Acta Mathematicae Applicatae Sinica, English Series最新文献

The Cauchy problem of the fifth-order nonlinear Schrödinger (foNLS) equation is investigated with nonzero boundary conditions in detailed. Firstly, the spectral analysis of the scattering problem is carried out. A Riemann surface and affine parameters are introduced to transform the original spectral parameter to a new spectral parameter in order to avoid the multi-valued problem. Based on Lax pair of the foNLS equation, the Jost functions are obtained, and their analytical, asymptotic, symmetric properties, as well as the corresponding properties of the scattering matrix are established systematically. For the inverse scattering problem, we discuss the cases that the scattering coefficients have simple zeros and double zeros, respectively, and we further derive their corresponding exact solutions via solving a suitable Riemann-Hilbert problem. Moreover, some interesting phenomena are found when we choose some appropriate parameters for these exact solutions, which are helpful to study the propagation behavior of these solutions.

A time-domain elastic scattering problem is considered in three dimensions. In problem setting, a rigid obstacle is immersed in an unbounded domain filled with homogeneous and isotropic elastic medium. In order to analyze the well-posedness of the target problem, we reduce the scattering problem into an initial boundary value problem in a bounded domain over a finite time interval by using a compressed coordinate transformation. The Galerkin method is adopted to prove the uniqueness results, and the energy method is used to prove the stability of the scattering problem. In addition, we derive a priori estimate with explicit time dependence.

This paper investigates the propagation dynamics of nonlocal dispersal cooperative systems within a shifting environment characterized by a contracting favorable region. We examine two distinct types of dispersal kernels. For thin-tailed kernels, we study the existence, uniqueness, and stability of forced waves using upper and lower solutions, the sliding method, and the dynamical systems approach. In the case of partially heavy-tailed kernels, considering compactly supported initial value functions, we demonstrate that for each species, the right side of the level sets exhibits accelerated rightward propagation, while transferability occurs. Conversely, the propagation on the left side does not move leftward but rather rightward, with a spreading speed equivalent to that of the shifting environment. Consequently, species with thin-tailed kernels inherently persist in a shifting habitat, provided they are part of a cooperative and irreducible system that includes at least one species with a heavy-tailed kernel, regardless of the magnitude of the shifting environment’s speed. This behavior markedly diverges from the dynamics observed in scalar equations.

This paper is concerned with the orbital stability of solitary waves in the mKdV-Schrödinger system with cubic-quintic nonlinear terms through detailed spectral analysis and abstract stability theorem. First, we derived the explicit solitary wave solutions by assuming the solution expression. Then, through using the orbital stability theory developed by Grillakis et al., we established general criteria for assessing the orbital stability of solitary waves of this system.

This paper concerns the existence and stability of solitary waves for the nonlinear Schrödinger equation with a partial confinement, which describes the limit case of the cigar-shaped model in Bose-Einstein condensate of dipolar quantum gases. More precisely, we applied a compactness argument, which comes from the confining potential x ¹₂ +x ²₂ , to overcome the lack of compactness caused by the translation invariance with respect to x₃. Then, since the mass supercritical character of this equation, we construct orbitally stable solutions by adapting a suitable localized minimization problem. Finally, the stability of solitary waves is obtained.

This paper is concerned with the energy decay rate of the total energy to a wave equation with p(x)-Laplacian damping (nonlinear strong damping) and nonlinear source. With some suitable restrictions on variable growth exponents r(x) and p(x), first, we prove that the local solution can be extended to exist globally. Second, by using suitable and weighted multiplier techniques, it is proved that the total energy decays logarithmically. The key and main difficulty is to give a prior estimate for the wighted integral (int_{Omega}chi^{p(x)-1}(tau)vertnabla u(tau)vert^{p(x)}dx) by some differential inequality techniques. In the proof of energy decay, the traditional method to eliminate the lower-order terms by exploiting the unique continuation and compactness arguments is not needed in our energy decay estimate.

In this paper, we establish the exponential stability of the one-dimensional Euler-Bernoulli beam equation with local viscosity. On the one hand, we demonstrate that the Euler-Bernoulli beam equation system is exponentially stable by employing the multiplier method, which relies on an appropriately constructed Lyapunov function. On the other hand, we discretize the Euler-Bernoulli beam equation system using the finite volume difference method. For the resulting semi-discrete system, we construct a discretized multiplier based on the discretized Lyapunov function. Finally, we prove that the semi-discrete Euler-Bernoulli beam equation system is also uniformly exponentially stable.

The Cauchy problem for non-isentropic compressible Navier-Stokes/Allen-Cahn system with degenerate heat-conductivity (kappa (theta) = tilde kappa {theta ^beta}) in 1-D is discussed in this paper. This system is widely used to describe the motion of immiscible two-phase flow with diffused interface. The well-posedness for strong solution of this problem is established with the H¹ initial data for density, temperature, velocity, and the H² initial data for phase field. The result shows that no discontinuity of the phase field, vacuum, shock wave, mass or heat concentration will be developed at any finite time in the whole space. From the hydrodynamic point of view, this means that no matter how complex the interaction between the hydrodynamic and phase-field effects, phase separation will not occur, but the phase transition is possible.

In this paper, we consider the time-harmonic electromagnetic scattering by a perfect conductor in a homogeneous chiral environment. For three-dimensional cylindrical structures, it can be simplified as a two-dimensional model problem, which can be modeled by two scalar Helmholtz equations via coupled boundary conditions. The boundary integral equation method is used to prove the unique existence of the weak solution to this problem. Then we apply the linear sampling method to recover the scatterer from one of the far field pattern of wave fields. Some numerical examples are shown to verify the correctness and effectiveness of the proposed method.

In this paper, we consider a Dirichlet-Laplacian with a drift problem. We prove existence of weak solutions of it on the Nehari manifold. Then, we show that the associated energy functional has a minimizer on a rearrangement class generated by a determined function. Finally, we prove a duality theorem for this boundary value problem.