1. Introduction.- 1.1 The Fermi-Pasta-Ulam Problem.- 1.2 Henon-Heiles Calculation.- 1.3 Discovery of Solitons.- 1.4 Dual Systems.- 2. The Lattice with Exponential Interaction.- 2.1 Finding of an Integrable Lattice.- 2.2 The Lattice with Exponential Interaction.- 2.3 Periodic Solutions.- 2.4 Solitary Waves.- 2.5 Two-Soliton Solutions.- 2.6 Hard-Sphere Limit.- 2.7 Continuum Approximation and Recurrence Time.- 2.8 Applications and Extensions.- 2.9 Poincare Mapping.- 2.10 Conserved Quantities.- 3. The Spectrum and Construction of Solutions.- 3.1 Matrix Formalism.- 3.2 Infinite Lattice.- 3.3 Scattering and Bound States.- 3.4 The Gel'fand-Levitan Equation.- 3.5 The Initial Value Problem.- 3.6 Soliton Solutions.- 3.7 The Relationship Between the Conserved Quantities and the Transmission Coefficient.- 3.8 Extensions of the Equations of Motion and the Kac-Moerbeke System.- 3.9 The Backlund Transformation.- 3.10 A Finite Lattice.- 3.11 Continuum Approximation.- 4. Periodic Systems.- 4.1 Discrete Hill's Equation.- 4.2 Auxiliary Spectrum.- 4.3 Relation Between ?j (k) and ?j (0).- 4.4 Related Integrals on the Riemann Surface.- 4.5 Solution to the Inverse Problem.- 4.6 Time Evolution.- 4.7 A Simple Example (A Cnoidal Wave).- 4.8 Periodic System of Three-Particles.- 5. Application of the Hamilton-Jacobi Theory.- 5.1 Canonically Conjugate Variables.- 5.2 Action Variables.- 6. Recent Advances in the Theory of Nonlinear Lattices.- 6.1 The KdV Equation as a Limit of the TL Equation.- 6.2 Interacting Soliton Equations.- 6.3 Integrability.- 6.4 Generalization of the TL Equation.- 6.5 Two-Dimensional TL.- 6.6 Bethe Ansatz.- 6.7 The Thermodynamic Limit.- 6.8 Hierarchy of Nonlinear Equations.- 6.9 Some Numerical Results.- Appendices.- Simplified Answers to Main Problems.- References.- List of Authors Cited in Text.

Theory of nonlinear lattices

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Geometric structures on filtered manifolds

Chern–Simons type Lagrangians in d = 3 dimensions are analysed from the point of view of their covariance and globality. We use the transgression formula to find out a new fully covariant and global Lagrangian for Chern–Simons gravity: the price for establishing globality is hidden in a bimetric (or biconnection) structure. Such a formulation allows us to calculate from a global and simpler viewpoint the energy–momentum complex and the superpotential both for Yang–Mills and gravitational examples.

http://iopscience.iop.org/article/10.1088/0305-4470/36/10/318/pdf

A covariant formalism for Chern–Simons gravity

The Drell-Hearn-Gerasimov and Bjorken sum rules are special examples of dispersive sum rules for the spin-dependent structure function G_1(\nu, Q^2) at Q^2=0 and \infty We generalize these sum rules through studying the virtual-photon Compton amplitudes S_1(\nu, Q^2) and S_2(\nu,Q^2) At small Q^2, we calculate the Compton amplitudes at leading-order in chiral perturbation theory; the resulting sum rules can be tested by data soon available from Jefferson Lab For Q^2>>\Lambda_{QCD}^2, the standard twist-expansion for the Compton amplitudes leads to the well-known deep-inelastic sum rules Although the situation is still relatively unclear in a small intermediate-Q^2 window, we argue that chiral perturbation theory and the twist-expansion alone already provide strong constraints on the Q^2-evolution of the G_1(\nu,Q^2) sum rule from Q^2=0 to Q^2=\infty

Generalized Sum Rules for Spin-Dependent Structure Functions of the Nucleon

We refer to Krupka's variational sequence, i.e. the quotient of the de Rham sequence on a finite order jet space with respect to a ‘variationally trivial’ subsequence. Among the morphisms of the variational sequence there are the Euler-Lagrange operator and the Helmholtz operator.

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Symmetries in finite order variational sequences

Covariant gauge-natural conservation laws

In the classical Lagrangian approach to conservation laws of gauge-natural field theories a suitable (vector) density is known to generate the so--called {\em conserved Noether currents}. It turns out that along any section of the relevant gauge--natural bundle this density is the divergence of a skew--symmetric (tensor) density, which is called a {\em superpotential} for the conserved currents. 
We describe gauge--natural superpotentials in the framework of finite order variational sequences according to Krupka. We refer to previous results of ours on {\em variational Lie derivatives} concerning abstract versions of Noether's theorems, which are here interpreted in terms of ``horizontal'' and ``vertical'' conserved currents. The gauge--natural lift of principal automorphisms implies suitable linearity properties of the Lie derivative operator. Thus abstract results due to Kol\'a\v{r}, concerning the integration by parts procedure, can be applied to prove the {\em existence} and {\em globality} of superpotentials in a very general setting.

Conservation laws and variational sequences in gauge-natural theories

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second variation is equal (up to horizontal differentials) to the vertical differential of the Euler–Lagrange morphism which turns out to be self-adjoint along solutions of the Euler–Lagrange equations. These two objects, respectively, generalize in an invariant way the Hessian morphism and the Jacobi morphism (which is then self-adjoint along critical sections) of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.

http://poincare.unisalento.it/vitolo/vitolo_files/publications/journals/14a_HiVarDGA.pdf

The Hessian and Jacobi morphisms for higher order calculus of variations

We investigate globality properties of conserved currents associated with local variational problems admitting global Euler–Lagrange morphisms. We show that the obstruction to the existence of a global conserved current is the difference of two conceptually independent cohomology classes: one coming from using the symmetries of the Euler–Lagrange morphism and the other from the system of local Noether currents.

Marcella Palese

Papers

Symmetries in finite order variational sequences

Covariant gauge-natural conservation laws

Conservation laws and variational sequences in gauge-natural theories

The Hessian and Jacobi morphisms for higher order calculus of variations

Local variational problems and conservation laws