Large-space and large-time asymptotics of the Camassa-Holm soliton gas

We investigate the large-space and large-time asymptotic behaviors of the soliton gas associated with the Camassa-Holm equation. Utilizing the nonlinear steepest descent method, we demonstrate that the soliton gas is slowly approaching an elliptic function with constant coefficients for $y\to -\infty$. In the regime $t\to +\infty$, we establish a global large-time asymptotic description of the Camassa-Holm soliton gas. The half-plane $\left\{\right(y,t):-\infty <y<+\infty ,t>0\}$ is divided into sharply separated regions with different asymptotics. To facilitate the large-time asymptotic analysis, we construct a series of g-functions. To reconstruct the solution, we control the behavior of g-functions when $k\to \frac{i}{2}$ and $k\to \infty$. The relevant asymptotic sector exists only if the g-function exists, we categorize the values of ${\eta }_1$ and ${\eta }_2$ into two distinct cases and rigorously prove the existence of these g-functions.