Nonhomogeneous Boundary-Value Problems for One-Dimensional Nonlinear Schr\"odinger Equations
By Jerry L. Bona, Shu-Ming Sun, Bingyu Zhang. 2015
100 citations recorded by the source corpus.
This paper is concerned with initial-boundary-value problems (IBVPs) for a class of nonlinear Schr\"odinger equations posed either on a half line $\mathbb{R}^+$ or on a bounded interval $(0, L)$ with nonhomogeneous boundary conditions. For any $s$ with $0\leq s \frac12$, compatibility between the initial and boundary conditions is also needed.) Global well-posedness is also discussed when $s \ge 1$. From the point of view of the well-posedness theory, the results obtained reveal a significant difference between the IBVP posed on $\mathbb{R}^+$ and the IBVP posed on $(0,L)$. The former is reminiscent of the theory for the pure initial-value problem (IVP) for these Schr\"odinger equations posed on the whole line $\mathbb{R}$ while the theory on a bounded interval looks more like that othe pure IVP posed on a periodic domain. In particular, the regularity demanded of the boundary data for the IBVP on $\mathbb{R}^+$ is consistent with the temporal trace results that obtain for solutions of the pure IVP on $\mathbb{R}$, while the slightly higher regularity of boundary data for the IBVP on $(0, L)$ resembles what is found for temporal traces of spatially periodic solutions.
Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/05ec561e58261a5f83f3c511fcc2a22fb6457333
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Control and Systems Engineering
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Long‐Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability (2015, methodology): “Well-posedness of the NLS equation on the half-line with data in Sobolev spaces was established independently and via different approaches by Holmer [42], Bona, Sun and Zhang [10] and Fokas, Himonas and the second author [33].”