Existence of a third self-similar solution of the Yang–Mills equation in high odd dimensions

Introduction

Continuing the study initiated in [1], we consider self-similar solutions of the Yang–Mills equation in high odd spatial dimensions. Under the spherically symmetric magnetic ansatz with gauge group $\mathrm{SO}(d)$ the field equation in $(d+1)$-dimensional Minkowski spacetime reduces to

The equation is invariant under scaling $\phi(t,r)\mapsto\phi(t/\lambda,r/\lambda)$ and its conserved energy scales as $\lambda^{d-4}$, hence it is energy-supercritical for $d\ge5$.

A self-similar solution takes the form

where the profile satisfies the ODE

The aim is to construct nonconstant profiles that are smooth for $0\le y\le1$, corresponding to the causal past of the blowup point $(T,0)$. Such solutions provide explicit examples of finite-time singularity formation from smooth initial data.

A first step in the classification of self-similar profiles was taken by Cazenave, Shatah, and Tahvildar-Zadeh [2], who showed that for odd $d\ge5$ smoothness at the light cone $y=1$ imposes the compatibility condition

where $c=u(1)$ and $P_m$ is a recursively constructed polynomial of degree $m=(d-5)/2$.

In their recent paper, Bizoń, Glogić, and Wasserman [1] used a shooting argument to prove that for $m \ge3$ each root of $P_m$ in $(0,1)$ determines a unique (modulo reflection) smooth self-similar profile. Their result reduces the classification of these profiles in odd spatial dimensions $d \ge11$ to the algebraic problem of counting roots of $P_m$ in $(0,1)$.

Two distinct roots in $(0,1)$ are given by the squared endpoint values of the explicit profiles

where

BGW conjectured that there is exactly one additional root, and hence a third smooth self-similar solution. They verified this conjecture by explicit computation for odd dimensions up to $d=35$.

In this paper, we establish the existence part of the BGW conjecture by proving the following result.

The classification theorem of [1] then gives:

The proof of Proposition 1.1 has two main steps. In Section 2, we show that $P_m$ has opposite signs at 0 and 1. Factoring out the two explicit roots preserves these signs, so the remaining factor has a root in $(0,1)$. To show that this root is distinct from both explicit ones, we first give a short irreducibility argument in Section 3. This applies whenever $3(d-2)(d-4)$ is not a perfect square and, in particular, covers $3 \le m \le11$. The remaining, exceptional dimensions form a sparse sequence beginning with $d=29,365,5045,\ldots$. To handle these cases, we prove in Section 4 that the explicit roots are simple for all $m \ge12$ and hence cannot be roots of the remaining factor.

The resonance polynomial and its endpoint signs

Introducing

we rewrite equation (2) as

Inserting the formal expansion,

into (4) and setting the coefficient of each power of $(x-1)$ to zero gives the recurrence relation

This recurrence determines the coefficients $a_1(c),\ldots,a_m(c)$ successively from $a_0=c$. At $n=m$ the coefficient of $a_{m+1}$ vanishes, leaving the compatibility condition $K_m(c)=0$, where

Thus $K_m(c)=0$ is precisely the condition for continuing the Taylor expansion through the resonant order. If this condition holds, then $a_{m+1}$ remains free.

The recurrence shows that $K_m$ is an odd polynomial with rational coefficients and vanishes at $c=0,\pm1$. Hence $K_m(c)/[c(1-c^2)]$ is a polynomial in $c^2$. We define the resonance polynomial $P_m$ by

The factor $(-1)^m$ is included for later convenience. The polynomial $P_m$ agrees, up to a nonzero constant factor, with the resonance polynomial of [1].

Evaluation at $c = 0$. Differentiating (8) at $c = 0$ gives

Setting $\eta= 0$ in (9), we obtain

The factors at $n = 0$ and $n = 1$ are $2m + 3$ and $3$, respectively. The remaining $m - 1$ factors are negative, since the quadratic is decreasing on $[2,m]$ and its value at $n = 2$ is $5 - 2m < 0$. The product therefore has sign $(-1)^{m-1}$, and hence

Evaluation at $c = 1$. Differentiating (8) at $c = 1$ gives

For $\eta= 1$, the factors in (9) simplify to

giving

Thus

$\square$

A third root

The explicit solutions (3) yield the two roots $z_{\pm} = u_{\pm}(1)^2$, given by

Factoring out these roots, we write

Since $0 < z_{\pm} < 1$, the quadratic factor $Q_m$ is positive at both endpoints. Proposition 2.1 therefore gives $S_m(0) < 0 < S_m(1)$, so $S_m$ has a root in $(0,1)$ by the intermediate value theorem.

To complete the proof of Proposition 1.1, it remains to show that the root of $S_m$ is distinct from both explicit roots. We divide the proof into two cases. First, we treat the nonexceptional dimensions, in which $3(d-2)(d-4)$ is not a perfect square, using an irreducibility argument. We then handle the exceptional dimensions using simplicity of the explicit roots.

Nonexceptional dimensions

Suppose that $3(d-2)(d-4)$ is not a perfect square. Then $Q_m$ is irreducible over the field $\mathbb{Q}$ of rational numbers. Write

where $R_m$ has rational coefficients and $k\geq1$ is maximal. Neither $z_-$ nor $z_+$ is a root of $R_m$: otherwise the irreducibility of $Q_m$ over $\mathbb{Q}$ would imply that $Q_m$ divides $R_m$, contradicting the maximality of $k$. Since $Q_m(0)>0$ and $Q_m(1)>0$, Proposition 2.1 implies

Thus $R_m$ has a root in $(0,1)$ distinct from both explicit roots, proving Proposition 1.1 in every nonexceptional dimension. In particular, this covers $3\leq m\leq11$.

Exceptional dimensions

The exceptional dimensions are those for which $3(d-2)(d-4)$ is a perfect square, which is equivalent to the Pell equation

for some integer $k$. Among odd dimensions $d\geq11$, this yields the sequence $d=29,365,5045,70229,\ldots$. Thus every exceptional dimension under consideration satisfies $m\geq12$. To handle these cases, we use the following general result. Since its proof may be of independent interest, we present it separately in § 4.

Proposition 3.1. For every $m\geq12$, the two explicit roots of $P_m$ are simple, i.e. $P_m'(z_\pm)\neq0$.

It follows that $S_m(z_\pm)\neq0$ for $m\geq12$. Thus the root of $S_m$ in $(0,1)$ is distinct from both explicit roots, completing the proof of Proposition 1.1.

Simplicity of the explicit roots

We prove simplicity of the two explicit roots for $m\geq12$ by showing that the corresponding linearized resonance obstructions are nonzero. The argument uses coefficient inequalities established inductively from a three-term recurrence. The initial inequalities are verified in Appendix § 6. Throughout, we assume that $m\geq12$, set $d=2m+5$, and write $r=\sqrt{3(d-2)(d-4)}$.

Linearized resonance and simplicity

For either solution branch (1.3), the explicit profile in the coordinate $x=1/y$ is

where

We write $c_\pm=v_\pm(1)$, so that $c_\pm^2=z_\pm$ and suppress the branch subscripts on $v$, $\alpha$, and $\beta$ to avoid cluttering the notation.

Linearizing (4) about $v$ we obtain

Let

Since

this is an invertible analytic change of variable near the light cone. Using

equation (11) becomes

Its indicial roots at $s=0$ are $0$ and $m+1$. For the exponent-zero expansion, write

Substitution into (13) gives

where

The recurrence uniquely determines $b_1,\ldots,b_m$. At $n=m$, the coefficient of $b_{m+1}$ vanishes, leaving the compatibility condition $\mathcal{E}_m=0$, where

The coefficients $b_n$ and the obstruction $\mathcal{E}_m$ are understood to depend on the chosen branch.

Propagation of the coefficient inequalities by induction

We rewrite (14) as

where

The initial inequalities, verified in Appendix § 6, are $G_8 > 0$ and

For either branch,

Together with $G_8 > 0$, this implies $G_n > 0$ for $8 \le n \le m$. Hence equation (17) gives, for $8 \le n < m$,

Starting from (4.9), induction yields

Since $D_m = 0$, the definition of $G_m$ gives

Thus $\mathcal{E}_m > 0$ for the plus branch and $\mathcal{E}_m < 0$ for the minus branch. By (16), for $m \ge12$

which proves Proposition 3.1.

Remark 4.2. Simplicity of $z_+$ also follows from the absence of a zero eigenvalue for the linearized perturbations about $u_+$, established by Glogić [4]. Indeed, if the resonance obstruction vanished, both local solutions at the light cone would be analytic, so the zero-mode solution regular at the origin would also be smooth at the light cone. We have given a direct proof to keep the exposition self-contained and to treat both branches uniformly.

Open problems

Having established the existence part of the BGW conjecture, we conclude with three open problems.

Uniqueness. The remaining part of the BGW conjecture is to prove that $P_m$ has exactly three distinct roots in $(0,1)$ for every $m \ge3$. Using AI-assisted exact root-counting computations, we have verified this for odd dimensions $11 \le d \le101$, but a proof of uniqueness for all odd $d \ge11$ remains open.

Even dimensions. Although our proof applies only to odd dimensions, numerical computations suggest that the third profile persists as $d$ varies continuously, just as the two explicit profiles do. We therefore conjecture that a third smooth self-similar profile exists in every dimension $d \ge11$. In even dimensions, a local solution near the light cone generally contains a term with a non-integer power, whose coefficient must vanish for the solution to be smooth. Unlike the logarithmic coefficient in odd dimensions, this coefficient is not determined by the endpoint value but is an independent local parameter. For a solution regular at the origin, its value must be determined by continuation to the light cone. Although existence is a connection problem in both odd and even dimensions, the shooting theorem of [1] reduces the odd-dimensional case to an algebraic condition on the profile value $u(1)$ at the past light cone. No analogous reduction is available in even dimensions, where one must instead control the coefficient of the noninteger-power term along the family of solutions regular at the origin. We shall address this connection problem in a forthcoming paper.

Stability and dynamics. The rigorous nonlinear stability results [3, 4], combined with the numerical evidence [5, 6], indicate that the self-similar profile $u_{+}$ governs generic spherically symmetric blowup. The dynamical roles of $u_{-}$ and $u_{*}$, on the other hand, remain to be understood. For odd dimensions, the numerical spectral analysis reported in [1] indicates that, apart from the time-translation mode, $u_{-}$ and $u_{*}$ have two and one unstable modes, respectively, for $11 \le d \le17$, with these counts reversed for $d \ge19$. The profile with a single unstable mode is expected to act as a critical solution whose codimension-one stable manifold separates blowup from dispersion.

A first step toward understanding these dynamics is to establish the spectral properties of $u_{-}$ and $u_{*}$ rigorously. For the explicit profile $u_{-}$, the spectral problem reduces to a connection problem for the Heun equation, analogous to that studied for $u_{+}$ in [6]. Treating $d$ as a real parameter, numerical computations indicate that $u_{-}$ and $u_{*}$ meet at $d \approx17.22278$ in what appears to be a transcritical bifurcation, exchanging one unstable mode. The Heun formulation provides a route to locating the associated zero eigenvalue on the explicit branch and establishing its crossing through zero. For $u_{*}$ a different approach is required, since the profile is not known in closed form.

Verification of the initial inequalities

For $m \ge12$, we establish the initial inequalities used in § 4:

Exact arithmetic calculations in this appendix were performed using Maple.

Parametrization and coefficient formulas

Set

Then

Since

the condition $m \ge12$ gives the ranges

Substitution into (18) yields on both intervals

Starting from $b_0=1$ and $b_{-1}=0$, successive application of (14) gives

and

where

Every factor in $D(h)$, as well as $17h^2-45$, is strictly positive for $25/9\le h^2<3$.

The plus branch

Put

Direct expansion shows that both

have strictly positive coefficients as polynomials in $t$. Hence $U(h)<0$ and $V(h)>0$ on the plus-branch interval. Since $h-3<0$, equations (A.2) and (A.3) imply $b_7<0$ and $b_8-b_7<0$, hence $b_8<b_7<0$.

The minus branch.

Put $t=h-5/3$, so that $0\le t<\sqrt{3}-5/3<1/15$. Direct expansion shows that $U(5/3+t)$ has strictly positive coefficients, and hence $U(h)>0$ on the minus-branch interval.

To determine the sign of $V(h)$, write

Expansion gives

Dropping the nonpositive linear and quadratic terms and bounding the higher-order terms at $t=1/15$, we obtain, for $0\le t\le1/15$,

Thus $V(h)<0$ throughout the minus-branch interval. Equations (A.2) and (A.3), together with $h-3<0$, yield $b_7>0$ and $b_8-b_7>0$, $b_8>b_7>0$.