Studies in Applied Mathematics最新文献

We study time-periodic solutions to stochastic discounted Hamilton–Jacobi equations driven by a cylindrical Wiener process in the sense of most probable paths. We first analyze the existence and regularity of solutions. Then, within a spectral Galerkin limit framework, we derive the Onsager–Machlup functional using a stopping-time localization argument and $�L^2$-type techniques. Finally, based on a minimization principle for this action functional, we characterize the system's most probable paths. By further combining weak KAM theory, we prove that there exist most probable paths with a time-periodic structure. This result extends the classical theory of time-periodic viscosity solutions for deterministic discounted Hamilton–Jacobi equations to the stochastic setting for the first time, and offers a new perspective on stochastic discounted Hamilton–Jacobi equations.

We systematically investigate breather solutions in the modified Korteweg–de Vries (mKdV) equation via numerical simulations. We show that the breather solutions are stable under a variety of perturbations, including amplitude changes, modifications of internal parameters, and perturbations to the nonlinearity of the equation. Our results are not only consistent with the known analytical studies on the stability of breathers in integrable systems, extending them further, but also provide numerical evidence that such breather-type structures can persist and remain stable over the simulated time scales in some nonintegrable settings with symmetric potentials.

The semiclassical limit of the derivative nonlinear Schrödinger equation with periodic initial conditions is studied analytically and numerically. The spectrum of the associated scattering problem for a certain class of initial conditions, referred to as periodic single-lobe potentials, is numerically computed, and it is shown that the spectrum becomes confined to the real and imaginary axes or the spectral parameter in the semiclassical limit. A formal Wentzel–Kramers–Brillouin expansion is computed for the scattering eigenfunctions, which allows one to obtain asymptotic expressions for the number, location, and size of the spectral bands and gaps. The results of these calculations suggest that, in the semiclassical limit, all excitations in the spectrum become effective solitons. Finally, the analytical predictions are compared with direct numerical simulations as well as with numerical calculations of the Lax spectrum, and the results are shown to be in excellent agreement.

The aim of this paper is to apply Hirota's bilinear method to the integrable discrete Manakov system in the focusing dispersion regime in order to construct and analyze soliton and breather solutions. After deriving the general bilinear form of the system, we show how to obtain fundamental solitons, as well as fundamental and composite breathers. We then obtain solutions exhibiting 2 solitons and 2 breathers and combinations of a soliton and a breather, and discuss all “two-body” interactions properties, with particular emphasis on explicit formulas, visualization, and long-time asymptotic behavior, thus rigorously confirming the highly nontrivial interaction properties of these coherent structures.

This paper is concerned with a diffusion interface model for phase separation of the anisotropic Cahn–Hilliard image inpainting with a concentration-dependent degenerate mobility in $��d�=�2�,�3�$ dimensions. We present global existence of weak solutions to the anisotropic non-degenerate Cahn–Hilliard image inpainting with a smooth double-well potential. We prove that as the value of the fidelity parameter $�{\lambda }_0$ outside of the inpainting region gets large, the existing region imposes a certain type of effective boundary conditions on the inpainting region, and we present a proof for the steady-state problem. Furthermore, we obtain the global existence and regularity of weak solutions to the anisotropic degenerate Cahn–Hilliard image inpainting system with a logarithmic potential.

This paper presents a unified and systematic approach to constructing the modified Kadomtsev–Petviashvili (mKP), BKP, and modified BKP (mBKP) hierarchies through the representation theory of polynomial Lie algebras. Based on the polynomial Lie algebras $�{\mathrm{gl}}_{\infty }^{�\left(�n�\right)�}$ and $�{\mathrm{so}}_{\infty }^{�\left(�n�\right)�}$, we naturally derive the complete structure of these integrable systems, including bilinear identities, tau functions, wave matrices, dressing operators, and Lax pairs, emerge naturally without external constraints. A key result is the automatic emergence of the BKP constraint $��L_{\text{BKP}}^{\ast }�=�-�\partial �L_{\text{BKP}}�{\partial }^{�-�1�}�$ from $�{\mathrm{so}}_{\infty }^{�\left(�n�\right)�}$ symmetry, contrasting with traditional approaches where such constraints are artificially imposed. The polynomial algebra framework provides a unifying principle where matrix structures arise intrinsically through fermionic Fock space decompositions, offering a powerful methodology for constructing and analyzing integrable hierarchies in higher dimensions.