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We analyze several types of soliton solutions to a family of Tzitzeica equations. To this end we use two methods for deriving the soliton solutions: the dressing method and Hirota method. The dressing method allows us to derive two types of soliton solutions. The first type corresponds to a set of 6 symmetrically situated discrete eigenvalues of the Lax operator $L$; to each soliton of the second type one relates a set of 12 discrete eigenvalues of $L$. We also outline how one can construct general $N$ soliton solution containing $N_1$ solitons of first type and $N_2$ solitons of second type, $N=N_1+N_2$. The possible singularities of the solitons and the effects of change of variables that relate the different members of Tzitzeica family equations are briefly discussed. All equations allow quasi-regular as well as singular soliton solutions.
The Sturm spirals which can be introduced as those plane curves whose curvature radius is equal to the distance from the origin are embedded in to one parameter family of curves. In this paper, we consider the spacelike and timelike Sturmian spirals in Lorentz-Minkowski plane.
We discuss the precanonical quantization of fields which is based on the De Donder-Weyl (DW) Hamiltonian formulation which treats the space and time variables on an equal footing. Classical field equations in DW Hamiltonian form are derived as the equations on the expectation values of precanonical quantum operators. This field-theoretic generalization of the Ehrenfest theorem demonstrates the consistency of three aspects of precanonical field quantization: (i) the precanonical representation of operators in terms of the Clifford (Dirac) algebra valued partial differential operators, (ii) the Dirac-like precanonical generalization of the Schrödinger equation without the distinguished time dimension, and (iii) the definition of the scalar product in order to calculate expectation values of operators using the precanonical wave functions.
We extend well-known results in group theory to gyrogroups, especially the isomorphism theorems. We prove that an arbitrary gyrogroup $G$ induces the gyrogroup structure on the symmetric group of $G$ so that Cayley's Theorem is obtained. Introducing the notion of L-subgyrogroups, we show that an L-subgyrogroup partitions $G$ into left cosets. Consequently, if $H$ is an L-subgyrogroup of a finite gyrogroup $G$, then the order of $H$ divides the order of $G$.
We consider complete integrability of the Hamiltonian of the geodesic flow of two particular solutions, the Kerr-Newman and the FLRW metrics of the Einstein equations in the sense of Liouville. We construct recursion operators using first integrals, and then obtain constants of motion of the geodesic flows by using the recursion operators.