For the one-dimensional focusing power nonlinear Schrödinger equation i∂tψ+∂x2ψ+∣ψ∣2kψ=0i\partial_t\psi+\partial_x^2\psi+|\psi|^{2k}\psi=0 in the L2L^2-supercritical regime k>2k>2, Chen and Moutinho (arXiv:2509.03637) proved that well-separated multi-solitons with distinct velocities are asymptotically stable on a codimension-mm Lipschitz center-stable manifold, but only under the technical restriction k>114k>\tfrac{11}{4}, which they explicitly do not claim to be optimal. The open problem is whether the same asymptotic stability conclusion holds for the full supercritical range k>2k>2 under the same hypotheses — equivalently, to determine the sharp exponent threshold — closing the gap 2<k≤1142<k\le\tfrac{11}{4} left between their theorem and the single-soliton case, where Krieger and Schlag settled all monic supercritical powers.

Background

The one-dimensional focusing nonlinear Schrödinger equation (NLS) i∂tψ+∂x2ψ+∣ψ∣2kψ=0,x∈R, k>2i\partial_t\psi+\partial_x^2\psi+|\psi|^{2k}\psi=0,\qquad x\in\mathbb{R},\ k>2 is L2L^2-supercritical, meaning the conserved mass ∥ψ∥L22\|\psi\|_{L^2}^2 sits above the scaling-critical level (the quintic power ∣ψ∣4ψ|\psi|^4\psi, i.e. k=2k=2, is L2L^2-critical in one dimension, so k>2k>2 is supercritical). It admits solitons — solitary wave solutions of the form eiα2t+iγϕα(x−y)e^{i\alpha^2 t+i\gamma}\phi_\alpha(x-y) with ϕα\phi_\alpha a localized ground-state profile — and, by superposition of mm solitons with distinct velocities v1>⋯>vmv_1>\cdots>v_m and well-separated centers min⁡ℓ(yℓ−yℓ+1)>L\min_\ell(y_\ell-y_{\ell+1})>L, approximate multi-soliton configurations whose exact counterparts can be constructed (Côte–Martel–Merle). In the supercritical regime each soliton is orbitally unstable (the linearized operator, a non-self-adjoint Schrödinger operator, has genuine unstable eigenvalues), so stability can only hold after removing finitely many unstable directions: one seeks a center-stable manifold, an invariant manifold of codimension equal to the number of unstable directions, such that solutions starting on it stay close to the multi-soliton and scatter, i.e. decompose as t→∞t\to\infty into a sum of modulated solitons plus free radiation eit∂x2fe^{it\partial_x^2}f — this is asymptotic stability. The classical benchmark is Krieger–Schlag (JAMS 2006), which proved asymptotic stability of a single soliton on a codimension-one center-stable manifold for all monic (pure-power) supercritical focusing 1D NLS, i.e. the full range k>2k>2 at m=1m=1. For several solitons the obstruction is the weak one-dimensional dispersion: linear solutions decay only like t−1/2t^{-1/2}, so multi-soliton interaction terms are barely time-integrable. Full-line multi-soliton asymptotic stability in 1D was previously known only for two solitons with strong relative speeds and structural assumptions on the nonlinearity (Perelman 1997), while in the stronger dispersion of higher dimensions it was known for NLS (Perelman 2004) and for nonlinear Klein–Gordon equations (Chen–Jendrej). Chen and Moutinho developed the necessary linear theory for matrix charge transfer models — linear Schrödinger equations with several moving potentials, as arise from linearizing around moving solitons — establishing dispersive estimates and scattering under distinct velocities and well-separated centers, with threshold resonances and unstable modes allowed. Building on it, they proved (arXiv:2509.03637, v1 2025, v4 July 2026): for k>114k>\tfrac{11}{4}, under hypotheses (H1) distinct ordered velocities and (H2) quantitative well-separation of centers, there is a codimension-mm Lipschitz center-stable manifold around the well-separated mm-soliton family such that solutions with data on it exist globally and converge in H1(R)H^1(\mathbb{R}) to a sum of mm modulated solitons plus a free radiation term, with the remainder in the modulation decomposition decaying at the sharp one-dimensional linear rate (1+t)−1/2(1+t)^{-1/2} in L∞L^\infty (Theorem 1.3) and the distance to the modulated multi-soliton plus radiation decaying like (1+t)−1/4(1+t)^{-1/4} in H1H^1 (Theorems 1.1 and 1.6). The restriction k>114k>\tfrac{11}{4} is technical: it enters through the time-integrability of the nonlinear interaction remainders against the weak 1D dispersive decay in the bootstrap argument, and the authors state verbatim that they do not claim it is optimal.

Problem Statement

For the one-dimensional focusing power nonlinear Schrödinger equation i∂tψ+∂x2ψ+∣ψ∣2kψ=0,x∈R,i\partial_t\psi+\partial_x^2\psi+|\psi|^{2k}\psi=0,\qquad x\in\mathbb{R}, in the L2L^2-supercritical regime, prove or disprove that the multi-soliton asymptotic stability theorem of Chen–Moutinho (arXiv:2509.03637, Theorems 1.1 and 1.6) extends from its stated range k>114k>\tfrac{11}{4} to the full supercritical range k>2k>2; or, more generally, determine the sharp exponent threshold for the following conclusion. Under hypotheses (H1) v1>⋯>vmv_1>\cdots>v_m distinct velocities and (H2) quantitative well-separation of the soliton centers min⁡ℓ(yℓ−yℓ+1)>L\min_\ell(y_\ell-y_{\ell+1})>L, there should exist, around the well-separated family of mm-soliton configurations, a codimension-mm Lipschitz center-stable manifold, invariant for t≥0t\ge0, such that every solution with initial data on it is global and converges in H1(R)H^1(\mathbb{R}) as t→∞t\to\infty to a sum of mm modulated solitons ∑ℓ=1mei(vℓ,∞x2−vℓ,∞2t4+αℓ,∞2t+γℓ,∞)ϕαℓ,∞(x−vℓ,∞t−Dℓ,∞)\sum_{\ell=1}^m e^{i(\tfrac{v_{\ell,\infty}x}{2}-\tfrac{v_{\ell,\infty}^2t}{4}+\alpha_{\ell,\infty}^2t+\gamma_{\ell,\infty})}\phi_{\alpha_{\ell,\infty}}(x-v_{\ell,\infty}t-D_{\ell,\infty}) plus free radiation eit∂x2fe^{it\partial_x^2}f with f∈H1(R)f\in H^1(\mathbb{R}), with the distance to the modulated multi-soliton plus radiation decaying at the sharp rate (1+t)−1/4(1+t)^{-1/4} in H1(R)H^1(\mathbb{R}). Equivalently, either the conclusion holds for all 2<k≤1142<k\le\tfrac{11}{4} and all m≥2m\ge2 (matching the m=1m=1 case of Krieger–Schlag), or there exists a sharp exponent k∗>2k_*>2 such that the conclusion holds for k>k∗k>k_* and fails for some 2<k≤k∗2<k\le k_*.

References

  1. Gong Chen and Abdon Moutinho, Asymptotic Stability of multi-solitons for 11d Supercritical NLS, preprint (v1 3 Sep 2025, v4 2 Jul 2026), arXiv:2509.03637, https://arxiv.org/abs/2509.03637
  2. Gong Chen and Abdon Moutinho, Dispersive analysis for one-dimensional charge transfer models, preprint (v1 May 2025, v2 Sep 2025), arXiv:2505.00867, https://arxiv.org/abs/2505.00867
  3. Gong Chen and Abdon Moutinho, Scattering Theory and dispersive estimates for general 11d Charge Transfer Models, preprint (v1 Oct 2025), arXiv:2510.11807, https://arxiv.org/abs/2510.11807
  4. Joachim Krieger and Wilhelm Schlag, Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension, Journal of the American Mathematical Society 19(4) (2006), 815–920, doi:10.1090/s0894-0347-06-00524-8, https://doi.org/10.1090/s0894-0347-06-00524-8
  5. Galina Perelman, Asymptotic Stability of Multi-soliton Solutions for Nonlinear Schrödinger Equations, Communications in Partial Differential Equations 29(7-8) (2004), 1051–1095, doi:10.1081/pde-200033754, https://doi.org/10.1081/pde-200033754
  6. Galina Perelman, Some results on the scattering of weakly interacting solitons for nonlinear Schrödinger equations, in: Spectral Theory, Microlocal Analysis, Singular Manifolds (Advances in Partial Differential Equations, Mathematical Topics 14), Akademie Verlag, Berlin, 1997, pp. 78–137, zbMATH Zbl 0931.35164, https://zbmath.org/1302118
  7. Raphaël Côte, Yvan Martel, and Frank Merle, Construction of multi-soliton solutions for the L2L^2-supercritical gKdV and NLS equations, Revista Matemática Iberoamericana 27(1) (2011), 273–302, doi:10.4171/rmi/636, https://doi.org/10.4171/rmi/636
  8. Gong Chen and Jacek Jendrej, Asymptotic Stability and Classification of Multi-solitons for Klein–Gordon Equations, Communications in Mathematical Physics 405(1), Article 7 (2024), doi:10.1007/s00220-023-04904-5, https://doi.org/10.1007/s00220-023-04904-5
  9. Yvan Martel, Frank Merle, and Tai-Peng Tsai, Stability in H1H^1 of the sum of KK solitary waves for some nonlinear Schrödinger equations, Duke Mathematical Journal 133(3) (2006), 405–466, doi:10.1215/s0012-7094-06-13331-8, https://doi.org/10.1215/s0012-7094-06-13331-8
  10. Abdon Moutinho, Collision of two solitons for 1d1d Nonlinear Schrodinger Equation with the same mass, preprint (v1 Mar 2024, v4 May 2025), arXiv:2403.12336, https://arxiv.org/abs/2403.12336