Studies in Applied Mathematics最新文献

In this paper, we first investigate the mathematical aspects of supersonic flow of a Chaplygin gas past a conical wing with diamond-shaped cross sections in the case of a shock wave detached from the leading edges. The flow under consideration is governed by the three-dimensional steady compressible Euler equations. For the Chaplygin gas, all characteristics are linearly degenerate, and shocks are reversible and characteristic. Using these properties, we can determine the location of the shock in advance and reformulate our problem as an oblique derivative problem for a nonlinear degenerate elliptic equation in conical coordinates. By establishing a Lipschitz estimate, we show that the equation is uniformly elliptic in any subdomain strictly away from the degenerate boundary, and then further prove the existence of a solution to the problem via the continuity method and vanishing viscosity method.

We parameterize elliptic function solutions to the derivative nonlinear Schrödinger (DNLS) equation with four independent parameters and generate two equivalent forms of $�N$-elliptic localized wave solutions to the DNLS equation through the Darboux–Bäcklund transformation. The $�N$-elliptic localized wave solutions are expressed as (the derivative of) the ratios of determinants with entries in terms of Weierstrass sigma functions. Moreover, the asymptotic behaviors of both forms of the $�N$-elliptic localized wave solutions are analyzed both along and between the propagation directions as $��t�\to �\pm �\infty �$, which verifies that the collisions between elliptic-solutions are elastic. We prove that the solution tends to a simple elliptic localized wave solution along each propagation direction. Between the propagation directions, the solution asymptotically approaches a shifted background. Furthermore, we establish sufficient conditions for strictly elastic collisions. The dynamic behaviors of the solutions are systematically investigated, with analytical results visualized through graphical illustrations. The asymptotic analysis of these solutions confirms that they exhibit the behaviors predicted by the generalized soliton resolution conjecture on the elliptic function background.

In this paper, we investigate periodic wave solution and the complete classifications of solutions for the modified Benjamin–Bona–Mahony (mBBM) equation, also commonly referred to as the modified regularized long wave (mRLW) equation with initial discontinuous Riemann problem. The nonlinear dispersive mBBM equation serves as an alternative model to the modified Korteweg–de Vries (mKdV) equation in physical phenomena such as long-crested waves in near-shore regions and unidirectional wave propagation in water channels. By employing hyperbolic theory, asymptotic analysis, Whitham modulation theory, and the dispersive shock wave (DSW) fitting method, along with direct numerical simulations, we systematically analyze the fundamental wave structures in the Riemann problem. These basic waves include linear wavetrains composed of one-phase and two-phase solutions, RWs, DSWs, and the two-phase modulation structures for DSW implosion. Due to the non-integrability of the mBBM equation and the non-convexity of its linear dispersion relation, its Riemann problem exhibits a richer variety of solutions compared to the mKdV equation. The most notable differences include the emergence of two-phase linear wavetrains and DSW implosion. Finally, based on the differences in wave structures, we classify the initial parameter space into 10 distinct regions.

The study has analyzed the diffraction and reflection phenomena of a weak shock interacting with a right-angled wedge in the context of a more realistic extended Chaplygin gas. Asymptotic solutions to the two-dimensional Euler system have been derived under appropriate boundary conditions that characterize the diffraction of a weak shock from the wedge. In this study, the effects of the specific gas considered are carefully modeled, and their influence on the overall flow configuration is examined. In particular, a detailed investigation of the local structure near a singular point is conducted, highlighting the significant role of the considered gas behavior in shaping the flow dynamics.

This paper presents a systematic study of lump solutions and lump chain solutions in the modified Kadomtsev–Petviashvili-I equation. Using the long-wave limit method, we derive both standard lump solutions and plane lump solutions from line-soliton solutions. Through detailed asymptotic analysis, it is shown that standard lump solutions exhibit normal scattering. We further investigate lump chain solutions and identify elastic interactions with explicit phase shifts, as well as distinctive resonant patterns, including a characteristic “Y”-shaped resonance and a unique parallel resonance phenomenon. An improved long - wave limit approach is introduced to construct higher-order lump solutions from lump chain solutions, yielding second-order lump solutions, second-order lump chain solutions, and semi-rational soliton solutions. Notably, the second-order lump solutions exhibit anomalous scattering, in which individual lumps propagate along curved trajectories despite sharing the same asymptotic velocity. Our analysis also establishes a connection between lump solutions and the root structure of Yablonskii–Vorob'ev polynomials. These results deepen the understanding of nonlinear wave interactions in (2 + 1)-dimensional integrable systems.

The paper is devoted to the dynamical stability of pullback attractors for non-autonomous Lagrangian-averaged Navier–Stokes equations with delays. First, using the Ascoli–Arzelà theorem, we prove the pullback asymptotic compactness of solution operators to establish the existence of pullback attractors. Second, we prove the pointwise upper semicontinuity of pullback attractors as the delay time approaches zero. Eventually, we further investigate their uniform upper semicontinuity when the delay time converges to zero. Combining these two results, the latter strengthens the former. To the best of our knowledge, this paper appears to be the first to address both types of upper semicontinuity for pullback attractors, particularly the interesting uniform case.

We study wave propagation through a one-dimensional array of subwavelength resonators with periodically time-modulated material parameters. Focusing on a high-contrast regime, we use a scattering framework based on Fourier expansions and scattering matrix techniques to capture the interactions between an incident wave and the temporally varying system. This way, we derive a formulation of the total energy flux corresponding to time-dependent systems of resonators. We show that the total energy flux is composed of the transmitted and reflected energy fluxes and derive an optical theorem which characterizes the energy balance of the system. We provide a number of numerical experiments to investigate the impact of the time-dependency, the operating frequency, and the number of resonators on the maximal attainable energy gain and energy loss. Moreover, we show the existence of lasing points, at which the total energy diverges. Our results lay the foundation for the design of energy dissipative or energy amplifying systems.