The critical mass of the Townes soliton

A laser beam in a Kerr medium collapses on itself once its power exceeds a threshold set by the Townes soliton. This soliton is the ground state of the two-dimensional cubic nonlinear Schrödinger equation. In dimensionless units the threshold is the soliton's squared $L^2$ norm, $N_c \approx 11.70$, and no closed form is known for it.

Problem

Let $Q$ be the unique positive radial solution of

$$Q''(r) + \frac{1}{r} Q'(r) - Q(r) + Q(r)^3 = 0, \qquad Q'(0) = 0, \qquad Q(r) \to 0 \text{ as } r \to \infty,$$

that is, of $\Delta Q - Q + Q^3 = 0$ in $\mathbb{R}^2$. Find a closed-form expression for the critical mass

$$N_c = \int_{\mathbb{R}^2} Q^2 \, dx = 2\pi \int_0^\infty Q(r)^2 \, r \, dr = 11.70089652455965387865\ldots$$

in terms of standard constants and special functions.

Background

Chiao, Garmire and Townes found this profile in 1964 as a self-trapped optical beam [1]. Kwong proved in 1989 that the positive solution is unique [3]. Weinstein showed in 1983 that $N_c$ gives the sharp constant in the two-dimensional Gagliardo–Nirenberg inequality,

$$\|f\|_{L^4}^4 \le \frac{2}{N_c} \, \|f\|_{L^2}^2 \, \|\nabla f\|_{L^2}^2 .$$

He also proved that solutions of $i u_t + \Delta u + |u|^2 u = 0$ with $\|u_0\|_{L^2}^2 < N_c$ exist for all time [2, 4]. At mass exactly $N_c$, the pseudoconformal transform of $e^{it} Q$ blows up in finite time [4]. So $N_c$ is the collapse threshold of the focusing cubic Schrödinger equation in the plane. In optics, the critical power for self-focusing in a Kerr medium is $P_c = N_c \lambda_0^2 / (8\pi^2 n_0 n_2)$, with $\lambda_0$ the wavelength and $n_0$, $n_2$ the linear and nonlinear refractive indices [5].

What is known

• $N_c = 11.7008965\ldots$, quoted by Lushnikov and Vladimirova (2014) [5]. HorizonMath computed the reference value $11.70089652455965387865397$ by high-order Taylor integration of the ODE, matched to the decaying Bessel solution $K_0$ at large $r$.
• The existence, uniqueness and variational characterization of $Q$ are classical [2, 3]. According to the source, no closed form for $N_c$ is known.

What would count as progress

• A closed-form expression that agrees with $N_c$ to 20 significant digits, HorizonMath's acceptance threshold. Its evaluator checks this at 100-digit precision and rejects numerical shortcuts such as ODE solvers, quadrature, root-finding, infinite series and fitted constants.
• A proof that an expression equals $N_c$. $Q$ itself is not known in closed form, so such a proof would have to evaluate $\int Q^2$ without solving for $Q$ explicitly.

References

1. R. Y. Chiao, E. Garmire and C. H. Townes, "Self-trapping of optical beams", Phys. Rev. Lett. 13 (1964), 479–482. doi:10.1103/PhysRevLett.13.479
2. M. I. Weinstein, "Nonlinear Schrödinger equations and sharp interpolation estimates", Comm. Math. Phys. 87 (1983), 567–576. doi:10.1007/BF01208265
3. M. K. Kwong, "Uniqueness of positive solutions of $\Delta u - u + u^p = 0$ in $\mathbb{R}^n$", Arch. Ration. Mech. Anal. 105 (1989), 243–266. doi:10.1007/BF00251502
4. B. Dodson, "The $L^2$ sequential convergence of a solution to the mass-critical NLS above the ground state", 2021. arXiv:2101.09172
5. P. M. Lushnikov and N. Vladimirova, "Nonlinear combining of laser beams", Opt. Lett. 39 (2014), 3429–3432. doi:10.1364/OL.39.003429

Source

Imported from HorizonMath townes_soliton, curated by Erik Y. Wang, Sumeet R. Motwani et al. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.