We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator $L$. The Lax representation is $\mathbb{Z}_2 \times \mathbb{Z}_2$ reduced and can be naturally associated with the symmetric space ${\rm SU(3)/S(U(1)\times U(2))}$. The simplest nontrivial equation in the hierarchy is a generalization of Heisenberg ferromagnetic model. We construct the $N$-soliton solutions for an arbitrary member of the hierarchy by using the Zakharov-Shabat dressing method with an appropriately chosen dressing factor. Two types of soliton solutions: quadruplet and doublet solitons are found. The one-soliton solutions of NLEEs with even and odd dispersion laws have different properties. In particular, the one-soliton solutions for NLEEs with even dispersion laws are not traveling waves while their velocities and amplitudes are time dependent. Calculating the asymptotics of the $N$-soliton solutions for $t\to\pm\infty$ we analyze the interactions of quadruplet solitons.

This is a survey on geometric quantum mechanics and some of its implications on general issues in quantum theory.

We present the results of studying free massive bosonic fields under the formalism of topological quantization. We identify certain harmonic maps as a geometric configuration equivalent to the classical system. We take as a concrete example the case of free massive bosonic fields in two dimensions, and construct the fiber bundle representing them and find its topological spectra. We found that the appearance of singularities in Euler's two form caused its integral to be dependent on the order in which the variables are integrated. We discuss the implications of this orientation dependency and formulate a well-defined expression for the Euler invariant emerging from it.

The idea of wave mechanics leads naturally to assume the well-known relation $E=\hbar \omega $ in the specific form $H=\hbar W $, where $H$ is the classical Hamiltonian of a particle and $W$ is the dispersion relation of the sought-for wave equation. We derive the expression of $H$ in a curved space-time with an electromagnetic field. Then we derive the Dirac equation from factorizing the polynomial dispersion equation corresponding with $H$. Conversely, summarizing a recent work, we implement the geometrical optics approximation into the canonical form of the Dirac Lagrangian. The Euler-Lagrange equations are thus obtained for the amplitude and the phase of the wave function. From them, one is led to define a four-velocity field which obeys exactly the classical equation of motion. The complete de Broglie relations are then derived as exact equations.

The determination of the interface shape between two fluids during the process of viscous fingering (Saffman--Taylor instability) in a Hele-Shaw cell is addressed here. The parametric equations describing this interface are obtained in an explicit analytic form. With these results, a number of interface shapes are presented, including shapes with opposite sides in contact that are considered as the onset of droplet pinch-off.

The Dyson model of a spinning ellipsoidal gas cloud expanding into a vacuum has been found to be Liouville integrable under certain additional assumptions, such as the absence of either vorticity or of angular momentum. Here we present a new formulation in the form of a 4x4 matrix equation, which generalizes a similar result obtained in rotationless cases. This implies to consider an extended affine space of seven dimensions, in which the seven coordinates of the point-mass representative of the cloud obey differential equations of the same general form as those defining the elliptic functions. This leads very directly to the linearization of the system in the so-called degenerate cases. We obtain also explicit expressions for the symmetry generators, a prerequisite in the task of constructing a Backlund transformation.

It is shown how one can apply the classification of the holonomy algebras of Lorentzian manifolds to solve some problems. In particular, a new proof to the classification of Lorentzian manifolds with recurrent curvature tensor is given and the classification of two-symmetric Lorentzian manifolds is explained. Then the conformally recurrent Lorentzian manifolds are classified and the recurrent symmetric bilinear forms on these manifolds are described.

Discussed are certain strange properties of the Galilei group, connected first of all with the property of mechanical energy-momentum covector to be an affine object, rather than the linear one. Its affine transformation rule is interesting in itself and dependent on the particle mass. On the quantum level this means obviously that we deal with the projective unitary representation of the group rather than with the usual representation. The status of mass is completely different than in relativistic theory, where it is a continuous eigenvalue of the Casimir invariant. In Galilei framework it is a parameter characterizing the factor of the projective representation, in the sense of V. Bargmann. This “pathology” from the relativistic point of view is nevertheless very interesting and it underlies the Weyl-Wigner-Moyal-Ville approach to quantum mechanics.

A Lorentzian surface in a four-dimensional manifold of neutral signature is called super-extremal if its reflector lift is horizontal. We give an elementary proof of a rigidity theorem for super-extremal surfaces in the space of constant curvature and neutral signature. As corollary, a characterization of the immersion of the Veronese type is given.

We show that the MIC-Kepler probrem is simply solved via the phase-space formulation of non-relativistic quantum mechanics. The MIC-Kepler problem is the Hamiltonian system behind the hydrogen atom subjected to the influence of the Dirac’s magnetic monopole field and the square inverse centrifugal potential force besides the Coulomb’s potential force. We get the energy spectrum of the bound states explicitly and construct the Green’s functions for $E \lt 0$ by means of the Moyal product, which is one of the “star” products denoted by “$\ast$”.

Deriving curves based on the other curves like involute-evolute curves or Bertrant curves is an old subject in geometry. In this paper, we have defined a ruled surface based on the involute curve of a given curve which is called the involutive B-scroll. We introduced the positions of the involutive B-scroll and the B-scroll relative to each other.

We prove that loop vortices are created by a point-like magnetic dipole in an infinite superconductor space. The geometry of the vortex system is obtained through analytic solutions of the linearized Ginzburg-Landau equation described in terms of Heun functions, generalizing the traditional hypergeometric behavior of such magnetic singularity.

We consider a Riemannian manifold $M$ (dim$M\geq 3$), which is flat or has negative sectional curvature. We suppose that there is a closed and connected subgroup $G$ of Iso$(M)$ such that dim$({M}/{G})=2$. Then we study some topological properties of $M$ and the orbits of the action of $G$ on $M$.

The modern geometrical approach to nonlinear PDEs is the outcome of a nontrivial synthesis of differential calculus over commutative algebras and cohomological algebra in the context of infinite jet spaces. In this paper we propose a very natural generalization of the notion of a jet space, which allows to treat the space of initial data of a nonlinear PDE on the same footing as the space of its solutions.

The Lie group method for construction of group-invariant solutions of partial differential equations is presented. The method is applied to a system of two coupled nonlinear Schrödinger equations. The so called reduced system of equations for translationally invariant solutions is obtained. Group-invariant solutions for the degenerate case of two decoupled Schrödinger equations are found.

Let $G$ be a semi-simple Lie group and $\pi$ some representation of $G$ belonging to the discrete series. We give interpretations of the constant $\pi (g)$, for $g \in Z(G)$, in terms of geometric concepts associated with the flag manifold $M$ of $G$. In particular, when $G$ is compact this constant is related to the action integral around closed curves in $M$. As a consequence, we obtain a lower bound for de cardinal of the fundamental group of Ham$(M)$, the Hamiltonian group of $M$. We also interpret geometrically the values of the infinitesimal character of $\pi$ in terms of quantization operators.

We give geometric picture for the Recursion Operators related to the Caudrey-Beals-Coifman linear problem and the Nonlinear Evolution Equations associated to it in the presence of $\mathbb{Z}_p$ reductions of Mikhailov type.