For every speed c∈(0,√2), there exists a nonconstant finite-energy profile psi:R^2→C such that the traveling wave Psi(t,x)=psi(x-ct e_1) solves the planar Gross–Pitaevskii equation ipartial_t Psi+Delta Psi+(1-|Psi|^2)Psi=0. Equivalently, psi solves -ic,partial_x_1psi+Delta psi+(1-|psi|^2)psi=0 and has finite energy E(psi)=frac12 int_R^2|nabla psi|^2,dx+frac14 int_R^2(1-|psi|^2)^2,dx<∞.

Existence of prescribed-speed planar Gross–Pitaevskii traveling waves

For every speed $c\in(0,\sqrt{2})$ , there exists a nonconstant finite-energy profile $\psi:\mathbb{R}^2\to\mathbb{C}$ such that the traveling wave $\Psi(t,x)=\psi(x-ct e_1)$ solves the planar Gross–Pitaevskii equation $\mathrm{i}\partial_t\Psi+\Delta\Psi+(1-|\Psi|^2)\Psi=0$ . Equivalently, $\psi$ solves $-\mathrm{i}c\,\partial_{x_1}\psi+\Delta\psi+(1-|\psi|^2)\psi=0$ and has finite energy $E(\psi)=\frac12\int_{\mathbb{R}^2}|\nabla\psi|^2\,dx+\frac14\int_{\mathbb{R}^2}(1-|\psi|^2)^2\,dx<\infty$ .

References

Primary source

arXiv

Additional references

Planar Gross--Pitaevskii traveling waves at every subsonic speed — arXiv

mathematical-physics nonlinear-analysis partial-differential-equations

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

An unrefereed September 2026 preprint claims to prove planar traveling waves at every subsonic speed, but the result has not been independently verified.

The problem asks whether finite-energy planar Gross–Pitaevskii traveling waves exist at every prescribed speed $c\in(0,\sqrt{2})$ . Earlier work established only almost-every-speed existence in two dimensions, leaving the complete planar range open.

Known results

Mariş proved existence for every subsonic speed in dimensions $d\ge 3$ .
Bellazzini–Ruiz proved existence for a full-measure subset of planar speeds.
Earlier constrained-minimization work produced branches of waves in dimensions two and three, but not at every prescribed planar speed.

September 2026 claimed full-range proof

A preprint titled Planar Gross–Pitaevskii traveling waves at every subsonic speed claims existence for every $c\in(0,\sqrt{2})$ , which would settle the two-dimensional problem. The retrieved evidence is an unrefereed preprint, with no independent verification or reported response yet.

Current status (as of September 2026): A preprint claims the full planar result for $c\in(0,\sqrt{2})$ , but the claim remains unverified.

Sources

No solutions have been posted yet.