Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions

We present the solution of the weak noise theory (WNT) for the Kardar-Parisi-Zhang equation in one dimension at short time for flat initial condition (IC). The nonlinear hydrodynamic equations of the WNT are solved analytically through a connection to the Zakharov-Shabat (ZS) system using its classical integrability. This approach is based on a recently developed Fredholm determinant framework previously applied to the droplet IC. The flat IC provides the case for a nonvanishing boundary condition of the ZS system and yields a richer solitonic structure comprising the appearance of multiple branches of the Lambert function. As a byproduct, we obtain the explicit solution of the WNT for the Brownian IC, which undergoes a dynamical phase transition. We elucidate its mechanism by showing that the related spontaneous breaking of the spatial symmetry arises from the interplay between two solitons with different rapidities.

Figure 1

Left: Plot of the coupling $g=g\left(H\right)$ of the $\{P,Q\}$ system (7) and (8) to be used to obtain $\mathrm{\Phi }\left(H\right)$ for a given $H$. It is obtained from $H=log{\mathrm{\Psi }}^{\prime }(z=-g)$. From left to right are the main branch, the first continuation, and the second continuation. The fields $H_{c1}=0$ and $H_{c2}=0.926581$ correspond to the limits of the three branches of solutions discussed in the text (with $g_{c1}=0$ and $g_{c2}=1/\sqrt{2e}$). Right: Schematic plot of the branches for $\mathrm{\Psi }\left(z\right)$ as $z=-g$ is varied, and the corresponding ranges of values for $H$. For $H<H_{c_1}=0$ one uses $\mathrm{\Psi }\left(z\right)={\mathrm{\Psi }}_0\left(z\right)$ given in Eq. (23). At $H=H_{c_1}=0$, one needs to turn around the branching point of ${\mathrm{\Psi }}_0\left(z\right)$ at $z=0$, and change the Riemann sheet. This leads to the continuation $\mathrm{\Psi }\left(z\right)={\mathrm{\Psi }}_0\left(z\right)+{\mathrm{\Delta }}_0\left(z\right)$ which, using Eqs. (24), determines $\mathrm{\Phi }\left(H\right)$ for all $H_{c1}=0<H<H_{c2}$ by decreasing $z$ from zero to $-g_{c2}$. A second continuation $\mathrm{\Psi }\left(z\right)={\mathrm{\Psi }}_0\left(z\right)+{\mathrm{\Delta }}_1\left(z\right)$ is obtained similarly by rotating around $z=-g_{c2}$, which, using Eqs. (24), determines $\mathrm{\Phi }\left(H\right)$ for all $H_{c2}<H$ by increasing $z$ from $-g_{c2}$ back to zero.

Figure 2

Left: The optimal height $h_{\mathrm{opt}}(x,t)$ for flat initial conditions plotted for various times $t$ and for two values of $H$ indicated by the black dots [one in the main branch, $H=-7.5266$ (dashed line), and the other in the third branch, $H=2.965$ (solid line)]. Right: Plot of the order parameter ${\mathrm{\Delta }}_h$ of the parity-breaking transition for the Brownian IC as a function of $H_B$ predicted here in Eq. (31), as compared to $-\mathrm{\Delta }/2$, where $\mathrm{\Delta }$ is defined and obtained numerically in Ref. [13]. Courtesy of B. Meerson for the data of the numerical solution of the WNT equations.

Figure 3

The phase $\phi \left(k\right)$ defined in Eq. (22) plotted versus $k$ for various values of $g$.

Figure 4

Plot of the function $A_t\left(x\right)$ for various positive times $t$ and coupling constants $g$ for the main and second branches.

Figure 5

Plot of the function $B_t\left(x\right)$ for various positive times $t$ and coupling constants $g$ for the main and second branches.

Figure 6

Plot of the function $A_t\left(x\right)$ for various positive times $t$ nad coupling constants $g$ for the main and second branches.

Figure 7

Plot of the function $A_t\left(x\right)$ for various positive times $t$ and coupling constants $g$ for the main and second branches.

Figure 8

The Lambert function $W$. The dashed red line corresponds to the branch $W_0$ whereas the blue line corresponds to the branch $W_{-1}$.