Multivariable Painleve'-II equation: connection formulas for asymptotic solutions

Abstract

It is shown that a generalization of the Painlevé-II equation (P-II) to a system of coupled equations with symmetry breaking terms is integrable. A Lax pair for this system is used to relate the asymptotic behavior of the solutions at different infinities via an asymptotically exact WKB approach. The analysis relies on an exact solution of the quantum mechanical Demkov-Osherov model (DOM). An application to the problem of unstable vacuum decay during a second order phase transition provides precise scaling of the number of excitations, including subdominant contributions.

Figures (4)

• Figure 1: Numerical test of Eqs. (\ref{['rho-fin2']})-(\ref{['phi2-fin2']}). Solid curves correspond to theoretical predictions for the dependence on the equation parameter $\varepsilon\in (0.05,5)$ (a) of the action variables $I_1$ (brown) and $I_2$ (blue); (b) of the angle-related variables $\sin \phi_1$ (brown) and $\sin \phi_2$ (blue). Discrete points correspond to results of numerical simulations. Both for analytical predictions and numerical calculations the initial conditions were chosen to be $\alpha_1=0.9$, $\alpha_2=0.8$, $\varphi_1=\pi/2$, $\varphi_2=\pi/3$. Simulations were performed in the interval $x\in(-5000,5000)$ with a discretization step $dx=0.00001$. The algorithm is described in Tyagi2025. Both theory and numerical simulations agree on that $\sigma=-1$ for all points here.
• Figure 2: Numerical test of Eqs. (\ref{['sigma-fin2']})-(\ref{['phi2-fin2']}). Solid curves correspond to theoretical predictions for the dependence on the angle $\varphi_1\in (0,\pi)$ (a,b) of the action variables $I_1$ (brown) and $I_2$ (blue), respectively; (c,d) of the angle-related variables $\sin \phi_1$ (brown) and $\sin \phi_2$ (blue), respectively. Discrete points correspond to results of numerical simulations. Both for analytical predictions and numerical calculations the initial conditions were chosen to be $\varepsilon=1$, $\alpha_1=0.8$, $\alpha_2=0.6$, $\varphi_2=\pi/2$. Simulations were performed in the interval $x\in(-5000,5000)$ with a discretization step $dx=0.00001$. Inset in (b) shows the test of Eq. (\ref{['sigma-fin2']}) for the sign $\sigma$ dependence on $\varphi_1$ (purple).
• Figure 3: An integration path ${P}$ (dashed arrow) with $x_0\rightarrow -\infty$ and $t\in (-t_0,t_0)$, where $t_0\gg x_0$, is deformed into the path ${P}_{\infty}$, such that the horizontal segment of $P_{\infty}$ lies at $x_0\rightarrow +\infty$ and $t\in (-t_0,t_0)$ (dotted arrows). This deformation does not change the evolution operator in Eq. (\ref{['u-tx']}), since the initial and final points of the path remain unchanged. The vertical legs of $P_{\infty}$ have $t=\pm t_0 \rightarrow \pm\infty$, which makes the evolution along them adiabatic.
• Figure 4: (a) Eigenvalues of the Hamiltonian (\ref{['h3']}) as functions of $t=\sqrt{|x|}\tau$ at large negative $x$ (blue curves). Here, $x=-16$; $\alpha_1=\alpha_2=0.1$, $\varphi_1=\pi/3$, $\varphi_2=3\pi/4$. The values of $u_{1}(x)$ and $u_2(x)$ are approximated by asymptotic formulas in Eq. (\ref{['u1m']}). The Hamiltonian (\ref{['h3']}) is written in the basis of diabatic states. The labels $0,\,1,\,2$ define the convention for the diabatic state indices. The diabatic states coincide asymptotically with the eigenstates as $t\rightarrow \pm \infty$. The red and green arrows show two semiclassical trajectories that originate on diabatic level $0$ as $t\rightarrow -\infty$ and terminate at level 1 as $t\rightarrow +\infty$. The dashed circle encloses the behavior of the energy levels near the pseudo-time point $t=\sqrt{|x|}/2$. (b) Near $t=\sqrt{|x|}/2$, the dynamics is described by the exactly solvable DOM for three levels that cross linearly at fixed couplings $g_{1}^+$ and $g_2^+$. The diagram of diabatic levels shows only the time dependence of diagonal elements of the Hamiltonian, which in the DOM are straight lines in the time-energy plot. (c) Eigenvalues of the Hamiltonian $H(t,x)+x^{3/2}(4\tau^2+1)\hat{1}$, as functions of $t=\sqrt{x}\tau$ at large positive $x$ (blue). Here, $x=10$, $A=0.2$, $\rho=0.12$, $\varepsilon=1.2$. A term proportional to $\hat{1}$ was added to the Hamiltonian to expose the spectrum better without changing its essential features. The values of $u_{1}(x)$ and $u_2(x)$ are approximated by asymptotic formulas in Eqs. (\ref{['u1-largex']}) and (\ref{['u2-largex']}), respectively. The red arrows show the unique path that connects diabatic levels $0$ and $1$. At large positive $x$, it passes through two avoided crossing points near times $t_0$ and $t_+$.