Establish existence of an interior maximizer for the dual NIE functional to obtain unconditional existence of traveling-wave solutions

Establish the existence, for any interval length L ∈ (0, ∞] and background parameter u_* ∈ ℝ, of a maximizer ν ∈ L^2(Ω) (Ω = (-L, L)) of the concave dual functional [ν] = ∫_{-L}^L ( - 1/2 · (a \bar w(x) − ν(x) − u_* Kν(x))^2 / (a + Kν(x)) ) dx such that the strict convexity condition a + Kν(x) > 0 holds on Ω, where Kν(x) = ∫_{x-1}^{x+1} ν(y) dy and a > 0. Demonstrate that such a maximizer lies in the interior of the domain of [ν], thereby yielding an unconditional existence result for smooth solutions of the nonlinear integral equation w + u_* K w + (1/2) K(w^2) = 0 (equivalently, for traveling-wave profiles of the semi-discrete Burgers system).

Background

In the dual variational formulation for the NIE, traveling-wave profiles are obtained as Euler–Lagrange equations at maximizers of a concave functional [ν], provided the strict convexity condition a + Kν > 0 holds. The authors prove conditional existence: if a maximizer exists in the interior of the functional’s domain (where a + Kν > 0), then a smooth solution of the traveling-wave NIE follows.

However, they do not currently have a proof that such interior maximizers exist for any values of L and u_*. Establishing this would convert the conditional existence result into an unconditional one, strengthening the theoretical foundation of the dual NIE approach.