Fifty Years of the Exact Solution of the Two-Dimensional Ising Model by Onsager

The exact solution of the two-dimensional Ising model by Onsager in 1944 represents one of the landmarks in theoretical physics. On the occassion of the fifty years of the exact solution, we give a historical review of this model. After briefly discussing the exact solution by Onsager, we point out some of the recent developments in this field. The exact solution by Onsager has inspired several developments in various other fields. Some of these are also briefly mentioned.Comment: 14 pages, no figure, revtex file, To be Published in "Current Science (India)", minor corrections made in sec.

Local Identities Involving Jacobi Elliptic Functions

We derive a number of local identities of arbitrary rank involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us recently, along with an extension to several new cyclic identities of arbitrary rank. Second, we obtain a generalization to cyclic identities in which successive terms have a multiplicative phase factor exp(2i\pi/s), where s is any integer. Third, we systematize the local identities by deriving four local ``master identities'' analogous to the master identities for the cyclic sums discussed by us previously. Fourth, we point out that many of the local identities can be thought of as exact discretizations of standard nonlinear differential equations satisfied by the Jacobian elliptic functions. Finally, we obtain explicit answers for a number of definite integrals and simpler forms for several indefinite integrals involving Jacobi elliptic functions.Comment: 47 page

Exact Solutions of the Two-Dimensional Discrete Nonlinear Schr\"odinger Equation with Saturable Nonlinearity

We show that the two-dimensional, nonlinear Schr\"odinger lattice with a saturable nonlinearity admits periodic and pulse-like exact solutions. We establish the general formalism for the stability considerations of these solutions and give examples of stability diagrams. Finally, we show that the effective Peierls-Nabarro barrier for the pulse-like soliton solution is zero

A facet is not an island: step-step interactions and the fluctuations of the boundary of a crystal facet

In a recent paper [Ferrari et al., Phys. Rev. E 69, 035102(R) (2004)], the scaling law of the fluctuations of the step limiting a crystal facet has been computed as a function of the facet size. Ferrari et al. use rigorous, but physically rather obscure, arguments. Approaching the problem from a different perspective, we rederive more transparently the scaling behavior of facet edge fluctuations as a function of time. Such behavior can be scrutinized with STM experiments and with numerical simulations.Comment: 3 page

Soliton Lattice and Single Soliton Solutions of the Associated Lam\'e and Lam\'e Potentials

We obtain the exact nontopological soliton lattice solutions of the Associated Lam\'e equation in different parameter regimes and compute the corresponding energy for each of these solutions. We show that in specific limits these solutions give rise to nontopological (pulse-like) single solitons, as well as to different types of topological (kink-like) single soliton solutions of the Associated Lam\'e equation. Following Manton, we also compute, as an illustration, the asymptotic interaction energy between these soliton solutions in one particular case. Finally, in specific limits, we deduce the soliton lattices, as well as the topological single soliton solutions of the Lam\'e equation, and also the sine-Gordon soliton solution.Comment: 23 pages, 5 figures. Submitted to J. Math. Phy

Dynamic and Static Excitations of a Classical Discrete Anisotropic Heisenberg Ferromagnetic Spin Chain

Using Jacobi elliptic function addition formulas and summation identities we obtain several static and moving periodic soliton solutions of a classical anisotropic, discrete Heisenberg spin chain with and without an external magnetic field. We predict the dispersion relations of these nonlinear excitations and contrast them with that of magnons and relate these findings to the materials realized by a discrete spin chain. As limiting cases, we discuss different forms of domain wall structures and their properties.Comment: Accepted for publication in Physica

Compacton Solutions in a Class of Generalized Fifth Order Korteweg-de Vries Equations

We study a class of generalized fifth order Korteweg-de Vries (KdV) equations which are derivable from a Lagrangian L(p,m,n,l) which has variable powers of the first and second derivatives of the field with powers given by the parameters p,m,n,l. The resulting field equation has solitary wave solutions of both the usual (non-compact) and compact variety ("compactons"). For the particular case that p=m=n+l, the solitary wave solutions have compact support and the feature that their width is independent of the amplitude. We discuss the Hamiltonian structure of these theories and find that mass, momentum, and energy are conserved. We find in general that these are not completely integrable systems. Numerical simulations show that an arbitrary compact initial wave packet whose width is wider than that of a compacton breaks up into several compactons all having the same width. The scattering of two compactons is almost elastic, with the left over wake eventually turning into compacton-anticompacton pairs. When there are two different compacton solutions for a single set of parameters the wider solution is stable, and this solution is a minimum of the Hamiltonian.Comment: 13 pages (8 embedded figures), RevTeX (plus macro), uses eps

Exact Moving and Stationary Solutions of a Generalized Discrete Nonlinear Schrodinger Equation

We obtain exact moving and stationary, spatially periodic and localized solutions of a generalized discrete nonlinear Schr\"odinger equation. More specifically, we find two different moving periodic wave solutions and a localized moving pulse solution. We also address the problem of finding exact stationary solutions and, for a particular case of the model when stationary solutions can be expressed through the Jacobi elliptic functions, we present a two-point map from which all possible stationary solutions can be found. Numerically we demonstrate the generic stability of the stationary pulse solutions and also the robustness of moving pulses in long-term dynamics.Comment: 22 pages, 7 figures, to appear in J. Phys.

New classes of quasi-solvable potentials, their exactly-solvable limit and related orthogonal polynomials

We have generated, using an sl(2,R) formalism, several new classes of quasi-solvable elliptic potentials, which in the appropriate limit go over to the exactly solvable forms. We have obtained exact solutions of the corresponding spectral problems for some real values of the potential parameters. We have also given explicit expressions of the families of associated orthogonal polynomials in the energy variable.Comment: 14 pages, 5 tables, LaTeX2