Nonlocal Reductions of a Generalized Heisenberg Ferromagnet Equation

We study nonlocal reductions of coupled equations in $1+1$ dimensions of the Heisenberg ferromagnet type. The equations under consideration are completely integrable and have a Lax pair related to a linear bundle in pole gauge. We describe the integrable hierarchy of nonlinear equations related to our system in terms of generating operators. We present some special solutions associated with four distinct discrete eigenvalues of scattering operator. Using the Lax pair diagonalization method, we derive recurrence formulas for the conserved densities and find the first two simplest conserved densities.Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:1711.0635

The Oscillatory Universe, phantom crossing and the Hubble tension

We investigate the validity of cosmological models with an oscillating scale factor in relation to late-time cosmological observations. We show that these models not only meet the required late time observational constraints but can also alleviate the Hubble tension. As a generic feature of the model, the Hubble parameter increases near the current epoch due to its cyclical nature exhibiting the phantom nature allowing to address the said issue related to late time acceleration.Comment: 10 pages, 7 figure

Integrable Zhaidary equations: reductions and gauge equivalence

The present work addresses the study and characterization of the integrability of some generalized spin systems (ISS) in 1+1 dimensions. Lax representations for these ISS are successfully obtained. The gauge equivalent counterparts of these integrable ISS are presented. Finally, we consider Zhanbota transcendents and some integrable Zhanbota equations. In particular, the gauge equivalence between some Zhanbota equations and the six Painleve equations is established.Comment: 17 page

Darboux transformation and solution of the modified Korteweg–de Vries equation

Darboux transformation and a comprehensive approach to construct exact solutions of the nonlinear differential equation are counted. It is applied to construct the explicit solutions of the (2+1)-dimensional modified Korteweg-de Vries (KdV) equation. In this work we derive one flod Darboux transformation of the modified KdV equation. Using the obtained Darboux transformation, the one-soliton solution is built from the «seed» solution. Further, we will construct other explicit solutions for this equation

Solvable K-essence Cosmologies and Modified Chaplygin Gas Unified Models of Dark Energy and Dark Matter

This paper is devoted to the investigation of modified Chaplygin gas model in the context of solvable k-essence cosmologies. For this purpose, we construct equations of state parameter of this model for some particular values of the parameter $n$. The graphical behavior of these equations are also discussed by using power law form of potential. The relationship between k-essence and modified Chaplygin gas model shows viable results in the dark energy scenario. We conclude that the universe behaves as a cosmological constant, quintessence phase or phantom phase depending upon $n$.Comment: 14 pages, 6 figure

Integrable Kuralay equations: geometry, solutions and generalizations

In this paper, we study the Kuralay equations, namely, the Kuralay-I equation (K-IE) and the Kuralay-II equation (K-IIE). The integrable motion of space curves induced by these equations is investigated. The gauge equivalence between these two equations is established. With the help of the Hirota bilinear method, the simplest soliton solutions are also presented. The nonlocal and dispersionless versions of the K-IE and K-IIE are considered.Comment: 19 page