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 \(C^\infty\)-solution of the sine-Gordon equation \(f_{xy}=\sin f\) in the square window \(W=[-L,L]^2\), and let \(\omega_n\) be the result of the \(n\)-fold application of the Backlund transformation with arbitrary real parameters \(a_1,\dots,a_n>0\) and arbitrary normalization constants \(c_1,\dots,c_n\in\mathbb R\). For each solution \(\omega\) of the sine-Gordon equation, define the smooth map \(r(\omega)\) of a surface of constant curvature \(K=-1\) (Chebyshev net, common initial frame at the origin; it is an immersion wherever \(\sin\omega\ne0\)). We prove the unconditional estimate\(\|r(\omega_n)-r((-1)^n f)\|_{L^\infty(W)} \le K_n\cdot \sum_{j=1}^{n}\frac1{a_j}, \qquad K_n=3L^2+(1+L)\cdot\pi(2T_*+\pi)\cdot 3^{n-1},\)where\(T_*:=\sup_{|x|\le L}\int_{-L}^{L}|f_y(x,y)|\,dy<\infty.\)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 \(T_*\le 2L^2\), which gives a fully explicit constant. Numerical experiments (several seeds, \(n\le3\)) illustrate the sharpness of the estimates; they are demonstrations and are not part of the proof.