Theoretical and Mathematical Physics最新文献

We consider the Cauchy problem for a nonclassical third-order partial differential equation with gradient nonlinearity (|nabla u(x,t)|^q). A solution of this problem is understood in a certain weak sense. Using Schauder estimates, we show that a local-in-time weak solution of the Cauchy problem has a certain smoothness with (q>N/(N-1)).

We consider the inverse variational problem for a nonautonomous dynamical system with one degree of freedom and prove several nonlocal existence theorems.

The process of high-frequency fluorescence of a two-level open quantum system with permanent electric dipole moment interacting with a pulse of nonresonant monochromatic electromagnetic (laser) field is modeled and studied by methods of nonequilibrium quantum statistical mechanics. The system in question is represented by a one-electron two-level asymmetric polar semiconductor quantum dot characterized by the electric dipole moment operator with unequal diagonal matrix elements in its ground and excited quantum states, and interacting with dissipative environment. The dot is permanently excited by incoherent pumping and is driven by an amplitude modulated pulse, whose monochromatic carrier frequency is much lower than the optical transition frequency of the quantum dot. We derive an analytical expression for the time-dependent fluorescence power spectrum as a function of the amplitude, duration, initial phase, and carrier frequency of the rectangular monochromatic driving pulse, as well as of the pumping intensity and observation time. We show that the pulse itself does not add any discernable amount of energy to the fluorescence intensity but rather facilitates redistribution of the energy incoherently pumped into the system over a plurality of spectral peaks that arise under the influence of the pulse instead of the initial stationary single-peaked fluorescence spectrum observed before the arrival of the pulse and after its departure.

We consider a system of three particles consisting of two identical fermions and one other particle on a one-dimensional lattice. The fermions interact via a nearest-neighbor potential of strength (mu_1inmathbb{R}), while the interaction between a fermion and one other particle is via an on-site potential with strength ({mu_2inmathbb{R}}). We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum (Kinmathbb{T}^1). Furthermore, we show that both the bound state (f_{mu_1mu_2}(K;,{cdot},{,},{cdot},)) and its corresponding eigenvalue (E_{mu_1mu_2}(K)) depend holomorphically on the quasimomentum.

Soliton equations with self-consistent sources (SESCSs) have extensive applications in physics. In this paper, we derive the Lakshmanan–Porsezian–Daniel equation with self-consistent sources (LPD-SCS). We construct (N)-fold Darboux transformations for SESCSs and explicitly obtain soliton solutions and breather solutions for LPD-SCS. Moreover, we construct the generalized Darboux transformations (GDT) for the LPD-SCS and obtain rogue wave solutions. The propagation of solutions for the LPD-SCS is influenced by the arbitrary function (C(t)) related to the time variable (t). We demonstrate such influence in this research. We also analyze the correlation between constant parameters and the propagation characteristics of solutions.

We present a family of matrix models such that their partition functions are tau functions of the universal character (UC) hierarchy. We find new matrix models associated with the product of two spheres with embedded graphs via a gluing matrix. We also generalize these studies to the multi-matrix model case, which corresponds to the multi-component UC hierarchy.

We solve a unique matrix spectral problem that encompasses eight distinct potentials and subsequently derive a corresponding soliton hierarchy using the zero curvature representation. Moreover, we establish a bi-Hamiltonian framework by applying the trace identity, thereby emphasizing the Liouville integrability of the derived soliton hierarchy. Two illustrative examples are provided, including generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations, to showcase the applicability and significance of the proposed methodology. Finally, we obtain some integrable reductions within the derived soliton hierarchy.

We discuss the problem of canonical quantization of electromagnetic field in the Schwarzschild spacetime. It is shown that a consistent procedure of canonical quantization of the field can be carried out without taking into account the internal region of the black hole. We prove that there exists a unitary gauge, which can be viewed as a combination of the Coulomb and Poincaré gauges and which is compatible with the field equations. We study solutions corresponding to stationary one-particle states of the electromagnetic field and obtain canonical commutation relations and the Hamiltonian of the quantized electromagnetic field.

We generalize the Burgers hierarchy to the case of an arbitrary positive fractional order. We introduce a nonlinear fractional differential operator generated by a fractional power of the recursion operator of the original hierarchy. We show that, as in the integer case, the fractional differential equations of the generalized hierarchy are linearized by the Cole–Hopf transformation. In particular, the fractional differential generalization of the Burgers equation is transferred by this transform into a fractional differential superdiffusion equation. We find recursion operators for these equations and construct higher symmetries, local and nonlocal, including fractional differential ones.

We consider discrete symmetries of the Kadomtsev–Petviashvili equation and compatibility of the corresponding differential–difference equations. A reduction of the general scheme to ((1+1))-dimensional case is presented.