Theoretical and Mathematical Physics最新文献

We reduce the (q)-KP hierarchy and the (q)-mKP hierarchy. By using the link between the (q)-KP hierarchy and the (q)-mKP hierarchy, we obtain a relation between the (q)-KdV and the (q)-mKdV hierarchies, as well as a relation between the (q)-Boussinesq and (q)-mBoussinesq equations. The connection between them can also be established by a gauge transformation. We verify that when in the limit (qto 1), these relations correspond to the results for classical systems.

In recent studies on holographic QCD, the consideration of five-dimensional Einstein–dilaton–Maxwell models has played a crucial role. Typically, one Maxwell field is associated with the chemical potential, while additional Maxwell fields are used to describe the anisotropy of the model. A more general scenario involves up to four Maxwell fields. The second field represents spatial longitudinal–transverse anisotropy, while the third and fourth fields describe anisotropy induced by an external magnetic field. We consider an ansatz for the metric characterized by four functions at zero temperature and five functions at nonzero temperature. The Maxwell field related to the chemical potential is treated with the electric ansatz, as is customary, whereas the remaining three Maxwell fields are treated with a magnetic ansatz. We demonstrate that in the fully anisotropic diagonal case, only six out of the seven equations are independent. One of the matter equations (either the dilaton or the vector potential equation) follows from the Einstein equations and the remaining matter equation. This redundancy arises due to the Bianchi identity for the Einstein tensor and the specific form of the energy–momentum tensor in the model. A procedure for solving this system of six equations is provided. This method generalizes previously studied cases involving up to three Maxwell fields. In the solution with three magnetic fields case, our analysis shows that the dilaton equation is a consequence of the five Einstein equations and the equation for the vector potential.

We investigate a general class of second-order integro–ordinary-differential equations with arbitrary-power nonlinear terms, which can be used as a mathematical model for a variety of important physical areas in mathematics, mathematical physics, and applied sciences. The exact smooth and nonsmooth solutions of the aforementioned integro–differential equation in terms of the Gauss hypergeometric function are obtained here for the first time using the rapidly convergent approximation method. The prerequisites for the existence of such solutions are outlined in a theorem. Additionally, a few theorems are presented that contain the conditions under which our derived nonsmooth solution can be viewed as a weak solution. Using the aforementioned results, we obtain exact smooth and nonsmooth solutions of the following nonlinear integro-partial differential equations: the ((1+1))-dimensional integro–differential Ito equation, the ((3+1))-dimensional Yu–Toda–Sasa–Fukuyama equation, and the Calogero–Bogoyavlenskii–Schiff equation.

We obtain the long-time asymptotic behavior and (N)th-order bound state soliton solutions of a generalized derivative nonlinear Schrödinger (g-DNLS) equation via the Riemann–Hilbert method. First, in the process of direct scattering, the spectral analysis of the Lax pair is performed, from which a Riemann–Hilbert problem (RHP) is established for the g-DNLS equation. Next, in the process of inverse scattering, different from traditional solution finding schemes, we give some Laurent expansions of related functions and use them to obtain solutions of the RHP for the reflection coefficients under different conditions, such as a single pole and multiple poles, where we obtain new (N)th-order bound state soliton solutions. Based on the originally constructed RHP, we use the (overline{partial})-steepest descent method to explicitly find long-time asymptotic behavior of the solutions of the g-DNLS equation. With this method, we obtain an accuracy of the asymptotic behavior of the solution that is currently not obtainable by the direct method of partial differential equations.

We study the dependence of the beta-function on the running coupling constant in holographic models supported by the Einstein–dilaton–Maxwell action for light and heavy quarks. The dilaton defines the running coupling of the models. Its dependence on boundary conditions leads to the running coupling dependence on them. However, the behavior of the (beta)-function as a function of the running coupling does not depend significantly on the boundary condition. The corresponding (beta)-functions are negative and monotonically decreasing functions, and have jumps at first-order phase transitions for both light and heavy quarks. In addition, we compare our holographic results for the (beta)-function as a function of the running coupling with perturbative results obtained within (2)-loop calculations.

We reformulate the time-independent Schrödinger equation as a Maurer–Cartan equation on the superspace of eigensystems of the former equation. We then twist the differential such that its cohomology becomes the space of solutions with a fixed energy. A perturbation of the Hamiltonian corresponds to a deformation of the twisted differential, leading to a simple recursive relation for the eigenvalue and eigenfunction corrections.

We introduce the notion of Darboux transformations for the discrete KP hierarchy and its strict version, and present an explicit form of these transformations for the solutions of discrete KP and discrete strict KP hierarchies constructed in our previous work.

An extended (2)-dimensional Toda lattice equation is investigated by means of the Cauchy matrix approach. We introduce a direction parameter in the extension and represented the equation as a coupled system in a (3)-dimensional space. The equation can also be considered as a negative-order member in one direction of the discrete Kadomtsev–Petviashvili equation. By introducing the (tau)-function and an auxiliary direction, the equation can be bilinearized in a (4)-dimensional space with a single (tau)-function.

We explore the dynamics of the Klein–Gordon oscillator in the presence of a cosmic string in the Som–Raychaudhuri space–time. The exact solutions for the free and oscillator cases are obtained and discussed. These solutions reveal the effects of the cosmic string and space–time geometry on bosonic particles. To illustrate these results, several figures and tables are included.

We show that the Lie algebraic approach with the perturbation method can be used to study the eigenvalues of the Hellmann Hamiltonian. The key element is the Runge–Lenz vector, which appears in problems with radial symmetry. This symmetry implies that the proper lie algebra for these Hamiltonians is (so(4)), which is a sum of two (so(3)) Lie algebras and requires symmetry of the angular momentum vector (vec{L}) and the Runge–Lenz vector (vec{M}), and therefore their cross products as (vec{W}=vec{L}timesvec{M}). Here, Yukawa potential is considered as a perturbation term, which is added to the Coulomb Hamiltonian to produce the Hellmann Hamiltonian. Lie algebraically, the perturbation term adds a magnitude of precession rate (Omega) to all three operators (vec{L}), (vec{M}), and (vec{W}). Topologically, we show that the appearance of this precession has a significant effect on the spectrum and the corresponding Lie algebra of the Hellmann potential. By using Lie algebraic properties of the Runge–Lenz vector and using the Kolmogorov method, we obtain the energy spectrum of this Hamiltonian.