On the Cauchy problem for the Tzitzéica equation: Soliton resolution conjecture and asymptotic analysis

Abstract

We study the Cauchy problem for the Tzitzéica equation, which is an important integrable model arising in affine differential geometry and characterizing proper affine spheres. Recently, Huang, Wang and Zhu (Math. Ann. \textbf{395}, 18 (2026)) reported the long-time asymptotic result for the Tzitzéica equation for the case of the purely continuous spectrum. Inspired by their work, we conduct in-depth research on the soliton resolution conjecture and asymptotic analysis of the Tzitzéica equation. Compared with the previous work, by developing the $\bar{\partial}$-nonlinear steepest descent method, we reveal that the solution of the Tzitzéica equation exhibits three different asymptotic regions depending on $ξ:=x/t$. For the region $ξ\in(-\infty,-1]\cup[1,\infty)$, where there are no stationary phase points, we rigorously prove that the solution of the Tzitzéica equation decays algebraically to zero with error $O(t^{-1})$. In the region $ξ\in(-1,1)$, the phase function $θ(z)$ has two stationary phase points. The corresponding asymptotic approximations can be characterized with an $N$-soliton solution as well as an interaction term between soliton solutions and the dispersion term with a residual error of order $O(t^{-3/4})$. In the transition regions $\{(x,t):1-\varepsilon<|ξ|<1\}$, the corresponding asymptotic approximation can be characterized by the $N$-soliton solution together with the interaction terms between the soliton solution and the dispersion term, with a residual error of order $O(t^{-3/4})$. Our results provide a rigorous analysis of the long-time asymptotic behavior of solutions of the Tzitzéica equation in different regions and confirm the soliton resolution conjecture for the Tzitzéica equation.