The Theory of Connections and the Problem of Existence of Backlund Transformations for Second Order Evolution Equations

Backlund transformations are used to search for solutions, particularly soliton solutions, of non-linear differential equations. In this paper we present an invariant geometrical theory of Backlund transformations for second order evolution equations with one space variable. The main concept is that of connection defining the representation of zero curvature for a given partial differential equation. The main result of this paper is a criterion of existence of Backlund transformations for second order evolution equations with one space variable. We find thegeneral form of a second order evolution equation that admits Backlund transformations. Furthermore, for a special important class of evolution equations we show that a Backlund transformation exists if and only if the equation has one of two special forms. An equation of the first of these types can be then reduced, by a change of variable, to the Burgers equation, and the equation of the second type can be reduced to a well-known linear equation. All differential-geometric considerations in this paper are local.Comment: Some of the results of this paper have been announced by the author in conference talk

Invariants of differential equations defined by vector fields

We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the second order. A result on the characterization of classes of these equations by the invariant functions is also given.Comment: 13 page

Notes on Lie symmetry group methods for differential equations

Fundamentals on Lie group methods and applications to differential equations are surveyed. Many examples are included to elucidate their extensive applicability for analytically solving both ordinary and partial differential equations.Comment: 85 Pages. expanded and misprints correcte

On the Harmonic Oscillation of High-order Linear Time Invariant Systems

Linear time invariant (LTI) systems are widely used for modeling system dynamics in science and engineering problems. Harmonic oscillation of LTI systems are widely used for modeling and analyses of periodic physical phenomenon. This study investigates sufficient conditions to obtain harmonic oscillation for high-order LTI systems. The paper presents a design procedure for controlling harmonic oscillation of singleinput single-output high-order LTI systems. LTI system coefficients are calculated by the solution of linear equation set, which imposes a stable sinusoidal oscillation solution for the characteristic polynomials of LTI systems. An example design is demonstrated for fourth-order LTI systems and the control of harmonic oscillations are discussed by illustrating Hilbert transform and spectrogram of oscillation signals.Comment: 8 Figures 12 page

Space-time as a structured relativistic continuum

It is well known that there are various models of gravitation: the metrical Hilbert-Einstein theory, a wide class of intrinsically Lorentz-invariant tetrad theories (of course, generally-covariant in the space-time sense), and many gauge models based on various internal symmetry groups (Lorentz, Poincare, ${\rm GL}(n,\mathbb{R})$, ${\rm SU}(2,2)$, ${\rm GL}(4,\mathbb{C})$, and so on). One believes usually in gauge models and we also do it. Nevertheless, it is an interesting idea to develop the class of ${\rm GL}(4,\mathbb{R})$-invariant (or rather ${\rm GL}(n,\mathbb{R})$-invariant) tetrad ($n$-leg) generally covariant models. This is done below and motivated by our idea of bringing back to life the Thales of Miletus idea of affine symmetry. Formally, the obtained scheme is a generally-covariant tetrad ($n$-leg) model, but it turns out that generally-covariant and intrinsically affinely-invariant models must have a kind of non-accidental Born-Infeld-like structure. Let us also mention that they, being based on tetrads ($n$-legs), have many features common with continuous defect theories. It is interesting that they possess some group-theoretical solutions and more general spherically-symmetric solutions. It is also interesting that within such framework the normal-hyperbolic signature of the space-time metric is not introduced by hand, but appears as a kind of solution, rather integration constants, of differential equations. Let us mention that our Born-Infeld scheme is more general than alternative tetrad models. It may be also used within more general schemes, including also the gauge ones.Comment: 41 page

Direct construction method for conservation laws of partial differential equations. Part II: General treatment

This paper gives a general treatment and proof of the direct conservation law method presented in Part I. In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.Comment: Published version; 19 pages; LaTe

On the integrability in Grassmann geometries: integrable systems associated with fourfolds Gr(3, 5)

We investigate dispersionless integrable systems in 3D associated with fourfolds in the Grassmannian Gr(3,5). Such systems appear in numerous applications in continuum mechanics, general relativity and differential geometry, and include such well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation, etc. We prove the equivalence of the four different approaches to integrability, revealing a remarkable correspondence with Einstein-Weyl geometry and the theory of GL(2,R) structures.Comment: This is an elaborated version of the main part concerning dispersionless integrable systems in 3D, reflected in the title of the paper. We omitted the last two sections on systems in 4D and higher dimensional Monge-Ampere equations that will be expanded and posted in arXiv separately in the near future. These parts, as well as supplementary materials, are still accessible via arXiv:1503.02274v

W-Symmetries of Ito stochastic differential equations

We discuss W-symmetries of Ito stochastic differential equations, introduced in a recent paper by Gaeta and Spadaro [J. Math. Phys. 2017]. In particular, we discuss the general form of acceptable generators for continuous (Lie-point) W-symmetry, arguing they are related to the (linear) conformal group, and how W-symmetries can be used in the integration of Ito stochastic equations along Kozlov theory for standard (deterministic or random) symmetries. It turns out this requires, in general, to consider more general classes of stochastic equations than just Ito ones.Comment: Preprint version; final (improved) version to appear in J. Math. Phy

Parametric Factorizations of Second-, Third- and Fourth-Order Linear Partial Differential Operators with a Completely Factorable Symbol on the Plane

Parametric factorizations of linear partial operators on the plane are considered for operators of orders two, three and four. The operators are assumed to have a completely factorable symbol. It is proved that ``irreducible'' parametric factorizations may exist only for a few certain types of factorizations. Examples are given of the parametric families for each of the possible types. For the operators of orders two and three, it is shown that any factorization family is parameterized by a single univariate function (which can be a constant function)

Canonical variables for multiphase solutions of the KP equation

The KP equation has a large family of quasiperiodic multiphase solutions. These solutions can be expressed in terms of Riemann-theta functions. In this paper, a finite-dimensional canonical Hamiltonian system depending on a finite number of parameters is given for the description of each such solution. The Hamiltonian systems are completely integrable in the sense of Liouville. In effect, this provides a solution of the initial-value problem for the theta-function solutions. Some consequences of this approach are discussed.Comment: 52 papes, 3 figures, uses psfig, latexsy