An explicitly solvable NLS model with discontinuous standing waves

TL;DR

This work analyzes a one-dimensional nonlinear Schrödinger equation on $\mathbb{R}$ with a point interaction at the origin that combines an attractive delta potential with a dipole condition, under focusing nonlinearity $|u|^{2\sigma}u$ with $0<\sigma\le2$, encoded by $u(0^+)=\tau u(0^-)$ and energy perturbation $-\frac{\alpha}{2}|u(0^-)|^2$. The authors provide a complete variational and spectral classification of stationary states, proving existence and uniqueness of Ground States for all masses in the subcritical regime and detailing mass-thresholds for excited states; in the critical case they obtain explicit threshold masses $\mu^\star$ and $\widetilde{\mu}$ (with explicit formulas) for both dipole and FT interactions, and they compute the optimal Gagliardo–Nirenberg constant $K_\tau$. All stationary states are explicitly computed, allowing a full bifurcation analysis of the branches as the interaction strength varies; the paper also establishes the existence of action minimizers at fixed frequency for $\omega>\frac{\alpha^2}{(\tau^2+1)^2}$. The results provide a transparent, solvable model illustrating the intricate interplay between nonlinear focusing dynamics and singular point interactions in one dimension, bridging delta-like and dipole/Dirac-type couplings on the line.

Abstract

We study the NLS Equation on the line with a point interaction given by the superposition of an attractive delta potential with a dipole interaction, in the cases of $L^2$-subcritical and $L^2$-critical nonlinearity. For a subcritical nonlinearity we prove the existence and the uniqueness of Ground States at any mass. If the mass exceeds an explicit threshold, then there exists a positive excited state too. For the critical nonlinearity we prove that Ground States exist only in a specific interval of masses, while in a different interval excited states exist. We provide the value of the optimal constant in the Gagliardo-Nirenberg estimate and describe in the dipole case the branches of the stationary states as the strength of the interaction varies. Since all stationary states are explicitly computed, ours is a solvable model involving a non-standard interplay of a nonlinearity with a point interaction.

Figures (7)

• Figure 1: Graph of the Ground State for $\tau=1.2$ on the left and $\tau=5$ on the right.
• Figure 2: Graph of the other stationary state for $\tau=1.2$ on the left and $\tau=5$ on the right.
• Figure 3: Geometric representation of the system \ref{['system:t+t-']} for $\tau > 1$, where the dots represent the solutions to the system for $\omega \rightarrow \infty$.
• Figure 4: A sketch of the stationary state $u^L$, in three different qualitative situations. It has always the profile of a tail of a soliton on the negative half-line, whereas on the positive half-line, depending on $\omega$, it can be a tail, a half-soliton or presents a bump.
• Figure 5: A sketch of the stationary state $u^R$. This stationary state, regardless of $\omega$, has always the profile of a tail of a soliton on the positive half-line and has a bump on the negative one.

Theorems & Definitions (31)

• Theorem 2.1: Stationary states for $0<\sigma<2$ and $\alpha>0$
• Theorem 2.2: Stationary states for $\sigma=2$ and $\alpha=0$
• Theorem 2.3: Stationary states for $\sigma=2$ and $\alpha>0$
• Lemma 3.1
• proof
• Remark 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof : Proof of Theorem \ref{['thm:gs-sub']}