calculation, both of which give the critical dynamical exponent $\text{z =}\text{1}\text{2}\text{d}$. The results are compared with the 1/n expansion. It is shown that for large n there is a non-asymptotic region characterized by an effective exponent $\text{z}\text{?}\text{= φ/2ν}$, where φ is the crossover exponent for a uniaxial perturbation, and ν the critical exponent of the correlation length.     

Representation of zero curvature for the system of nonlinear partial differential equations x_{\alpha,z\bar z} = \exp (kx)_\alpha and its integrability

Representation of zero curvature for the system of essentially nonlinear partial differential equations \(x_{\alpha ,z\bar z} = exp\left( {\sum\nolimits_{\beta = 1}^r {k_{\alpha \beta } x_\beta } } \right),1 \leqslant \alpha \leqslant r\) , with an arbitrary numeral matrix k is constructed in an explicit form. On the basis of this representation we give an invariant integration method for the system when k coincides with the Cartan matrix of simple Lie algebra of rank r. The final solutions depend on 2r arbitrary functions.     

Orthogonal polynomial solutions of the Fokker-Planck equation

We have tabulated the form of the coefficientsg ₁(x) andg ₂(x) as well as the boundary values [a, b] of the Fokker-Planck equation $$\frac{{\partial P(x, t)}}{{\partial t}} = - \frac{\partial }{{\partial x}}[g_1 (x)P(x, t)] + \frac{{\partial ^2 }}{{\partial x^2 }}[g_2 (x)P(x, t)],a \leqslant x \leqslant b$$ for which the solution can be written as an eigenfunction expansion in the classical orthogonal polynomials. We also discuss the problem of finding solutions in terms of the discrete classical polynomials for the differential difference equations of stochastic processes.     

On Methods of Finding Bäcklund Transformations in Systems with More than Two Independent Variables

Abstract

Bäcklund transformations, which are relations among solutions of partial differential equations–usually nonlinear–have been found and applied mainly for systems with two independent variables. A few are known for equations like the Kadomtsev-Petviashvili equation [1], which has three independent variables, but they are rare. Wahlquist and Estabrook [2] discovered a systematic method for searching for Bäcklund transformations, using an auxiliary linear system called a prolongation structure. The integrability conditions for the prolongation structure are to be the original differential equation system, most of which systems have just two independent variables. This paper discusses how the Wahlquist-Estabrook method might be applied to systems with larger numbers of variables, with the Kadomtsev-Petviashvili equation as an example. The Zakharov-Shabat method is also discussed. Applications to other equations, such as the Davey-Stewartson and Einstein equation systems, are presented.