Theoretical and Mathematical Physics最新文献

This article investigates the existence, uniqueness, and regularity of solutions to the Muskat equation describing the motion of two immiscible fluids with constant densities in an incompressible porous medium, where the velocity is governed by Darcy’s law. The initial data and its first and second derivatives are assumed to belong to different Lebesgue spaces.

We obtain necessary and sufficient conditions for Darboux integrability of hyperbolic partial differential equations that admit non-autonomic first-order integrals along one of the characteristics. Based on these conditions, we find a family of Darboux integrable equations, which is probably new.

When quantizing field-theoretical models with gauge symmetries, quantum anomalies are often encountered. It is commonly believed that the cause of these anomalies lies in the infinite numbers of degrees of freedom, which requires the field system to be completed within a suitable regularization and renormalization scheme. We present an example of a finite-dimensional Hamiltonian system with first-class constraints, whose quantization leads to unavoidable quantum anomalies. These anomalies arise due to the nontrivial topology of the reduced phase space of the system.

The CK direct method is employed to investigate exact solutions of the ((2+1))-dimensional extended Bogoyavlenskii–Kadomtsev–Petviashvili equation, which usually describes the propagation of nonlinear waves in various fields, such as fluid dynamics and plasma physics. It is extremely challenging to derive exact solutions for the eBKP equation. We have found that at present there is no research in the scientific literature on the application of the CK method to the eBKP equation due to tedious and complex calculations and inherent difficulty in determining explicit expressions for (beta) and (z). To address these limitations, we adopt a separation-of-equations approach to find concrete expressions for (beta) and (z). Through an extensive series of complex calculations, we successfully obtain new similarity reductions and new exact solutions for the eBKP equation, including Painlevé-type reductions, Weierstrass elliptic function solutions, and rational solutions that have not been reported in prior studies. Solutions of the eBKP equation can successfully degenerate into those of the BKP equation. From a physical perspective, through the analysis of the new solutions to the BKP equation, we find that as (t) gradually increases, wave BKP solutions develop progressive instability and exhibit a tendency toward collapse. We find that introducing extended dispersion terms in the BKP equation enhances the amplitude of wave solutions and induces a tilting effect on wave propagation along the crest line.

We construct and study transport solutions of the biquaternion wave equation, which is a biquaternion generalization of the Dirac and Maxwell equations. These equations describe the electromagnetic fields of electromagnetic and electro-gravimagnetic wave sources moving in a fixed direction with a constant speed that is less than the speed of wave propagation in an electromagnetic medium (speed of light). We construct fundamental and generalized transport solutions describing fields of moving objects at subluminal speeds. Using the Fourier transform of distributions, we construct a biquaternion Green function (bifunction) in a moving coordinate system. This function describes the field generated by a moving point source on the (z)-axis. We find the energy density and the Poynting vector of this field. The influence of the speed of motion on the field characteristics is studied.

We consider a one-dimensional bosons model with point-wise interaction. We calculate the overlaps of Bethe vectors corresponding to different coupling constants. We obtain a sum formula for the overlap of off-shell Bethe vectors. A new formula for the overlap of eigenvectors of different Hamiltonians is also obtained.

This study conducts an approximate symmetry analysis of the singularly Kuramoto–Sivashinsky perturbed version of the modified Gardner equation, renowned for modeling the super-nonlinear propagation of ion–acoustic waves and quantum electron–positron–ion magnetoplasmas, and the Camassa–Holm equation, which serves as a critical model for nonlinear wave dynamics in cylindrical axially symmetric hyperelastic rods. The analysis employs perturbative expansion of infinitesimal generators resulting in the derivation of the approximate infinitesimal generators, which are further systematically utilized in Olver’s optimal theory for constructing an optimal system of Lie subalgebras. Furthermore, elements of the derived system are employed to reduce the governing problems to ordinary differential equations, facilitating the determination of exact invariant solutions by appropriate solution methods. Diverse wave phenomena arise from the intricate interplay of dispersion, nonlinearity, and perturbation effects. Accordingly, graphical depictions of the solutions highlight key nonlinear wave phenomena.

We explore isotropic charged compact stars within the framework of (f(R,mathcal Tmkern1.5mu)) gravity, employing a novel approach grounded in conformal Killing vectors to model strange stars and analyze their physical viability and stability. Utilizing the simplified MIT bag equation of state (EOS) for quark matter, we derive exact solutions to the Einstein field equations for observed masses of strange star candidates, with LMC X-(4) as a representative case. The parameter (varpi) (ranging from (-1.6) to (1.6)) governs modifications in the (f(R,mathcal Tmkern1.5mu)) gravity formalism, enabling systematic investigation of key properties. Our results confirm singularity-free metric potentials, monotonically decreasing effective energy density ((rho^{mathrm{ef}})) and pressure ((p^{mathrm{ef}})), and adherence to energy conditions across all (varpi) values. The mass–radius relationship, analyzed for a fixed bag constant (mathcal B=83,mathrm{MeV}/mathrm{fm}^3), reveals that maximum mass points increase with (varpi). A prescribed density profile (free of central singularities) combined with the MIT bag EOS yields exact solutions to the modified Tolman–Oppenheimer–Volkoff equations, circumventing numerical complexities. Stability analysis demonstrates equilibrium via force balance, with an emergent force (F_{mathrm m}) in (f(R,mathcal Tmkern1.5mu)) gravity: repulsive (outward) for (varpi<0) and attractive (inward) for (varpi>0). Stability is further validated by subluminal sound speeds ((v_{mathrm s}^2in[0,1])) and adiabatic indices ((Gamma>4/3)). High surface/central densities and redshifts ((sim 0.23)–(0.36)) align with strange quark star characteristics, while all (2M/mathcal R) values remain below the Buchdahl limit. The results establish a robust, stable stellar model for strange stars, leveraging conformal symmetries and (f(R,mathcal Tmkern1.5mu)) gravity, and provide a foundation for future studies on alternative density profiles in modified gravity.

We consider the linear Stark effect in the five-dimensional (SU(2)) Yang–Coulomb problem. We show that a constant homogeneous electric field completely removes the degeneracy of energy levels in terms of both the orbital quantum number and the isospin of the system. We obtain an explicit expression for the additional integral of motion for the (SU(2)) Yang–Coulomb problem in the presence of a constant homogeneous electric field.

We focus on investigating the negative-order modified Korteweg–de Vries equation with a special source. Based on the inverse scattering transform method, we derive the temporal dependency of scattering data for the Dirac operator with moving eigenvalues. We derive the multisoliton solution to considered problem using matrix triplet method and illustrate the wave behavior of solutions in both presence and absence of a source.