Hierarchy Of Integrable Nonlinear Equations Research Articles

SYMMETRIES IN THE GROSS-NEVEU PHASE DIAGRAM

The existence of crystalline condensates in the temperature and chemical potential phase diagram of the Gross-Neveu models can be traced to intricate symmetries of the associated inhomogeneous gap equation. The gap equation based on the Ginzburg-Landau expansion is precisely the mKdV or AKNS hierarchy of integrable nonlinear equations for the Gross-Neveu model with discrete or continuous chiral symmetry, respectively. The former model also has a dense-dilute symmetry that is due to the energy-reflection duality of the underlying quasi-exactly soluble spectral operators.

Quasigraded lie algebras, Kostant-Adler scheme, and integrable hierarchies

Using special “anisotropic” quasigraded Lie algebras, we obtain a number of new hierarchies of integrable nonlinear equations in partial derivatives admitting zero-curvature representations. Among them are an anisotropic deformation of the Heisenberg magnet hierarchy, a matrix and vector generalization of the Landau—Lifshitz hierarchies, new types of matrix and vector anisotropic chiral-field hierarchies, and other types of “anisotropic” hierarchies.

Deformations of loop algebras and integrable systems: hierarchies of integrable equations

Using special quasigraded Lie algebras, that could be viewed as deformations of loop algebras, we obtain new hierarchies of integrable nonlinear equations admitting zero-curvature representations. In particular, we obtain integrable hierarchies that generalize the Heisenberg magnet, Landau–Lifshitz, and anisotropic chiral field hierarchies. We also obtain a new type of so(3) anisotropic chiral field equation along with its higher rank generalization.

Hierarchies of integrable equations associated with hyperelliptic Lie algebras

Using a family of special quasigraded Lie algebras on hyperelliptic curves we construct new hierarchies of integrable nonlinear equations admitting zero-curvature representations. We show that in the case of the rational degeneration of the curve they coincide with Heisenberg magnet hierarchies.

SUPERLOOP EQUATIONS AND TWO-DIMENSIONAL SUPERGRAVITY

We propose a discrete model whose continuum limit reproduces the string susceptibility and the scaling dimensions of (2, 4m) minimal superconformal models coupled to 2D supergravity. The basic assumption in our presentation is a set of super-Virasoro constraints imposed on the partition function. We recover the Neveu-Schwarz and Ramond sectors of the theory, and we are also able to evaluate all planar loop correlation functions in the continuum limit. We find evidence to identify the integrable hierarchy of nonlinear equations describing the double scaling limit as a supersymmetric generalization of KP studied by Rabin.

THE $W^{(2)}_3$ CONFORMAL ALGEBRA AND THE BOUSSINESQ HIERARCHY

The classical [Formula: see text] algebra Polyakov is shown to be equivalent to the second Poisson structure of a new integrable hierarchy of nonlinear equations. The hierarchy is related to the Boussinesq hierarchy by interhcanging the roles of the space and time variables x and t in the Boussinesq equation. From this relation the Miura map, relating the new hierarchy to its modified version, can be derived systematically. It is found to be equivalent to the known free field representation of the [Formula: see text] algebra.