Soliton resolution conjecture for the cubic-quintic nonlinear Schrödinger equation on $\mathbb{R}^3$

The cubic-quintic nonlinear Schrödinger equation $i\partial_t u+\Delta u+|u|^2u-|u|^4u=0$ on $\mathbb{R}^3$ is a canonical globally well-posed, blow-up-free, non-integrable dispersive model with a rich family of stable and unstable solitons. Its static soliton theory is now essentially complete — uniqueness of energy minimizers, resolution of the Killip–Oh–Pocovnicu–Vişan monotonicity conjectures, and a sharp orbital-stability threshold — and scattering is known for data in and beyond the positive-virial region of the mass/energy plane. The remaining central question is dynamical: the soliton resolution conjecture, i.e., whether every finite-energy solution decomposes, as $t\to\pm\infty$, into finitely many decoupled modulated solitons plus a scattering radiation term. This problem is explicitly stated to be open in the source paper (arXiv:2306.13950) and reaffirmed open in its November 2025 successor (arXiv:2511.00471); no known result resolves or refutes it.

Background

The cubic-quintic nonlinear Schrödinger equation (NLS) on $\mathbb{R}^3$ is

$$i\partial_t u+\Delta u+|u|^2u-|u|^4u=0,\qquad (t,x)\in\mathbb{R}\times\mathbb{R}^3,$$

where the cubic term $|u|^2u$ is focusing (attractive, concentrating mass) and the quintic term $-|u|^4u$ is defocusing (repulsive, spreading mass); the equation models, for example, light propagation in saturating optical media and Bose–Einstein condensates with attractive two-body and repulsive three-body interactions. The defocusing quintic term is energy-critical in three dimensions (its potential-energy density $|u|^6$ scales like the kinetic-energy density $|\nabla u|^2$ under the natural dilation) and precludes finite-time blow-up, so the equation is globally well-posed in the energy space $H^1(\mathbb{R}^3)$ (Xiaoyi Zhang, 2006; restated as Theorem 1.1 of Killip, Oh, Pocovnicu, and Vişan, 2017): every finite-energy solution exists for all time and blow-up cannot occur. The conserved mass and energy are $M(u)=\int_{\mathbb{R}^3}|u|^2\,dx$ and $E(u)=\int_{\mathbb{R}^3}\bigl(\tfrac12|\nabla u|^2-\tfrac14|u|^4+\tfrac16|u|^6\bigr)\,dx$.

Standing waves, or solitons, are solutions of the form $u(t,x)=e^{i\omega t}Q_\omega(x)$ with $Q_\omega$ the unique positive, radially symmetric, decaying solution of the elliptic equation $-\Delta Q+\omega Q-|Q|^2Q+|Q|^4Q=0$; they exist exactly for frequencies $\omega\in(0,\tfrac{3}{16})$. Along this family the soliton mass diverges as $\omega$ approaches either endpoint of the interval, the energy tends to $0$ as $\omega\to0$ and to $-\infty$ as $\omega\to\tfrac{3}{16}$ (so some soliton has zero energy), and the mass attains a positive minimum at an interior frequency. Combined with the equation's phase, translation, and Galilean symmetries, these form the full soliton family (a Galilean-boosted soliton is $e^{i(v\cdot x-|v|^2t+\omega t)}Q_\omega(x-2vt-x_0)$, i.e., a soliton translated to move rigidly at velocity $2v$). A solution is said to scatter (in $H^1$) if it converges in norm, as $t\to\pm\infty$, to a solution of the free linear Schrödinger equation $i\partial_t w+\Delta w=0$ — pure dispersion with no persistent localized structure. A soliton is orbitally stable if every solution starting near it remains, for all time, close to its orbit under the symmetry group; this constrains only the shape, not the asymptotic phase. The variational and spectral theory of these solitons is now complete: Killip–Oh–Pocovnicu–Vişan (2017) posed two monotonicity conjectures about the mass curves of the solitons and of their rescalings, and Zhang–Zhu (2023, arXiv:2306.13950) together with the successor Zhang–Wang–Zhu (2025, arXiv:2511.00471) resolved both and established the sharp stability threshold: at a unique critical frequency $\omega_*$, which is the frequency of the minimal-mass soliton, the soliton $Q_\omega e^{i\omega t}$ is orbitally unstable for $\omega<\omega_*$ and orbitally stable for $\omega>\omega_*$ (the two papers attribute the endpoint $\omega=\omega_*$ differently: arXiv:2306.13950 includes it in the stable regime, arXiv:2511.00471 in the unstable regime). On the dynamical side, scattering holds for mass/energy data in the positive-virial region of the mass/energy plane and beyond it (Killip–Oh–Pocovnicu–Vişan 2017; Killip–Murphy–Vişan 2021), where the virial functional — a quantity proportional to the second time derivative of a localized moment of inertia, whose sign forces dispersion — is positive; Ardila–Murphy (2023) classified the possible dynamics on the soliton-only part of the boundary of that region, identifying a special threshold solution that scatters in one time direction and converges exponentially to a soliton in the other. Notably, the boundary is partially delineated by rescalings of solitons that are not themselves solitons, which shows an intrinsic limitation of virial-based methods. Separately, multi-soliton solutions — solutions that converge as $t\to\infty$ to a prescribed sum of widely separated solitons — can be constructed for equations of this type by the compactness methods of Martel–Merle–Tsai, Côte–Martel–Merle, and Côte–Le Coz, and such sums are orbitally stable in $H^1$; these are individual solutions consistent with the conjecture, not statements about arbitrary data.

The soliton resolution conjecture, in the general formulation of Tao's 2009 survey, predicts that for dispersive equations whose nonlinearity is too weak to cause finite-time blow-up, a solution with generic initial data should eventually resolve, as $t\to\pm\infty$, into a finite number of solitons moving apart at different rates plus a radiative term that disperses to zero. The conjecture has been proved only for equations treatable by the inverse scattering method (the exact-solution technique available for integrable dispersive equations, such as KdV and the one-dimensional cubic NLS) and, very recently, for a few special models: the one-dimensional Calogero–Moser derivative NLS (a self-dual, mass-critical equation that is integrable through a Lax pair but whose soliton resolution was proved without inverse scattering) and radial NLS with spatially localized nonlinearities in five and more dimensions under an assumed uniform $H^1$ bound. For genuinely non-integrable equations in physically natural dimensions — including the cubic-quintic NLS on $\mathbb{R}^3$ — it is open. Both the source paper arXiv:2306.13950 and its 2025 successor state explicitly that the soliton resolution conjecture for this equation is still open, citing Tao's survey; in three dimensions even asymptotic stability of a single soliton (convergence of nearby solutions to exactly one soliton plus dispersion, which is the first perturbative case of resolution) is not known, the best adjacent result being Martel's asymptotic stability of small standing waves for the one-dimensional cubic-quintic NLS. Like the general conjecture, the specialization to this equation is not a single formalized statement: pinning down the exact solution class (all data, or all data outside a precisely characterized exceptional set), the topology of convergence, and whether the decomposition holds continuously in time rather than only along sequences of times is part of the problem itself.

Problem Statement

For the cubic-quintic nonlinear Schrödinger equation

$$i\partial_t u+\Delta u+|u|^2u-|u|^4u=0\quad\text{on }\mathbb{R}\times\mathbb{R}^3,$$

prove or refute the soliton resolution conjecture: every finite-energy ($H^1(\mathbb{R}^3)$) solution — or every solution outside a precisely characterized exceptional set (characterized independently of the resolution property itself, not circularly as the data whose solutions fail to resolve, and genuinely exceptional — small in the sense that it cannot be enlarged to absorb the soliton region or any other part of the mass/energy plane where resolution is not already known), should exotic non-resolving solutions exist — admits, as $t\to+\infty$ (and analogously as $t\to-\infty$), a decomposition

$$u(t,x)=\sum_{j=1}^{J} e^{i\theta_j(t)}\,Q_{\omega_j}\bigl(x-x_j(t)\bigr)+r(t,x),$$

where $Q_\omega$ denotes the unique positive radial solution of the elliptic equation $-\Delta Q+\omega Q-|Q|^2Q+|Q|^4Q=0$, which exists exactly for frequencies $\omega\in(0,\tfrac{3}{16})$; each summand is such a soliton, possibly Galilean-boosted (multiplied by a phase $e^{iv_j\cdot x}$, so that its center moves at constant velocity $2v_j$), with phase and translation parameters modulated so that the centers of distinct summands separate at positive relative velocities as $t\to+\infty$; and the remainder $r$ scatters, i.e., there exists $r_+\in H^1(\mathbb{R}^3)$ with $\|r(t)-e^{it\Delta}r_+\|_{H^1(\mathbb{R}^3)}\to0$ as $t\to+\infty$. The endpoint cases $J=0$ (pure scattering) and $r=0$ (multi-soliton solutions) are included. A resolution must handle the full continuum of soliton frequencies, the zero-energy and rescaled solitons discovered by Killip–Oh–Pocovnicu–Vişan, and data anywhere in the mass/energy plane, including the soliton region where virial-based scattering arguments fail; and it must state explicitly the solution class and topology for which the decomposition holds, and whether it holds continuously in time or only along sequences of times, with the decomposition displayed above understood as a limit as $t\to+\infty$, so that a statement valid only along a sequence of times is the corresponding weaker sub-claim.

References

1. Jian Zhang and Shuihui Zhu, "The monotonicity conjecture and stability of solitons for the Cubic-Quintic NLS on R^3", 2023, arXiv:2306.13950, https://arxiv.org/abs/2306.13950
2. Rowan Killip, Tadahiro Oh, Oana Pocovnicu, and Monica Visan, "Solitons and Scattering for the Cubic–Quintic Nonlinear Schrödinger Equation on R^3", Archive for Rational Mechanics and Analysis 225(1) (2017) 469–548, DOI 10.1007/s00205-017-1109-0, https://doi.org/10.1007/s00205-017-1109-0 (preprint arXiv:1409.6734, https://arxiv.org/abs/1409.6734)
3. Xiaoyi Zhang, "On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations", Journal of Differential Equations 230(2) (2006) 422–445, DOI 10.1016/j.jde.2006.08.010, https://doi.org/10.1016/j.jde.2006.08.010
4. Terence Tao, "Why are solitons stable?", Bulletin of the American Mathematical Society 46(1) (2009) 1–33, DOI 10.1090/S0273-0979-08-01228-7, https://doi.org/10.1090/S0273-0979-08-01228-7
5. Rowan Killip, Jason Murphy, and Monica Visan, "Scattering for the Cubic-Quintic NLS: Crossing the Virial Threshold", SIAM Journal on Mathematical Analysis 53(5) (2021) 5803–5812, DOI 10.1137/20m1381824, https://doi.org/10.1137/20m1381824
6. Alex H. Ardila and Jason Murphy, "Threshold solutions for the 3d cubic-quintic NLS", Communications in Partial Differential Equations 48(5) (2023) 819–862, DOI 10.1080/03605302.2023.2212477, https://doi.org/10.1080/03605302.2023.2212477 (preprint arXiv:2208.08510, https://arxiv.org/abs/2208.08510)
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8. Raphaël Côte, Yvan Martel, and Frank Merle, "Construction of multi-soliton solutions for the L^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
9. Raphaël Côte and Stefan Le Coz, "High-speed excited multi-solitons in nonlinear Schrödinger equations", Journal de Mathématiques Pures et Appliquées 96(2) (2011) 135–166, DOI 10.1016/j.matpur.2011.03.004, https://doi.org/10.1016/j.matpur.2011.03.004
10. Jian Zhang, Chenglin Wang, and Shihui Zhu, "Monotonicity Conjectures and Sharp Stability for Solitons of the Cubic-Quintic NLS on R^3", 2025, arXiv:2511.00471, https://arxiv.org/abs/2511.00471
11. Avy Soffer and Xiaoxu Wu, "Soliton resolution for nonlinear Schrödinger type equations in the radial case", 2023, arXiv:2304.04245, https://arxiv.org/abs/2304.04245
12. Taegyu Kim and Soonsik Kwon, "Soliton resolution for Calogero–Moser derivative nonlinear Schrödinger equation", 2024, arXiv:2408.12843, https://arxiv.org/abs/2408.12843
13. Yvan Martel, "Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation", Inventiones mathematicae 237(3) (2024) 1253–1328, DOI 10.1007/s00222-024-01270-4, https://doi.org/10.1007/s00222-024-01270-4