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Nonlinear dispersive PDE · energy-critical waves · soliton resolution · concentration compactness

Soliton Resolution for the Focusing Energy-Critical Wave Equation

Collaboration beta

At late times, should every bounded non-scattering wave split cleanly into free radiation and finitely many coherent solitons, continuously in time? The general nonradial full-time problem remains open.

u→(t)=u→L(t)+∑j=1J(Q→ℓjj)λj(t),xj(t)+ϵ→(t),|ϵ→(t)|H→0
Known results and sources
A dark wave field with outward cobalt radiation, four separated mint soliton cores, and a continuous late-time ribbon interrupted by one amber unresolved seam.
Soliton resolution separates radiation from finitely many coherent profiles; the unresolved general nonradial issue is convergence at every late time.

Research problem

Exact mathematical statement

For 3≤d≤5, consider

∂t2u-Δu=|u|4d-2u,supt≥0|u→(t)|H<∞.\partial_t^2u-\Delta u=|u|^{\frac4{d-2}}u, \qquad \sup_{t\ge0}\|\vec u(t)\|_{\mathcal H}<\infty.

The full-time soliton-resolution target is a decomposition

u→(t)=u→L(t)+∑j=1J(Q→ℓjj)λj(t),xj(t)+ϵ→(t),|ϵ→(t)|H→0(t→+∞),\vec u(t)=\vec u_L(t)+\sum_{j=1}^{J} \left(\vec Q_{\ell_j}^{\,j}\right)_{\lambda_j(t),x_j(t)} +\vec\varepsilon(t), \qquad \|\vec\varepsilon(t)\|_{\mathcal H}\to0 \quad(t\to+\infty),

with finitely many traveling stationary states and pairwise pseudo-orthogonal parameters. The essential quantifier is every late time, not only a sequence.

Problem infographic

Problem at a glance

Three-zone soliton-resolution explainer with the focusing wave and stationary equations, a full-time radiation-plus-solitons decomposition, a scoped radial versus nonradial status table, and the incoming-shell diagnostic.
Known radial full-time results and general sequence-of-times resolution do not settle nonradial full-time convergence. The packet's incoming shell removes one energy-only collar shortcut.

Current mathematical picture

Where work on Soliton Resolution for the Focusing Energy-Critical Wave Equation stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureEnergy-only exterior-to-hyperboloid trace

A smooth radial incoming free-wave shell has uniformly bounded horizontal energy, conformal trace squared growing like its advanced-time parameter, and a rescaling with vanishing energy but nonzero collar trace. A genuine weighted/no-incoming radiation theorem or the finite-slab characteristic current may still supply the required collar.

Route status · Narrowed route
Main reductionCurrent reduction

In the restricted d=5 compact-support, separated-ground-state sector, transform the forward cone to the unit ball. Physical solitons become shrinking critical bubbles and the equation becomes an autonomous damped gradient system. A corrected collar/bridge must first provide a full-ball phase tube; only then can calibration and a finite-length Lyapunov argument be applied.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeDerive the exact linear incoming-flux norm corresponding to the ball collar trace and its P/E amplification.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Soliton Resolution for the Focusing Energy-Critical Wave Equation in numbers

5.5kretained lines of mathematical investigation5,479 in the current working snapshot
Argument development
4,533 · 83%
Explored or eliminated routes
90 · 2%
Computational analysis
53 · 1%
Open obligations
384 · 7%
Definitions and setup
419 · 8%
9selected mapped statements1routes investigated3open questions3contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 14, 2026
Source recordedAug 14, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

13 selected steps

Scroll horizontally to explore the route

Working route overview for Soliton Resolution for the Focusing Energy-Critical Wave EquationA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Full-time, not merely sequential, soliton resolution. — Depends on missing premiseFull-time, not merelysequential, solitonresolution.Current reduction — Depends on missing premiseCurrent reductionEnergy-critical equation — Depends on missing premiseEnergy-critical equationFull-time decomposition — Depends on missing premiseFull-time decompositionRestricted dimension-five target — Depends on missing premiseRestricted dimension-fivetargetClosing target — Depends on missing premiseClosing targetDamped-ball Lyapunov identity — Depends on missing premiseDamped-ball LyapunovidentityIncoming-shell trace obstruction — Depends on missing premiseIncoming-shell traceobstructionSequential limitation — Depends on missing premiseSequential limitationEnergy-only exterior-to-hyperboloid trace — stoppedEnergy-onlyexterior-to-hyperboloidtraceDerive the exact linear incoming-flux norm corresponding to the ball collar trace and its P/E amplification. — OpenDerive the exact linearincoming-flux normcorresponding…Decide whether the canonical future radiation class excludes sparse incoming conformal tails under the actual compact-support dynamics. — OpenDecide whether the canonicalfuture radiation classexcludes…Close the finite-slab characteristic collar and compact graphwise full-phase bridge without hiding endpoint velocity or collar assumptions. — OpenClose the finite-slabcharacteristic collar andcompact…
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routeEnergy-only exterior-to-hyperboloid trace

A smooth radial incoming free-wave shell has uniformly bounded horizontal energy, conformal trace squared growing like its advanced-time parameter, and a rescaling with vanishing energy but nonzero collar trace. A genuine weighted/no-incoming radiation theorem or the finite-slab characteristic current may still supply the required collar.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Derive the exact linear incoming-flux norm corresponding to the ball collar trace and its P/E amplification.Suggested move: Start from the null stress-energy identity on pq=E², compute the weighted initial-data norm, and replay the incoming-shell stress test before proposing a theorem.
Ready to work on
02
Decide whether the canonical future radiation class excludes sparse incoming conformal tails under the actual compact-support dynamics.Suggested move: Test one fixed finite-energy free radiation state in the linear problem; if it violates the needed decay, reject the pointwise collar route and switch immediately.
Ready to work on
03
Close the finite-slab characteristic collar and compact graphwise full-phase bridge without hiding endpoint velocity or collar assumptions.Suggested move: Keep the localized quadratic incoming term explicit, seek cancellation tied to the actual radiation solution, and prove outer position and velocity trace at the collision endpoints.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen problem

The unrestricted nonradial continuous-in-time conjecture remains open in the checked source scope. Sequential general results, full-time radial results, and the 2026 restricted compact-support theorem are material advances with explicit hypotheses, not a proof of the unrestricted statement.

[1][2][3]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedFor compactly supported nonradial data in dimensions three through five, under explicit ground-state traveling-wave and distinct-velocity assumptions, a full-time-versus-collision dichotomy is proved and the…[4]
  2. Peer reviewedFull-time soliton resolution is proved for bounded radial solutions in the respective odd-dimensional and dimension-at-least-four settings of these papers.[2][3]
  3. Peer reviewedGeneral type-II solutions in dimensions three through five are resolved along a sequence of times.[1]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusSoliton resolution for the focusing energy-critical wave equation
Weaker or relaxed formresolution along a sequence of times

Sequential-in-time resolution for general type-II solutions is weaker than continuous-in-time nonradial resolution.

[1]
Solved special caseradial full-time soliton resolution

Full-time resolution is established under radial symmetry in the dimensions covered by these papers.

[2][3]
Solved special caserestricted compact-support nonradial case with at most two solitons

The 2026 paper rules out its collision alternative when there are at most two ground-state traveling solitons with distinct velocities, for compactly supported data in dimensions three through five.

[4]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetNo checked formal statement alignment for type-II solutions, radiation, or multi-soliton decomposition was located.
  • Formalization targetNo formal library support for the nonlinear dispersive PDE estimates and profile decompositions was assessed.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements4 proposed statements3 open questions1 narrowed routes
Statements by mathematical role9 selected mapped statements
  • theorem candidate1 of 91
  • reduction2 of 92
  • lemma2 of 92
  • equivalence2 of 92
  • negative result1 of 91
  • counterexample1 of 91
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 1 route included
  • retained route statementFull-time, not merely sequential, soliton resolution.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementEnergy-critical equationintermediate
  • retained route statementFull-time decompositionintermediate
  • retained route statementSequential limitationintermediate
  • retained route statementRestricted dimension-five targetintermediate
  • retained route statementDamped-ball Lyapunov identityintermediate
  • retained route statementIncoming-shell trace obstructionintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureEnergy-only exterior-to-hyperboloid tracereported failure
  • Research targetDerive the exact linear incoming-flux norm corresponding to the ball collar trace and its P/E amplification.open
  • Research targetDecide whether the canonical future radiation class excludes sparse incoming conformal tails under the actual compact-support dynamics.open
  • Research targetClose the finite-slab characteristic collar and compact graphwise full-phase bridge without hiding endpoint velocity or collar assumptions.open
  • Narrowed routeEnergy-only exterior-to-hyperboloid traceA smooth radial incoming free-wave shell has uniformly bounded horizontal energy, conformal trace squared growing like its advanced-time parameter, and a rescaling with vanishing energy but nonzero collar trace. A genuine weighted/no-incoming radiation theorem or the finite-slab characteristic current may still supply the required collar.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeDerive the exact linear incoming-flux norm corresponding to the ball collar trace and its P/E amplification.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

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Prepared starting pointDerive the exact linear incoming-flux norm corresponding to the ball collar trace and its P/E amplification.

Soliton Resolution for the Focusing Energy-Critical Wave Equation · ready to start

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Research contextPrepared context for any AI agent

At late times, should every bounded non-scattering wave split cleanly into free radiation and finitely many coherent solitons, continuously in time? The general nonradial full-time problem remains open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Soliton resolution along a sequence of times for the focusing energy critical wave equationpeer reviewed result · Thomas Duyckaerts, Hao Jia, Carlos Kenig, Frank Merle · Geometric and Functional Analysis · 2017 · ARXIV 1601.01871 · DOI 10.1007/s00039-017-0418-7 · accessed Aug 14, 2026
  2. 2
    Soliton resolution for the radial critical wave equation in all odd space dimensionspeer reviewed result · Thomas Duyckaerts, Carlos E. Kenig, Frank Merle · Acta Mathematica · 2023 · ARXIV 1912.07664 · DOI 10.4310/ACTA.2023.v230.n1.a1 · accessed Aug 14, 2026
  3. 3
    Soliton resolution for the energy-critical nonlinear wave equation in the radial casepeer reviewed result · Jacek Jendrej, Andrew Lawrie · Annals of PDE · 2023 · ARXIV 2203.09614 · DOI 10.1007/s40818-023-00159-4 · accessed Aug 14, 2026
  4. 4
    Non-radial soliton resolution for the energy critical wave equation for initial data with compact supportpeer reviewed result · Charles Collot, Thomas Duyckaerts, Carlos Kenig, Frank Merle · Discrete and Continuous Dynamical Systems · 2026-06-08 · DOI 10.3934/dcds.2026115 · accessed Aug 14, 2026

Important qualifications

  • This was a bounded source check, not a systematic literature review.
  • The general-open conclusion is an inference from the latest checked primary result's explicit restrictions and dichotomy.
  • Radial results and the 2026 compact-support result have hypotheses that must not be elided in reader copy.
  • No formalization, proof artifact, software, or independently reproduced computation was identified or executed.

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