Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold

Abstract

In this paper, we consider the following nonlinear Schrödinger equation: $$\begin{array}{} \displaystyle \begin{cases}iu_t+{\it\Delta}_g u+ia(x)u-|u|^{p-1}u=0\qquad (x,t)\in \mathcal{M} \times (0,+\infty), \cr u(x,0)=u_0(x)\qquad x\in \mathcal{M},\end{cases} \end{array}$$(0.1) where (𝓜, g) is a smooth complete compact Riemannian manifold of dimension n(n = 2, 3) without boundary. For the damping terms −a(x)(1 − Δ)−1a(x)ut and $\begin{array}{} \displaystyle ia(x)(-{\it\Delta})^{\frac12}a(x)u, \end{array}$ the exponential stability results of system (0.1) have been proved by Dehman et al. (Math Z 254(4): 729-749, 2006), Laurent. (SIAM J. Math. Anal. 42(2): 785-832, 2010) and Cavalcanti et al. (Math Phys 69(4): 100, 2018). However, from the physical point of view, it would be more important to consider the stability of system (0.1) with the damping term ia(x)u, which is still an open problem. In this paper, we obtain the exponential stability of system (0.1) by Morawetz multipliers in non Euclidean geometries and compactness-uniqueness arguments.

Families of Solutions of Multitemporal Nonlinear Schrödinger PDE

“…The exact solutions of multitime NLSE (with oblique derivative) are closely related to the orbits of the direction vector field h. Our techniques for finding these solutions started from this important idea applied to PDEs that contain directional derivative. We follow a different route than those in the papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and carry out the calculations to obtain significative exact solutions of multitime NLSE. This computational paper and the obtained results show that our methods are simple, efficient, straightforward and powerful.…”

“…The recent mathematical literature dedicated to the context insists on the following topics: The papers [2][3][4][5] give properties of nonlinear Schrödinger equation; [6,7] underline an exact solution of the single-time Schrödinger equation; [8] introduces and studies the multitime Schrödinger equation; [1,[9][10][11][12] introduce and describe the multitime solitons as solutions of multitime PDEs. The paper [13] refers to solving the time-dependent Schrödinger equation via Laplace transform on t. A single-time Schrödinger equation in Riemannian setting is studied in the paper [14]. The paper [15] comes closest to our style by offering "methods for constructing complex solutions of nonlinear PDEs using simpler solutions".…”

Structural stability for the Boussinesq equations interfacing with Darcy equations in a bounded domain

“…We want to know whether a slight change in the coefficients in the equations or boundary data, or even the equation itself, will lead to drastic changes in the solution. For a review of the nature of the structural stability, refer [13][14][15], Scott [16], Scott and Straughan [17], Payne et al [18][19][20][21] and some related papers [22][23][24]. The previous publications of structural stability usually study one fluid in a bounded domain.…”

On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term

“…(1.6), and then obtained a threshold value separating the existence and nonexistence of solutions. For the other types of second-order Schrödinger equation, many experts have paid attention to their qualitative properties and obtained abundant theoretical results, we refer the reader to see [30,19,2,36,32,7,6,1] and the papers cited therein.…”