Static multi-soliton solutions in the affine su(N+1) Toda models

We study some static multi-soliton configurations in the su(N + 1) Toda models. Such configurations exist for N > 1. We construct explicitly a multi-soliton solution for any N and study conditions for having such solutions. The number of static solitons is limited by the rank of the su(N + 1) Lie algebra. We give some examples of non-static multi-soliton solutions with static components.

💡 Research Summary

The paper investigates static multi‑soliton configurations in the affine su(N + 1) Toda field theories defined on a (1+1)‑dimensional Minkowski space. While the sine‑Gordon model (the su(2) case) does not admit two solitons at rest, the authors show that for any N > 1 static multi‑soliton solutions do exist.

Starting from the standard Toda action, the scalar fields φᵃ (a = 1,…,N) are assembled into a vector φ using the simple roots αᵃ and fundamental weights λᵃ. The equations of motion become
∂_μ∂^μφ = 4i ∑{j=0}^N α_j e^{iα_j·~φ}.
To solve these equations the Hirota bilinear method is employed. One introduces N + 1 τ‑functions τ_j (j = 0,…,N) and writes
φ_a = i ln(τ_a/τ_0), ~φ = i ∑{k=0}^N (2α_k/α_k²) ln τ_k.
The τ‑functions satisfy the Hirota bilinear equations
∂+∂-τ_j τ_j − (∂+τ_j)(∂-τ_j) = τ_j² − τ_{j‑1}τ_{j+1},
where the indices are understood modulo N + 1. These equations are directly linked to the extended Cartan matrix K_{jk}=2α_j·α_k/α_k².

Solving the eigenvalue problem K v^{(n)} = λ_n v^{(n)} yields
λ_n = 4 sin²