Studies in Applied Mathematics最新文献

We investigate rogue wave formation and spectral downshifting in the higher-order nonlinear Schrödinger (HONLS) equation and its dissipative extensions: the nonlinear mean-flow damping model (NLD-HONLS) and the viscous damping model (V-HONLS). Using Floquet spectral analysis, we characterize (i) the organization of the underlying dynamical background and (ii) the structure of the emergent rogue waves, distinguishing soliton-like rogue waves (SRWs)—which are sharply localized and spectrally coherent—from more diffuse rogue-wave events that lack coherent localization. In the conservative HONLS, SRWs arise only for sufficiently steep initial data, with the evolution intermittently switching between SRW formation and disordered multi-mode dynamics, while moderately steep initial data produce only broader, less coherent rogue waves.

Nonlinear mean-flow damping in the NLD-HONLS model suppresses disorder and promotes a sustained, well-organized Floquet spectrum that supports persistent soliton-like states from which SRWs emerge. In contrast, viscous damping in the V-HONLS model generates a disordered Floquet spectral evolution, broader rogue wave events, and enhanced phase variability. Furthermore, the NLD-HONLS exhibits a close link between rogue wave events and the time of permanent spectral downshift, whereas these phenomena appear decoupled in the V-HONLS model.

These results clarify how dissipation type and wave steepness interact to regulate coherence, rogue wave formation, and spectral downshifting in near-integrable wave systems. Moreover, the persistent coherent states identified in the NLD-HONLS align with the focused, highly organized wave groups observed immediately prior to wave breaking in laboratory experiments, providing a Floquet spectral framework that is conceptually relevant to pre-breaking wave dynamics.

This paper presents a detailed examination of the propagation of shock waves in a medium that exhibits mixed-type weak nonlinearity. The governing equations are the one-dimensional Euler equations, supplemented by the van der Waals equation of state. We consider an undisturbed state in which weak nonlinearity causes the fundamental derivative to change sign twice, leading to the appearance of quartic nonlinearity. This behavior leads to the formation of both expansion and compression waves in a single pulse. We conduct a comprehensive analytical study of the interactions between these waves that reveals several interesting and unconventional results. These include the formation of multiple precursor waves that can extinguish in finite time, the occurrence of a centered wave fan during the interaction, and the splitting of a single shock wave into two distinct shock waves, which are not seen in cases where quartic nonlinearity is absent. Additionally, we investigate how the parameters of non-ideal fluids affect various interaction times and the overall propagation of waves.

In this paper, we consider a gravity-driven unsaturated two-phase flow with nonequilibrium formalism proposed by Hassanizadeh and coworkers. The nonequilibrium effect stems from the dynamic capillary pressure in porous media flow. Consequently, the resulting model is a nonlinear, degenerate pseudo-parabolic diffusion–convection equation. We delve into illustrating and analyzing how the dynamic capillary effect and gravity introduce qualitative differences in the local behavior of wetting saturation fronts in the vicinity of the interface. The underlying idea is to analyze the traveling wave solutions and subsequently perform an asymptotic analysis on solutions with initial support in a half line.

This work is devoted to numerical analysis and computation of the time fractional diffusion-wave equations with the $�\psi$-Caputo derivative of order $��\alpha �\in �(�1�,�2�)�$. The $�\psi$-Caputo derivative, characterized by its adaptive integral kernel function $��\psi �\left(�t�\right)�$, offers a unified framework for modeling complex memory effects in anomalous diffusion. We first discuss the existence, uniqueness, regularity, and decay of the solution to the considered model. Subsequently, an efficient fully discrete scheme is developed by combining the H2N2 discretization in time with finite element method in space. Stability and convergence of the proposed scheme are rigorously analyzed. The proposed scheme turns out to be of $��(�3�-�\alpha �)�$th order convergence in time and $��(�r�+�1�)�$th order convergence in space. Numerical experiments are conducted to corroborate the theoretical findings.

This paper is devoted to the global well-posedness for small initial data and the large-time behavior of solutions to the $�n$-dimensional ($��n�\ge �2�$) incompressible Boussinesq equations with fractional dissipation. We first establish the asymptotic stability of the system by proving that the $�L^2$-norm of the solutions decays to zero over time. Subsequently, we prove the local existence of solutions via a mollification approach and the Picard theorem, and then establish a series of a priori estimates that allow us to extend these solutions globally in time using a continuity argument. Furthermore, for initial data lying in negative Sobolev spaces, we demonstrate the global well-posedness and propagation of regularity in these spaces. A key contribution of this work is the detailed analysis of the large-time behavior, where we derive both upper and lower bounds for the decay rates of the solutions and their higher-order derivatives. The fact that these bounds coincide establishes the sharpness (optimality) of the decay rates. To the best of our knowledge, this work provides the first comprehensive study on the stability and large-time dynamics of the multi-dimensional Boussinesq equations with general fractional dissipation. By introducing novel techniques in Fourier analysis and energy methods, we not only extend several previous results to the $�n$-dimensional case but also improve upon others, particularly by relaxing the restrictions on the fractional exponents and the initial data.

We study propagation of a sum of two electromagnetic waves of the same frequency in a plane shielded waveguide with anisotropic nonlinear permittivity. Since the permittivity is nonlinear, then the sum of waves forms a coupled nonlinear wave. The main problem is to prove that the waveguide supports coupled nonlinear guided waves. The problem is formulated for Maxwell's equations and then is reduced to a specific type of nonlinear eigenvalue problems. Existence of the guided waves is proved using a nonlinear perturbation approach. Using this approach it is proved existence of solutions without linear counterparts. We also present numerical results that, on the one hand, illustrate the theoretical findings and, on the other hand, show that there are solutions that cannot be found using the developed approach.

The existence of solitary wave solutions of the one-dimensional version of the fractional nonlinear Schrödinger (fNLS) equation was analyzed by the authors in a previous work. In this paper, the asymptotic decay of the solitary waves is analyzed. From the formulation of the differential system for the wave profiles as a convolution, these are shown to decay algebraically to zero at infinity, with an order which depends on the parameter determining the fractional order of the equation. Some numerical experiments illustrate the result.

In this paper, we study the eigenvalue problem for cooperative systems where the eigenvalue parameter appears on both the equation and the boundary. By utilizing a series of one-parameter eigenvalue problems, we give a sufficient condition for the existence of the positive eigenvalue, which corresponds to the positive eigenfunction, and prove that it is unique when the system is symmetric. Then, we apply the theoretical result to investigate the existence and stability of non-constant solutions for a general reaction-diffusion model with nonlinear boundary conditions. In addition, the influence of nonlinear boundary conditions on the long-time behavior of the solution is illustrated by numerical simulations.