Classical perturbation theory and resonances in some rigid body systems

We consider the system of a rigid body in a weak gravitational field on the zero level set of the area integral and study its Poincaré sets in integrable and nonintegrable cases. For the integrable cases of Kovalevskaya and Goryachev–Chaplygin we investigate the structure of the Poincaré sets analytically and for nonintegrable cases we study these sets by means of symbolic calculations. Based on these results, we also prove the existence of periodic solutions in the perturbed nonintegrable system. The Chaplygin integrable case of Kirchhoff’s equations is also briefly considered, for which it is shown that its Poincaré sets are similar to the ones of the Kovalevskaya case.     

A New Integrable Case for the Kirchhoff Equation

A new integrable case is found for the Kirchhoff equation. The additional integral of motion is a fourth-degree polynomial, the principal metric is diagonal with the eigenvalues a ₁ = a ₂ = 1 and a ₃ = 2, and the other two metrics are nondiagonal.     

Lax Pairs for the Deformed Kowalevski and Goryachev–Chaplygin Tops

We consider a quadratic deformation of the Kowalevski top. This deformation includes a new integrable case for the Kirchhoff equations recently found by one of the authors as a degeneration. A 5×5 matrix Lax pair for the deformed Kowalevski top is proposed. We also find similar deformations of the two-field Kowalevski gyrostat and the so(p,q) Kowalevski top. All our Lax pairs are deformations of the corresponding Lax representations found by Reyman and Semenov-Tian-Shansky. A similar deformation of the Goryachev–Chaplygin top and its 3×3 matrix Lax representation is also constructed.     

Minimal hexagonal chains with respect to the Kirchhoff index

Let G be a connected graph. The resistance distance between any two vertices of G is equal to the effective resistance between them in the corresponding electrical network constructed from G by replacing each edge with a unit resistor. The Kirchhoff index of G is defined as the sum of resistance distances between all pairs of vertices. Hexagonal chains are graph representations of unbranched catacondensed benzenoid hydrocarbons. It was shown in Yang and Klein (2014) [30] that among all hexagonal chains with n hexagons, the linear chain $L_n$ is the unique chain with maximum Kirchhoff index. However, for hexagonal chains with minimum Kirchhoff index, it was only claimed that the minimum Kirchhoff index is attained only when the hexagonal chain is an “all–kink” chain. In this paper, by standard techniques of electrical networks and comparison results on Kirchhoff indices of $S,T$-isomers, “all-kink” chains with maximum and minimum Kirchhoff indices are characterized. As a consequence, hexagonal chains with minimum Kirchhoff indices are singled out.     

Kirchhoff Indexes of a network

In this work we define the effective resistance between any pair of vertices with respect to a valueλ?0and a weightω on the vertex set. This allows us to consider a generalization of the Kirchhoff Index of a finite network. It turns out that λ is the lowest eigenvalue of a suitable semi-definite positive Schrödinger operator and ω is the associated eigenfunction. We obtain the relation between the effective resistance, and hence between the Kirchhoff Index, with respect to λ and ω and the eigenvalues of the associated Schrödinger operator. However, our main aim in this work is to get explicit expressions of the above parameters in terms of equilibrium measures of the network. From these expressions, we derive a full generalization of Foster’s formulae that incorporate a positive probability of remaining in each vertex in every step of a random walk. Finally, we compute the effective resistances and the generalized Kirchhoff Index with respect to a λ and ω for some families of networks with symmetries, specifically for weighted wagon-wheels and circular ladders.     

Resistance distances and the Kirchhoff index in Cayley graphs

In this paper, closed-form formulae for the Kirchhoff index and resistance distances of the Cayley graphs over finite abelian groups are derived in terms of Laplacian eigenvalues and eigenvectors, respectively. In particular, formulae for the Kirchhoff index of the hexagonal torus network, the multidimensional torus and the t-dimensional cube are given, respectively. Formulae for the Kirchhoff index and resistance distances of the complete multipartite graph are obtained.     

Asymptotic behaviour for the Kirchhoff equation

In this paper we study the asymptotic behaviour of the solution u of the Kirchhoff eqation with small data. More precisely we show that {fx293-01} every k ε Nwhere v is a suitable solution of an appropriate wave equation. Moreover we give some estimates on $$\mathop {\lim }\limits_{t \to \infty } \parallel \nabla u\parallel _2$$ .     

Existence of positive solutions to Kirchhoff type problems with zero mass

The existence of positive solutions depending on a nonnegative parameter λ to Kirchhoff type problems with zero mass is proved by using variational method, and the new result does not require usual compactness conditions. A priori estimate and a Pohozaev type identity are used to obtain the bounded Palais–Smale sequences for constant coefficient nonlinearity, while a cut-off functional and Pohozaev type identity are utilized to obtain the bounded Palais–Smale sequences for the variable-coefficient case.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

Existence and multiplicity of solutions for nonlocal systems with Kirchhoff type

Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of Fadell and Rabinowitz. We also consider the critical case and prove existence of positive least energy solution when the parameter β is sufficiently large.     

Kirchhoff index of composite graphs

Let G₁+G₂, G₁°G₂ and G₁{G₂} be the join, corona and cluster of graphs G₁ and G₂, respectively. In this paper, Kirchhoff index formulae of these composite graphs are given.     

Factorization of the Loop Algebra and Integrable Toplike Systems

With any Lie algebra of Laurent series with coefficients in a semisimple Lie algebra and its decomposition into a sum of the subalgebra consisting of the Taylor series and a complementary subalgebra, we associate a hierarchy of integrable Hamiltonian nonlinear ODEs. In the case of the so(3) Lie algebra, our scheme covers all classical integrable cases in the Kirchhoff problem of the motion of a rigid body in an ideal fluid. Moreover, the construction allows generating integrable deformations for known integrable models.     

On the Kirchhoff index of some toroidal lattices

The resistance distance is a novel distance function on a graph proposed by Klein and Randi? [D.J. Klein and M. Randi?, Resistance distance, J. Math. Chem. 12 (1993), pp. 81–85]. The Kirchhoff index of a graph G is defined as the sum of resistance distances between all pairs of vertices of G. In this article, based on the result by Gutman and Mohar [I. Gutman and B. Mohar, The quasi-Wiener and the Kirchhoff indices coincide, J. Chem. Inf. Comput. Sci. 36 (1996), pp. 982–985], we compute the Kirchhoff index of the square, 8.8.4, hexagonal and triangular lattices, respectively.     

An attractor for a nonlinear dissipative wave equation of Kirchhoff type

In this paper we prove the existence and some absorbing properties of an attractor in a local sense for the initial-boundary value problem of a quasilinear wave equation of Kirchhoff type with a standard dissipation ut.     

Separation of variables for some systems with a fourth-order integral of motion

We construct separation variables for Yehia’s integrable deformations of the Kovalevskaya top and the Chaplygin system on a sphere. In the general case, the corresponding quadratures are given by the Abel-Jacobi map on a two-dimensional submanifold of the Jacobian of a genus-three algebraic curve, which is not hyperelliptic.     

Global attractor for the Kirchhoff type equation with a strong dissipation

The paper studies the longtime behavior of the Kirchhoff type equation with a strong dissipation utt−M(‖∇u^‖2)Δu−Δut+h(ut)+g(u)=f(x). It proves that the related continuous semigroup S(t) possesses in the phase space with low regularity a global attractor which is connected. And an example is shown.     

Asymptotic stability for anisotropic Kirchhoff systems

We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative anisotropic Kirchhoff systems, involving the p(x)-Laplacian operator, governed by time-dependent nonlinear damping forces and strongly nonlinear power-like variable potential energies. This problem had been considered earlier for potential energies which arise from restoring forces, whereas here we allow also the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. The results are further extended to the more delicate problem involving higher order damping terms.     

Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation

The paper studies the existence of the finite-dimensional global attractors and exponential attractors for the dynamical system associated with the Kirchhoff type equation with a strong dissipation utt−M(‖∇u^‖2)Δu−Δut+h(ut)+g(u)=f(x). It proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor. For application, the fact shows that for the concerned viscoelastic flow the permanent regime (global attractor) can be observed when the excitation starts from any bounded set in phase space, and the dimension of the attractor, that is, the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow, is finite.     

A second-order Cauchy problem in a scale of Banach spaces and application to Kirchhoff equations

In a scale of Banach spaces we study the Cauchy problem for the equation u^′′=A(Bu(t),u), where A is a bilinear operator and B is a completely continuous operator. Obtained results are applied to prove existence of solutions in the Gevrey class for Kirchhoff equations.     

The Sokolov case,integrable Kirchhoff elasticae,and genus 2 theta functions via discriminantly separable polynomials

We use the discriminantly separable polynomials of degree 2 in each of three variables to integrate explicitly the Sokolov case of a rigid body in an ideal fluid and integrable Kirchhoff elasticae in terms of genus 2 theta functions. The integration procedure is a natural generalization of the one used by Kowalevski in her celebrated 1889 paper. The algebraic background for the most important changes of variables in this integration procedure is associated to the structure of the two-valued groups on an elliptic curve. Such two-valued groups have been introduced by V.M. Buchstaber.