Asymptotic stability of small solitary waves for the one-dimensional NLS with an attractive delta potential in the mass-subcritical regime

For the one-dimensional nonlinear Schrodinger equation with an attractive point defect, $i\partial_t u = (-\tfrac12\partial_x^2 + q\delta(x))u + \mu|u|^p u$ with $q<0$, Masaki, Murphy, and Segata proved that every sufficiently small $H^1$ solution decomposes into a small solitary wave pinned at the defect plus a radiation term that scatters to a linear state, for even integer powers $p \ge 4$ (a range that already includes the mass-critical exponent $p=4$ in one dimension). They explicitly posed the extension to $L^2$-subcritical nonlinearities ($p<4$), anticipating that stronger, weighted assumptions on the data would be needed. In the later literature, the extension to arbitrary real $p>4$ (for all solitary-wave sizes, under a spectral condition) was achieved by the authors' 2023 follow-up, while the subcritical regime remains open in full strength. Known partial results delimit rather than close it: Cuccagna and Maeda proved locally-averaged decay and ground-state selection for $p>1$ without any scattering claim; Chen proved the cubic case ($p=2$) with modified scattering for smooth trapping potentials, noting the delta potential as a limiting case; Murphy and Nakanishi proved that scattering to a solitary wave is impossible for $0<p\le 1$. The problem asks for the asymptotic stability of the small defect-bound solitary waves across the mass-subcritical range $0<p<4$ - the question explicitly posed by the source authors - together with the function-space assumptions on the initial data that make it true; the known obstructions sit at the two thresholds $p=2$ and $p=1$ (linear scattering can occur only for $2<p<4$, modified scattering is expected for $1<p\le 2$, and for $0<p\le 1$ the radiation cannot converge to a linear wave).

Background

The one-dimensional nonlinear Schrodinger equation (NLS) with a point defect is

$i\partial_t u = Hu + \mu|u|^p u, \qquad H = -\tfrac12\partial_x^2 + q\,\delta(x), \quad q < 0,$

where $\delta$ is the Dirac delta distribution, the coupling $q<0$ makes the defect attractive (trapping), and $\mu \ne 0$ selects the focusing ($\mu<0$) or defocusing ($\mu>0$) sign. The operator $H$ acts as $-\tfrac12\partial_x^2$ on functions in $H^1(\mathbb{R}) \cap H^2(\mathbb{R}\setminus\{0\})$ obeying the jump condition $\partial_x u(0^+) - \partial_x u(0^-) = 2q\,u(0)$; it is self-adjoint with essential spectrum $[0,\infty)$ and a single negative eigenvalue $-q^2/2$ whose eigenfunction $\phi_0(x) = |q|^{1/2}e^{-|q||x|}$ is localized at the defect. The equation is a standard model for resonant nonlinear propagation of light through optical waveguides with a localized defect. Bifurcation from the linear eigenvalue produces, for every power $p>0$, a gauge-covariant family of small nonlinear bound states $Q[z] = z\phi_0 + O(z^2)$ with $z \in \mathbb{C}$ small, giving solitary waves $e^{-iE[|z|]t}Q[z]$ pinned at the defect; for power nonlinearities these states are given by explicit formulas.

Two notions of stability are relevant. Orbital stability - the solution remains close for all time to the family of bound states modulo the phase symmetry - was settled for this model by Fukuizumi, Ohta, and Ozawa: with an attractive delta potential the bound states are orbitally stable in the mass-subcritical ($p<4$) and mass-critical ($p=4$) cases, while in the mass-supercritical case ($p>4$) the branch is stable only near the linear eigenvalue and unstable deeper. Here 'mass-(sub/super)critical' refers to the $L^2$-scaling critical exponent $p = 4$ of the nonlinearity $|u|^p u$ in one dimension. Asymptotic stability is strictly stronger: a small solution should converge to a single solitary wave,

$u(t) = Q[z(t)] + v(t), \qquad v(t) - e^{-itH}v_+ \to 0 \ \text{in}\ H^1 \ \text{as}\ t \to \infty,$

where the radiation $v$ scatters to a linear solution ($v_+$ lies in the continuous spectral subspace of $H$) and the modulation parameter $z(t)$ converges up to phase. Masaki, Murphy, and Segata proved this for every even integer $p \ge 4$ - a range that includes the mass-critical exponent $p = 4$ - for initial data with sufficiently small $H^1$ norm, together with global space-time bounds from Strichartz estimates (mixed space-time $L^q_t L^r_x$ bounds on solutions of the linear equation governed by $e^{-itH}$). Working with $H^1$ data alone forces $p \ge 4$ through the Duhamel estimates for $|v|^p v$, in exact analogy with small-data scattering for the free one-dimensional power NLS, which is why the authors expect the subcritical extension to require stronger, weighted integrability assumptions on the initial data.

One further boundary is known. In one dimension the nonlinearity $|u|^p u$ is short-range only for $p > 2$; for long-range powers the radiation cannot converge to a purely linear wave. Murphy and Nakanishi proved for this delta-potential model that for $0 < p \le 1$ no solution can decompose into a localized wave (solitary or otherwise) plus a nonzero linear scattering state, so in that range the modulation parameter $z(t)$ can at best converge up to phase, and at and below the threshold $p = 2$ the expected asymptotics are those of modified scattering, in which the radiation converges to a linear wave only after a logarithmic phase correction.

Problem Statement

Consider the initial-value problem for the one-dimensional nonlinear Schrodinger equation with an attractive delta potential,

$i\partial_t u = \Big(-\tfrac12\partial_x^2 + q\,\delta(x)\Big)u + \mu|u|^p u, \qquad q < 0, \quad \mu \in \mathbb{R}\setminus\{0\}, \quad 0 < p < 4,$

where $p$ ranges over the mass-subcritical powers. Determine whether the family of small solitary waves $Q[z]$, $|z|$ small, is asymptotically stable, and under what assumptions on the initial data. This is one scientific objective - the question explicitly posed by Masaki, Murphy, and Segata for $L^2$-subcritical nonlinearities, for both signs of $\mu$ - which the known results delimit but do not close; the known obstructions are tied to the two thresholds $p=2$ (the long-range threshold in one dimension) and $p=1$ (the Murphy-Nakanishi impossibility threshold):

1. Short-range subcritical regime, $2 < p < 4$: here the target is full scattering stability - prove or disprove that there is a function space $X \subset H^1(\mathbb{R})$ carrying additional spatial decay or integrability (the source authors expect stronger integrability conditions of this kind to be needed) and nondegenerate, in that it contains data whose solutions have a nonzero radiation component (excluding trivial choices such as $X = \{0\}$), such that every initial datum $u_0 \in X$ of sufficiently small norm yields a global solution admitting, for all $t \ge 0$, a decomposition $u(t) = Q[z(t)] + v(t)$ with $v(t) - e^{-itH}v_+ \to 0$ in $H^1$ for some $v_+$ in the continuous spectral subspace of $H$, convergence of $z(t)$ up to phase, and global space-time bounds on $v$.

2. Long-range subcritical regime, $1 < p \le 2$: linear scattering is impossible for the free equation in this range and is expected to fail here as well, so the target here is asymptotic stability with modified scattering of the radiation; the cubic case with smooth trapping potentials was established by Chen.

3. Very weak regime, $0 < p \le 1$: scattering to any localized wave is impossible (Murphy-Nakanishi), so the target in this range is selection of $z(t)$ up to phase for small data; Cuccagna-Maeda proved selection for $p>1$ and explicitly leave $p \le 1$ open.

The scientific objective is the source's single question: the asymptotic stability of the small defect-bound solitary waves across the mass-subcritical range $0<p<4$, for both signs of $\mu$, together with the function-space assumptions on the initial data that make it hold.

References

1. Satoshi Masaki, Jason Murphy, Jun-ichi Segata, Stability of small solitary waves for the one-dimensional NLS with an attractive delta potential, Analysis & PDE 13 (2020), no. 4, 1099-1128, DOI 10.2140/apde.2020.13.1099, https://doi.org/10.2140/apde.2020.13.1099 (source paper; preprint arXiv:1807.01423)
2. Satoshi Masaki, Jason Murphy, Jun-ichi Segata, Asymptotic stability of solitary waves for the 1d NLS with an attractive delta potential, Discrete and Continuous Dynamical Systems 43 (2023), no. 6, 2137-2185, DOI 10.3934/dcds.2023006, https://doi.org/10.3934/dcds.2023006 (preprint arXiv:2008.11645)
3. Scipio Cuccagna, Masaya Maeda, On stability of small solitons of the 1-D NLS with a trapping delta potential, SIAM Journal on Mathematical Analysis 51 (2019), no. 6, 4311-4331, DOI 10.1137/19M1258402, https://doi.org/10.1137/19M1258402 (preprint arXiv:1904.11869)
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