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Abstract We obtain rigorous large time asymptotics for the Landau–Lifshitz (LL) equation in the soliton free case by extending the nonlinear steepest descent method to genus 1 surfaces. The methods presented in this paper pave the way to a rigorous analysis of other integrable equations on the torus and enable asymptotic analysis on different regimes of the LL equation. 

The third Painlevé equation in its generic form, often referred to as Painlevé-III($$D_6$$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x} \right)^2-\frac{1}{x} \frac{{\rm d}u}{{\rm d}x} + \frac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \qquad \alpha,\beta \in \mathbb C. $$ Starting from a generic initial solution $$u_0(x)$$ corresponding to parameters $$\alpha$$, $$\beta$$, denoted as the triple $$(u_0(x),\alpha,\beta)$$, we apply an explicit Bäcklund transformation to generate a family of solutions $$(u_n(x),\alpha + 4n,\beta + 4n)$$ indexed by $$n \in \mathbb N$$. We study the large $$n$$ behavior of the solutions $$(u_n(x), \alpha + 4n, \beta + 4n)$$ under the scaling $x = z/n$ in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution $$u_n(z/n)$$. Our main result is a proof that the limit of solutions $$u_n(z/n)$$ exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III($$D_8$$), $$ \frac{{\rm d}^2U}{{\rm d}z^2} =\frac{1}{U}\left(\frac{{\rm d}U}{{\rm d}z}\right)^2-\frac{1}{z} \frac{{\rm d}U}{{\rm d}z} + \frac{4U^2 + 4}{z}.$$ A notable application of our result is to rational solutions of Painlevé-III($$D_6$$), which are constructed using the seed solution $(1,4m,-4m)$ where $$m \in \mathbb C \setminus \big(\mathbb Z + \frac{1}{2}\big)$$ and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at $z = 0$ when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both $$D_6$$ and $$D_8$$ at $z = 0$. We also deduce the large $$n$$ behavior of the Umemura polynomials in a neighborhood of $z = 0$.