Studies in Applied Mathematics最新文献

We investigate the Cauchy problem of an integrable focusing fifth-order modified Korteweg–de Vries (KdV) equation, which contains the fifth-order dispersion and relevant higher order nonlinear terms. The long-time asymptotics of solution is established in the case of initial conditions that lie in some low regularity weighted Sobolev spaces and allow for the presence of discrete spectrum. Our method is based on a $�\overline{\partial }$ generalization of the nonlinear steepest descent method of Deift and Zhou. We show that the solution decomposes in the long time into three main regions: (i) an expanding oscillatory region where solitons and breathers travel with positive velocities, the leading order term has the form of a multisoliton/breather and soliton/breather–radiation interactions; (ii) a Painlevé region, which does not have traveling solitons and breathers, the asymptotics can be characterized with the solution of a fourth-order Painlevé II equation; (iii) a region of breathers traveling with negative velocities. Employing a global approximation via PDE techniques, the asymptotic behavior of solution is extended to lower regularity spaces with weights.

We consider the self-dual Yang–Mills equation and its reduction, the Manakov–Zakharov system. We discuss three- and four-dimensional generalizations of the chiral field equations, and explain methods for constructing their exact solutions.

Wave propagation in the form of fronts or kinks, a common occurrence in a wide range of physical phenomena, is studied in the context of models defined by their Hamiltonian structure. Motivated, for dispersive wave evolution equations such as a strongly nonlinear model of two-layer internal waves in the Boussinesq limit, by the symmetric properties of a class of front-propagating solutions, known as conjugate states or solibores, a generalized formulation based purely on the dispersionless reduction of a system is introduced, and a class of undercompressive shock solutions, here referred to as “Hamiltonian shocks,” is defined. This analysis determines whether a Hamiltonian shock, representing locally a kink for the parent dispersive equations, will interact with a sufficiently smooth background wave without inducing loss of regularity, which would take the form of a classical dispersive shock for the parent equations. This property is also related to an infinitude of conservation laws, drawing a parallel to the case of completely integrable systems.

When extending the complex Ginzburg–Landau equation (CGLE) to more than one spatial dimension, there is an underlying question of whether one is capturing all the interesting physics inherent in these higher dimensions. Although spatial anisotropy is far less studied than its isotropic counterpart, anisotropy is fundamental in applications to superconductors, plasma physics, and geology, to name just a few examples. We first formulate the CGLE on anisotropic, time-varying media, with this time variation permitting a degree of control of the anisotropy over time, focusing on how time-varying anisotropy influences diffusion and dispersion within both bounded and unbounded space domains. From here, we construct a variety of exact dissipative nonlinear wave solutions, including analogs of wavetrains, solitons, breathers, and rogue waves, before outlining the construction of more general solutions via a dissipative, nonautonomous generalization of the variational method. We finally consider the problem of modulational instability within anisotropic, time-varying media, obtaining generalizations to the Benjamin–Feir instability mechanism. We apply this framework to study the emergence and control of anisotropic spatiotemporal chaos in rectangular and curved domains. Our theoretical framework and specific solutions all point to time-varying anisotropy being a potentially valuable feature for the manipulation and control of waves in anisotropic media.

In this study, the dynamics of a reaction–diffusion equation on a bounded interval at the first dynamic transition point is investigated. The main difference of this study from the existing ones in the dynamic transition literature is that in this work, no specific form of the nonlinear operator is assumed except that it is an analytic function of $�u$ and $�u_x$. The linear operator is also assumed to be a general operator with sinusoidal eigenvectors. Moreover, we assume that there is a first transition as a real simple eigenvalue changes sign. With this general framework, we make a rigorous analysis of the dependence of the first transition dynamics on the coefficients of the Taylor expansion of the nonlinear operator. The main tool we use in this study is the center manifold reduction combined with the classification of the dynamic transition theory. This study is a generalization of a recent work where the nonlinear operator contains only low-order (quadratic and cubic) nonlinearities. The current generalization requires certain technical difficulties such as the validity of the genericity conditions on the Taylor coefficients of the nonlinear operator and a bootstrapping argument using these genericity conditions. The results of this work can be generalized in various directions, which are discussed in the conclusions section.

The methodology based on the so-called global relation, introduced by the first author, has recently led to the derivation of a novel nonlinear integral-differential equation characterizing the classical problem of the Saffman–Taylor fingers with nonzero surface tension. In the particular case of zero surface tension, this equation is satisfied by the explicit solution obtained by Saffman and Taylor. Here, first, for the case of zero surface tension, we present a new nonlinear integrodifferential equation characterizing the Saffman–Taylor fingers. Then, by using the explicit Saffman–Taylor solution valid for the particular case of zero surface tension, we show that the above equations give rise to sets of remarkable integral trigonometric identities.

The study of $��-�1�$ orthogonal polynomials viewed as $��q�\to �-�1�$ limits of the $�q$-orthogonal polynomials is pursued. This paper presents the continuous polynomials part of the $��-�1�$ analog of the $�q$-Askey scheme. A compendium of the properties of all the continuous $��-�1�$ hypergeometric polynomials and their connections is provided.

This paper investigates the stability problem and large time behavior of solutions to the three-dimensional magnetohydrodynamic equations with horizontal velocity dissipation and magnetic diffusion only in the $�x_2$ direction. By applying the structure of the system, time-weighted methods, and the method of bootstrapping argument, we prove that any perturbation near the background magnetic field (1, 0, 0) is globally stable in the Sobolev space $��H^3��\left(�R^3�\right)��$. Furthermore, explicit decay rates in $��H^2��\left(�R^3�\right)��$ are obtained. Motivated by the stability of the three-dimensional Navier–Stokes equations with horizontal dissipation, this paper aims to understand the stability of perturbations near a magnetic background field and reveal the mechanism of how the magnetic field generates enhanced dissipation and helps stabilize the fluid.