Studies in Applied Mathematics最新文献

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.

In this paper, we concentrate on the high-order rogue waves of the Davey–Stewartson I equation and investigate their dynamics in great detail. The approximate trajectories of curvy rogue waves in the intermediate stage and certain lumps at large time are determined by a special polynomial which is expressed by some Schur polynomials and their derivatives. Besides, the number of lumps at large time is illustrated by using a positive integer partition and its corresponding Young diagram. The true and estimated results show excellent agreement.

In this paper, we revisit the eigenvalue problem of the one-dimensional Schrödinger equation for smooth single well potentials. In particular, we provide a new interpretation of the Bohr–Sommerfeld quantization formula. A novel aspect of our results, which are based on recent work of the authors on the turning point problem based upon dynamical systems methods, is that we cover all eigenvalues $��E�\in �[�0�,�O�(�1�\left)�\right]�$ and show that the Bohr–Sommerfeld quantitization formula approximates all of these eigenvalues (in a sense that is made precise). At the same time, we provide rigorous smoothness statements of the eigenvalues as functions of $�\epsilon$. We find that whereas the small eigenvalues $��E�=�O�\left(�\epsilon �\right)�$ are smooth functions of $�\epsilon$, the large ones $��E�=�O�\left(�1�\right)�$ are smooth functions of $��n�\epsilon �\in ��[�c_1�,�c_2�]��,��0�<�c_{}$