Nonlinear mirror image method for nonlinear Schrödinger equation: Absorption/emission of one soliton by a boundary

TL;DR

This work extends the nonlinear mirror image method to the nonlinear Schrödinger equation on the half-line with time-dependent boundary conditions, by fixing the Bäcklund transformation at infinity and coupling it to a time-dependent reflection matrix $K(t,\lambda)$. It reveals two main solution classes: Robin-like soliton reflections and a novel regime where a single soliton can be absorbed or emitted by the boundary, a phenomenon unique to time-dependent integrable BCs. The authors develop a two-step BT framework to implement folding symmetry, map the IBVP to a full-line problem with precise symmetry constraints on scattering data, and provide explicit multisoliton constructions that illustrate the absorption/emission mechanisms. The results advance the unification of integrable boundary-condition formalisms and open avenues for connections with the Fokas method and quantum boundary phenomena.

Abstract

We perform the analysis of the focusing nonlinear Schrödinger equation on the half-line with time-dependent boundary conditions along the lines of the nonlinear method of images with the help of Bäcklund transformations. The difficulty arising from having such time-dependent boundary conditions at $x=0$ is overcome by changing the viewpoint of the method and fixing the Bäcklund transformation at infinity as well as relating its value at $x=0$ to a time-dependent reflection matrix. The interplay between the various aspects of integrable boundary conditions is reviewed in detail to brush a picture of the area. We find two possible classes of solutions. One is very similar to the case of Robin boundary conditions whereby solitons are reflected at the boundary, as a result of an effective interaction with their images on the other half-line. The new regime of solutions supports the existence of one soliton that is not reflected at the boundary but can be either absorbed or emitted by it. We demonstrate that this is a unique feature of time-dependent integrable boundary conditions.

Figures (5)

• Figure 1: 2D-contour plots of $|u(x,t)|$ corresponding to two solitons reflected with time-dependent BCs \ref{['BCQn']} for $\alpha=2$ and $\beta=1$. The same zeros $\lambda_1=1+2i$ and $\lambda_2=(1+5i)/2$ are used for both plots and with $\varepsilon=1$ in \ref{['BT2relationnormingC']}. The norming constants are $c(\lambda_1)= -4e^{-20},\ c(\lambda_2)=5e^{5}$ on the left and $c(\lambda_1)= -4e^{4},\ c(\lambda_2)=5e^{-10}$ on the right.
• Figure 2: Line representation of the (quantum) reflection equation depicting a two-particle process being factorised into two possible successions of particle-particle interactions and particle-boundary interactions. The consistency of the two possibilities which must yield the same physical scattering matrix requires the reflection equation.
• Figure 3: 2D-contour plots of $|u(x,t)|$ corresponding to two solitons, one reflected and one absorbed with time-dependent BCs \ref{['BCQn']} for $\alpha=4$, $\beta=2$. The same zeros $\lambda_0=1+2i$ and $\lambda_1=(1+5i)/2$ are used for both plots. The norming constants are $c(\lambda_0)= 1,\ c(\lambda_1)= 5e^{15}$ on the left and $c(\lambda_0)= e^{20},\ c(\lambda_1)= 5$ on the right.
• Figure 4: 2D-contour plots of $|u(x,t)|$ corresponding to two solitons, one reflected and one absorbed with time-dependent BCs \ref{['BCQn']} for $\alpha=6$, $\beta=1$. The same zeros $\lambda_0=(1+6i)/2$ and $\lambda_1=(2+5i)/2$ are used for both plots. The norming constants are $c(\lambda_0)= 4e^{16},\ c(\lambda_1)= 5$ on the left and $c(\lambda_0)= 4e^{-8},\ c(\lambda_1)= 5e^{15}$ on the right.
• Figure 5: Line representation of two-soliton solutions when one is absorbed.

Theorems & Definitions (18)

• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Lemma 4.1
• Definition 4.2
• Lemma 4.3
• Lemma 4.4
• Proposition 4.5
• Lemma 4.6
• Lemma 4.7