Characters of Explicit Solutions for a Semidiscrete Integrable Coupled Equation

A semidiscrete integrable coupled system is obtained by embedding a free function into the discrete zero-curvature equation. Then, explicit solutions of the first two nontrivial equations in this system are derived directly by the Darboux transformation method. Finally, in order to compare the solutions before and after coupling intuitively, their structure figures are presented and analyzed.

On Riemannian Geometries of Visual Space

This is a study of the phenomenal geometries constructed in the Riemannian geometry framework from simulated iso-disparity conics in the horizontal visual plane of the binocular system with the asymmetric eyes (AEs). The iso-disparity conic's arcs in the Cyclopean direction are the frontal visual geodesics. For the eyes' resting vergence posture, which depends on parameters of the AE, the iso-disparity conics are frontal straight lines in physical space. For all other fixations, the iso-disparity conics consist of families of the ellipses or hyperbolas depending on both the AE's parameters and the bifoveal fixation. An assumption underlying the relevant architecture of the human visual system is combined with results from simulated iso-disparity straight lines, giving the relative depth as a function of the distance. This establishes the metric tensor in binocular space of fixations for the eyes' resting vergence posture. The resulting geodesics in the gaze direction, give the distance to the horizon and zero curvature. For all other fixations, only the sign of the curvature can be inferred from the global behavior of the simulated iso-disparity conics.

Two Isospectral-Nonisospectral Super-Integrable Hierarchies and Related Invariant Solutions

In this article, we adopt two kinds of loop algebras corresponding to the Lie algebra B(0,1) to introduce two line spectral problems with different numbers of even and odd superfunctions. Through generalizing the time evolution λt to a polynomial of λ, two isospectral-nonisospectral super integrable hierarchies are derived in terms of Tu scheme and zero-curvature equation. Among them, the first super integrable hierarchy is further reduced to generalized Fokker–Plank equation and special bond pricing equation, as well as an explicit super integrable system under the choice of specific parameters. More specifically, a super integrable coupled equation is derived and the corresponding integrable properties are discussed, including the Lie point symmetries and one-parameter Lie symmetry groups as well as group-invariant solutions associated with characteristic equation.

A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

Algebro-geometric integration of a modified shallow wave hierarchy

By introducing two sets of Lenard recursion relations, we derive a hierarchy of modified shallow wave equations associated with a 3 × 3 matrix spectral problem with three potentials from the zero-curvature equation. The Baker–Akhiezer function and two meromorphic functions are defined on the trigonal curve which is introduced by utilizing the characteristic polynomial of the Lax matrix. Analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions at two infinite points, we arrive at the explicit algebro-geometric solutions for the entire hierarchy in terms of the Riemann theta function by showing the explicit forms of the normalized Abelian differentials of the third kind.

A nonisospectral integrable model of AKNS hierarchy and KN hierarchy, as well as its extended system

In this paper, we first introduce a nonisospectral problem associate with a loop algebra. Based on the nonisospectral problem, we deduce a nonisospectral integrable hierarchy by solving a nonisospectral zero curvature equation. It follows that the standard AKNS hierarchy and KN hierarchy are obtained by reducing the resulting nonisospectral hierarchy. Then, the Hamiltonian system of the resulting nonisospectral hierarchy is investigated based on the trace identity. Additionally, an extended integrable system of the resulting nonisospectral hierarchy is worked out based on an expanded higher-dimensional Loop algebra.

DIRECT AND INVERSE PROBLEMS FOR THERMAL GROOVING BY SURFACE DIFFUSION WITH TIME DEPENDENT MULLINS COEFFICIENT

We consider the Mullins’ equation of a single surface grooving when the surface diffusion is not considered as very slow. This problem can be formed by a surface grooving of profiles in a finite space region. The finiteness of the space region allows to apply the Fourier series analysis for one groove and also to consider the Mullins coefficient as well as slope of the groove root to be time-dependent. We also solve the inverse problem of finding time-dependent Mullins coefficient from total mass measurement. For both of these problems, the grooving side boundary conditions are identical to those of Mullins, and the opposite boundary is accompanied by a zero position and zero curvature which both together arrive at self adjoint boundary conditions.

Theory of the Stationary Self-Consistent Universe

Since the creation of GR and subsequent works in cosmology, the question of the curvature of space in the Universe is considered one of the most important and debated to this day. This is evident, because the curvature of space depends whether the Universe expands, contracts or is static. These discussions allowed the author to propose a paradoxical idea: simultaneous existence in the Universe of three interconnected space-times (positive, negative and zero curvature) and on this basis, to develop a theory in which each space-time plays its own role and develops in a strict accordance with its sign of curvature. The three space-time model of the structure of the Universe, proposed by the author, allows to solve many fundamental problems of modern cosmology and theoretical physics and creates the basis for building a unified physical theory (including one that unites GR and quantum physics).

Classically Integrable Non-Linear Sigma Models and their Geometric Properties

General classes of non-linear sigma models originating from a specified action are developed and studied. Models can be grouped and considered within a single mathematical structure this way. The geometrical properties of these models and the theories they describe are developed in detail. The zero curvature representation of the equations of motion are found. Those representations which have a spectral parameter are of importance here. Some new models with Lax pairs which depend on a spectral parameter are found. Some particular classes of solutions are worked out and discussed.