Theoretical and Mathematical Physics最新文献

We consider a periodic boundary value problem for the convective variant of the Cahn–Hilliard equation in the case where solutions depend on two spatial variables and time. For this boundary value problem, we study the existence and properties of traveling waves. These issues are reduced to the analysis of local bifurcations of an auxiliary boundary value problem. The possibility of the existence of traveling wave packets is demonstrated. We determine the stability of the invariant manifolds of the evolutionary boundary value problem under consideration, as well as the stability of the traveling waves forming them. Asymptotic formulas for these solutions are obtained.

The following results on differential-difference matrix Lax representations (MLRs) and on discrete Miura-type transformations (MTs) for differential-difference equations are obtained:

(bullet) A necessary and sufficient condition under which a given MLR can be simplified via gauge transformations.

(bullet) A technique for constructing MTs (including nonautonomous examples) via MLRs and gauge transformations.

(bullet) A criterion for checking whether a given MLR is gauge-inequivalent to a trivial MLR.

We consider the general ((1+1))-dimensional evolutionary differential-difference case when an MLR can depend on any shifts of dependent variables and can be nonautonomous. As applications and illustrations of the presented general theory, we construct several new two-component integrable equations (with new MLRs) connected by new MTs to integrable equations obtained by S. Konstantinou-Rizos, A. V. Mikhailov, and P. Xenitidis, and to those introduced by E. Mansfield, G. MaríBeffa, and J. P. Wang.

We present a nonlocal matrix coupled dispersionless system arising from a reverse-space-time symmetry reduction of a general matrix integrable system. A Lax pair is proposed and the Darboux–Bäcklund transformation is constructed on the solutions of the system. An (N)-fold transformation formula in terms of quasideterminants is derived; furthermore, the system reduces to the scalar reverse-space-time nonlocal coupled dispersionless system, a nonlocal sine-Gordon equation, and a nonlocal Maxwell–Bloch system. Explicit solutions are constructed and shown to exhibit dark soliton-like structures, a dark rogue wave, and a bright rogue wave with a four-armed star pattern, all illustrated graphically.

Collinear factorization and the leading-twist operator product expansion (OPE) in perturbative QCD express suitably inclusive observables in scale-separated kinematics as composites of perturbative short-distance coefficients with universal long-distance nonperturbative correlators such as parton distribution functions (PDFs), up to controlled power corrections. A persistent structural feature is the nonuniqueness of presentation: coefficients and correlators are not individually physical, but are defined only up to finite factorization-scheme redefinitions induced by collinear subtractions and renormalized-operator mixing. We formalize this redundancy categorically by introducing an interface algebra object encoding admissible finite collinear counterterms/mixing kernels and by organizing coefficient data and hadronic data as right/left modules over this algebra in a symmetric monoidal category encoding the chosen recomposition calculus. Our main result, the Core Representation Theorem, identifies the universal scheme-invariant carrier: the functor of balanced (scheme-invariant) pairings is represented by the relative tensor product (Cotimes_A f), which is terminal among all quotients of the naive composite (Cotimes f) that preserve scheme-invariant semantics. Finally, we show how standard physics inputs (symmetry constraints, locality/OPE, and a stated accuracy truncation) canonically induce the interface algebra and module structures, and we prove a minimal closure principle for completing a generating set of long-distance operators/correlators to an (A)-stable sector.

We study the behavior of solutions to a nonlinear second-order differential equation with delay in the case where the feedback is of an impulsive type. The cases where the solutions of the linear part oscillate and do not oscillate are considered separately. In each case, the problem is reduced to studying the dynamics of a discrete mapping that is constructed explicitly. An analysis of the dynamical properties of the constructed mapping is performed, and the results are transferred to the original differential equation. It is shown that, depending on the parameters, the solution can monotonically tend to zero (starting from a certain moment of time) or tend to periodic or chaotic behavior.

We consider a hierarchy of integrable nonlinear equations depending on functions (p) and (u). For ({uequiv0}), this hierarchy transforms into the hierarchy of the derivative nonlinear Schrödinger equation. For ({unotequiv0}), the constructed hierarchy equations describe the propagation of interacting nonlinear waves (p) and (u). Some examples of solutions to the new integrable nonlinear equations are considered.

The paper considers an initial-boundary value problem for a diffusion equation with modular nonlinearity. This problem was identified as a simplified formulation of a tumor growth model. In the present work, it was reduced to a single equation in the one-dimensional case. A distinctive feature of the problem is that the front is formed not between two saddle equilibrium positions, but between a saddle and a stable focus, or between a saddle and a stable node. The paper provides a justification for the existence of a solution to the model equation. The result of the study allows one to roughly estimate the velocity of the tumor propagation front, its dependence on the oxygen concentration, and the cell density in the proliferative layer. The methods employed in the study can be applied to a system of equations in the three-dimensional case for the purpose of justifying the general model.

We prove that the full symmetric Toda system is integrable in the sense of the Lie–Bianchi criterion. Specifically, we show that there exists a solvable Lie algebra of vector fields of dimension (N=dim M) on the phase space (M) of this system such that the system is invariant under the action of these fields. The proof is based on the symmetries of the full symmetric system constructed earlier, and the emergence of the stochastic Lie algebra structure in their description.

We investigate the onset of deterministic spatiotemporal chaos in a weakly perturbed coupled nonlinear Schrödinger (CNLS) system that admits an explicit homoclinic breather in the Hamiltonian limit. Our analysis identifies a geometric mechanism organized by an internal (1{:}1) resonance near the symmetric equilibrium. A resonant amplitude-phase reduction reveals a two-scale spectral splitting that produces a one-dimensional unstable manifold persisting smoothly under perturbation. This codimension-one saddle structure allows homoclinic splitting to be characterized by an explicit scalar Melnikov function, whose zeros define a persistence surface in parameter space predicting transversal invariant-manifold intersections and chaotic transport. High-resolution numerical simulations, including maximal Lyapunov exponents, correlation-dimension estimates, and Poincaré sections, confirm that regions of enhanced Lyapunov instability align with the Melnikov persistence surface. The results provide a unified geometric interpretation of chaos in the perturbed CNLS system, showing that irregular dynamics is organized by resonant saddle geometry and global homoclinic splitting rather than by purely local linear instability.

A family of parametric Korteweg–de Vries hierarchies is introduced. For fixed values of the family parameters, the classical Korteweg–de Vries hierarchy is obtained, which has been described in terms of recursions in well-known works (Lenard, Gelfand–Dickey). A polynomial isomorphism is constructed between the studied family of parametric Korteweg–de Vries hierarchies and an explicitly defined sequence of infinite-dimensional polynomial dynamical systems.