n-fold Backlund transformation and mirror surface: an unconditional estimate for an arbitrary seed of the sine-Gordon equation (Theorem with Proof)

Let \(f\) be an arbitrary \(C^∞\)-solution of the sine-Gordon equation \(fxy=sin f\) in the square window \(W=[-L,L]^2\), and let \(ω_n\) be the result of the \(n\)-fold application of the Backlund transformation with arbitrary real parameters \(a_1,,a_n>0\) and arbitrary normalization constants \(c_1,,c_n∈ R\). For each solution \(ω\) of the sine-Gordon equation, define the smooth map \(r(ω)\) of a surface of constant curvature \(K=-1\) (Chebyshev net, common initial frame at the origin; it is an immersion wherever \(sinω≠0\)). We prove the unconditional estimate\(\|r(ω_n)-r((-1)^n f)\|L^∞(W) ≤ K_n· ∑ⱼ₌₁ⁿ1{a_j}, K_n=3L^2+(1+L)·π(2T_*+π)· 3ⁿ⁻¹,\)where\(T_*:=|x|≤ L∫-LL|f_y(x,y)|\,dy<∞.\)The estimate contains no restrictions on the size of the parameters \(a_j\), on their order and ratios, on the constants \(c_j\), or on the size of the window; the constant depends only on \((L,n,T_*)\). The method is: reduction to a cascade of forced pendulum equations, an integral Lyapunov estimate of the phase distance to equilibria, and operator estimates for the orthogonal propagator of the Gauss--Weingarten frame. For the Amsler seed \(f(x,y)=Y(xy)\), we prove \(T_*≤ 2L^2\), which gives a fully explicit constant. Numerical experiments (several seeds, \(n≤3\)) illustrate the sharpness of the estimates; they are demonstrations and are not part of the proof.