[Paper Review] On the tritronquée solutions of P$_I^2$

Тамара Грава, Klein, C.+1 more|arXiv (Cornell University)|Jun 26, 2013

Nonlinear Waves and Solitons32 references3 citations

TL;DR

This paper establishes the existence and asymptotic behavior of tritronquée solutions to the second member of the Painlevé I hierarchy, P$_\mathrm{I}^2$, using Riemann-Hilbert problem techniques. It identifies two distinct 0-parameter solutions, $U_0(x,t)$ and $V_0(x,t)$, with uniform algebraic asymptotics in specific complex sectors, proves their exponential difference on the negative real axis, and derives the large-$n$ asymptotics of formal expansion coefficients.

ABSTRACT

For equation P$_I^2$, the second member in the P$_I$ hierarchy, we prove existence of various degenerate solutions depending on the complex parameter $t$ and evaluate the asymptotics in the complex $x$ plane for $|x| o\infty$ and $t=o(x^{2/3})$. Using this result, we identify the most degenerate solutions $u^{(m)}(x,t)$, $\hat u^{(m)}(x,t)$, $m=0,...,6$, called {\em tritronquée}, describe the quasi-linear Stokes phenomenon and find the large $n$ asymptotics of the coefficients in a formal expansion of these solutions. We supplement our findings by a numerical study of the tritronquée solutions.

Motivation & Objective

• To prove the existence and uniqueness of tritronquée solutions to the P$_\mathrm{I}^2$ equation, which are globally meromorphic and exhibit quasi-stationary behavior.
• To characterize the asymptotic behavior of these solutions in the complex $x$-plane for $|x| \to \infty$ and $t = o(x^{2/3})$, identifying the sectors of uniform algebraic decay.
• To analyze the quasi-linear Stokes phenomenon and the exponentially small difference between distinct tritronquée solutions in overlapping sectors.
• To compute the large-$n$ asymptotics of the coefficients in the formal power series expansion of the tritronquée solutions.
• To provide a numerical validation of the analytical results, including the location of poles and the behavior near Stokes lines.

Proposed method

• Employing the Riemann-Hilbert problem approach to analyze the monodromy data and construct solutions to P$_\mathrm{I}^2$ via inverse scattering methods.
• Using the isomonodromic deformation theory to relate the P$_\mathrm{I}^2$ equation to a linear system with rational coefficients and monodromy preserving deformations.
• Applying the steepest descent method to the Riemann-Hilbert problem to derive the asymptotic behavior of solutions in different sectors of the complex $x$-plane.
• Deriving the large-$n$ asymptotics of the coefficients in the formal expansion of the tritronquée solutions using the connection between the Riemann-Hilbert problem and the recurrence relations of the expansion.
• Performing numerical integration of the P$_\mathrm{I}^2$ equation with asymptotic initial data on complex lines to verify the absence of poles in the predicted regular sectors.
• Using analytic continuation of the cubic root $x^{1/3}$ in specific sectors to ensure consistent branch choices in the numerical and analytical treatments.

Experimental results

Research questions

• RQ1 What are the precise sectors in the complex $x$-plane where the tritronquée solutions of P$_\mathrm{I}^2$ exhibit uniform algebraic asymptotics $u \sim -\sqrt[3]{6}\,x^{1/3}$?
• RQ2 How do the two distinct tritronquée solutions $U_0(x,t)$ and $V_0(x,t)$ differ in their asymptotic behavior and analytic continuation properties?
• RQ3 What is the nature of the quasi-linear Stokes phenomenon in the context of P$_\mathrm{I}^2$, and how does it affect the exponentially small difference between solutions?
• RQ4 What are the large-$n$ asymptotics of the coefficients in the formal power series expansion of the tritronquée solutions?
• RQ5 How do numerical simulations confirm the analytical predictions regarding the location of poles and the behavior of solutions near Stokes lines?

Key findings

• The solution $U_0(x,t)$ is proven to have uniform algebraic asymptotics $u \sim -\sqrt[3]{6}\,x^{1/3}$ in the union of two sectors: $\arg x \in \left[-\frac{3\pi}{7} - \frac{3}{7}\arctan\frac{1}{\sqrt{5}}, \frac{3\pi}{7} + \frac{3}{7}\arctan\frac{1}{\sqrt{5}}\right] \cup \left[3\pi - \frac{3}{7}\arctan\frac{1}{\sqrt{5}}, 3\pi + \frac{3}{7}\arctan\frac{1}{\sqrt{5}}\right]$.
• The solution $V_0(x,t)$ is shown to have uniform algebraic asymptotics in the sector $\arg x \in \left[3\pi - \frac{6\pi}{7} + \frac{3}{7}\arctan\frac{1}{\sqrt{5}}, 3\pi + \frac{6\pi}{7} - \frac{3}{7}\arctan\frac{1}{\sqrt{5}}\right]$, differing from $U_0$ by branch choice of $x^{1/3}$.
• The difference between $U_0(x,t)$ and $V_0(x,t)$ is exponentially small on the negative real axis, decreasing linearly in a logarithmic plot until the rounding error level.
• The large-$n$ asymptotics of the coefficients in the formal expansion of the tritronquée solutions are derived, providing quantitative control over the series' growth.
• Numerical simulations confirm the absence of poles in the predicted regular sectors and validate the transition to trigonometric behavior near the Stokes lines.
• The numerical study shows that as $t$ increases, poles of the solution approach the real axis, reducing the maximal $|b|$ for which the solution remains analytic on lines parallel to the real axis.