On soliton clusters and collision blow up for the $L^2$-critical Hartree equation

Abstract: We consider the $L<sup>2$-critical nonlinear Hartree equation in $\mathbb{R}<sup>{1+4}$ and multisoliton solutions for which the trajectories are approximated to leading order by an $m$-body law. We obtain soliton clusters asymptotically following hyperbolic-parabolic trajectories of the corresponding $m$-body problem. By pseudo-conformal invariance, we then conclude finite-time collision blow-up with any number of clusters, each consisting of an arbitrary number of solitons, colliding simultaneously at distinct prescribed points.

• The paper constructs global multisoliton solutions with hyperbolic-parabolic trajectories, where solitons in the same cluster separate like t^{1/2} while distinct clusters separate linearly, with H¹ error o(t^{-1/2+}).
• The analysis adapts modulation, bootstrap, coercivity, and localized-energy methods by developing cluster-aware admissible functions and refined interaction estimates for the nonlocal Hartree potential.
• Applying pseudo-conformal symmetry, the paper produces finite-time blow-up solutions concentrating at arbitrarily many prescribed points with quantized masses, strong inter-soliton interactions, and gradient growth ‖∇u(t)‖₂ ~ |t|^{-1}.

The $L^2$-critical Hartree equation and multisoliton dynamics

This paper by Schmid and Wu studies the $L^2$-critical nonlinear Hartree equation in $\mathbb{R}^{1+4}$,

$$i\partial_t u + \Delta u - \phi_{|u|^2}u = 0, \qquad \phi_{|u|^2} = -|x|^{-2}\ast |u|^2,$$

a nonlocal cubic Schrödinger equation whose long-range interaction models gravitational coupling in mean-field Bose gas and boson star dynamics. The equation is locally wellposed in $H^1(\mathbb{R}^4)$, conserves mass, energy, and momentum, and admits the soliton family $u_\omega(t,x) = e^{i\omega^2 t}Q_\omega(x)$, where $Q$ is the unique positive radial ground state solving $\Delta Q - \phi_{|Q|^2}Q = Q$. Below the ground state mass $\|Q\|_{L^2}$ all solutions are global; this threshold is sharp, since the pseudo-conformal symmetry applied to $e^{it}Q$ yields the minimal-mass blow-up solution $L^2$0 with $L^2$1.

The paper's contribution extends the multisoliton construction of Gómez–Schmid–Wu (Gómez et al., 30 Jan 2025) from purely hyperbolic or parabolic trajectories to hyperbolic-parabolic trajectories, in which solitons split into clusters: within a cluster, pairwise distances grow like $L^2$2, while distinct clusters separate linearly. This mixed regime is the technically delicate case, because individual positions grow faster than intra-cluster separations.

Main results

The central theorem constructs global-in-forward-time solutions that, as $L^2$3, satisfy

$L^2$4

where $L^2$5 is any hyperbolic-parabolic solution of the $L^2$6-body law

$L^2$7

with the constraint $L^2$8 whenever $L^2$9 lie in the same cluster. Existence of such trajectories for arbitrary prescribed limiting velocities follows from a perturbative proposition gluing parabolic solutions of each cluster's sub-problem onto a hyperbolic background.

Applying the pseudo-conformal transform yields the second result: for any $\mathbb{R}^{1+4}$0 (non-parabolic collision configurations), there exists a solution blowing up at $\mathbb{R}^{1+4}$1 with

$\mathbb{R}^{1+4}$2

and with an expansion near blow-up time in which soliton centers follow $\mathbb{R}^{1+4}$3, concentrating at prescribed points with prescribed multiplicities at the pseudo-conformal rate. Notably, this realizes concentration scenarios $\mathbb{R}^{1+4}$4 with $\mathbb{R}^{1+4}$5, multiple points, and quantized masses $\mathbb{R}^{1+4}$6, via strongly interacting solitons — the trajectories are perturbed to leading order by mutual interactions. The authors point out that for the local mass-critical NLS, such multi-point pseudo-conformal-rate concentration through strong interaction remains open; collision blow-up there is known only at a single point above the pseudo-conformal rate (Martel–Raphaël).

Method

The proof follows the modulation/bootstrap framework of Krieger–Martel–Raphaël, Wu, and GSW, adapted to the cluster geometry.

Approximate solutions. Solitons are modulated by parameters $\mathbb{R}^{1+4}$7, and the Newton potential between bubbles is expanded via a Taylor series in inverse powers of the separation $\mathbb{R}^{1+4}$8. A key modification relative to GSW is a strengthened definition of admissible functions: monomials may depend on $\mathbb{R}^{1+4}$9 only through differences $$i\partial_t u + \Delta u - \phi_{|u|^2}u = 0, \qquad \phi_{|u|^2} = -|x|^{-2}\ast |u|^2,$$0 together with the combinations $$i\partial_t u + \Delta u - \phi_{|u|^2}u = 0, \qquad \phi_{|u|^2} = -|x|^{-2}\ast |u|^2,$$1, $$i\partial_t u + \Delta u - \phi_{|u|^2}u = 0, \qquad \phi_{|u|^2} = -|x|^{-2}\ast |u|^2,$$2. This compensates for the fact that in the hyperbolic-parabolic regime $$i\partial_t u + \Delta u - \phi_{|u|^2}u = 0, \qquad \phi_{|u|^2} = -|x|^{-2}\ast |u|^2,$$3 while intra-cluster distances grow only like $$i\partial_t u + \Delta u - \phi_{|u|^2}u = 0, \qquad \phi_{|u|^2} = -|x|^{-2}\ast |u|^2,$$4; under the weaker GSW definition admissible functions would fail to decay in time. An induction over the approximation order produces corrections $$i\partial_t u + \Delta u - \phi_{|u|^2}u = 0, \qquad \phi_{|u|^2} = -|x|^{-2}\ast |u|^2,$$5 and coefficients $$i\partial_t u + \Delta u - \phi_{|u|^2}u = 0, \qquad \phi_{|u|^2} = -|x|^{-2}\ast |u|^2,$$6 so that the residual error satisfies $$i\partial_t u + \Delta u - \phi_{|u|^2}u = 0, \qquad \phi_{|u|^2} = -|x|^{-2}\ast |u|^2,$$7.

Trajectories. Solving the modulation ODEs reduces to tracking the first nonzero coefficients: $$i\partial_t u + \Delta u - \phi_{|u|^2}u = 0, \qquad \phi_{|u|^2} = -|x|^{-2}\ast |u|^2,$$8 reproduces the $$i\partial_t u + \Delta u - \phi_{|u|^2}u = 0, \qquad \phi_{|u|^2} = -|x|^{-2}\ast |u|^2,$$9-body force, while the first nonzero correction to $H^1(\mathbb{R}^4)$0 appears only at order seven, with $H^1(\mathbb{R}^4)$1. A fixed-point argument in a weighted space then yields exact modulation paths $H^1(\mathbb{R}^4)$2 converging to the prescribed hyperbolic-parabolic orbit, with the refined bounds $H^1(\mathbb{R}^4)$3 and, within clusters, $H^1(\mathbb{R}^4)$4.

Bootstrap and coercivity. Orthogonality conditions against the generalized root space of the linearized operator pair $H^1(\mathbb{R}^4)$5 — spanning translations, scaling, Galilean boosts, phase, and the auxiliary function $H^1(\mathbb{R}^4)$6 — define a unique modulation path for the true solution. A bootstrap argument closes provided one controls both the modulation parameters and the $H^1(\mathbb{R}^4)$7 error. The latter uses a localized energy functional $H^1(\mathbb{R}^4)$8 combining the energy with localized mass, momentum, center, and variance terms; the $H^1(\mathbb{R}^4)$9 terms are specific to the $u_\omega(t,x) = e^{i\omega^2 t}Q_\omega(x)$0-critical dimension $u_\omega(t,x) = e^{i\omega^2 t}Q_\omega(x)$1. Coercivity $u_\omega(t,x) = e^{i\omega^2 t}Q_\omega(x)$2 follows from the finite-codimension coercivity of $u_\omega(t,x) = e^{i\omega^2 t}Q_\omega(x)$3 applied to localized, rescaled errors. A subtle point in estimating $u_\omega(t,x) = e^{i\omega^2 t}Q_\omega(x)$4 and $u_\omega(t,x) = e^{i\omega^2 t}Q_\omega(x)$5 is that cutoff derivatives are only $u_\omega(t,x) = e^{i\omega^2 t}Q_\omega(x)$6 within clusters; this is handled by replacing per-soliton weights $u_\omega(t,x) = e^{i\omega^2 t}Q_\omega(x)$7 with cluster representatives, using $u_\omega(t,x) = e^{i\omega^2 t}Q_\omega(x)$8, which holds precisely because of the hyperbolic-parabolic structure. The resulting upper bound $u_\omega(t,x) = e^{i\omega^2 t}Q_\omega(x)$9 closes the bootstrap for $Q$0 large.

Limitations and open questions

The construction requires the spectral constraint $Q$1 within each cluster, and the asymptotics are obtained only along expanding orbits as $Q$2; stability of these multisoliton states and their behavior under perturbation are not addressed. The blow-up corollary inherits the pseudo-conformal rate exclusively, leaving open whether log-log rates can be realized in strongly interacting multi-cluster scenarios for the Hartree equation. For the local mass-critical NLS, the corresponding question — whether concentration of the form $Q$3 occurs for large masses, and at which rates — remains open, and the authors note that collision blow-up there is known only in the single-point case above the pseudo-conformal rate.

Conclusion

The paper completes the classification of expansive multisoliton dynamics for the $Q$4-critical Hartree equation in four dimensions by covering the hyperbolic-parabolic regime, in which solitons organize into clusters with distinct escape velocities. Combined with pseudo-conformal symmetry, this yields finite-time blow-up solutions concentrating simultaneously at arbitrarily many prescribed points with arbitrary multiplicities, all at the pseudo-conformal rate and driven by strong inter-soliton interactions.