Tsunami Solitons Emerging from Superconducting Gap

Abstract

We propose a classical integrable system exhibiting tsunami-like solitons with a rocky-desert-like disordered stationary background. One of the Lax operators describing this system is interpretable as a Bogoliubov--de Gennes Hamiltonian in parity-mixed superconductors. The family of integrable equations is generated from this seed operator using Krichever's method, whose pure $s$-wave limit includes the coupled Schrödinger--Boussinesq hierarchy applied to plasma physics. A linearly unstable finite background with a superconducting gap supports the tsunami-soliton solution, where the propagation of the step structure turns back at a certain moment, accompanied with the oscillation on the opposite side. In addition, the equation allows inhomogeneous stationary solutions with an arbitrary number of bumps at arbitrary positions, which we term \textit{the Korteweg--de Vries (KdV) rocks}. In the Zakharov--Shabat scheme, the tsunami solitons are created from the Bogoliubov quasiparticles in the energy gap and the KdV rocks from normal electrons/holes. The unexpected large space of stationary solutions originates from the non-coprime Lax pair and the multivalued Baker--Akhiezer functions on the Riemann surface, formulated in terms of higher-rank holomorphic bundles by Krichever and Novikov. Furthermore, the concept of \textit{isodispersive phases} is introduced to characterize quasiperiodic multi-tsunami backgrounds and consider their classification.

Figures (3)

• Figure 1: (Color online) Dispersion relation $\epsilon(k)$ for plane-wave eigenfunctions $\hat{L}\phi=\epsilon\phi,\ \phi \propto e^{ikx}$, with a uniform background $(u,v,q,\xi)=(u_0,0,q_0,0)$. The red solid and blue dashed lines represent the relations with ($q_0\ne0$) and without ($q_0=0$) the $s$-wave superconducting gap, respectively. The triangle and square markers show exponentially divergent seed solutions used to construct soliton solutions.
• Figure 2: (Color online) One-tsunami solution. The background parameters are $(u_0,q_0)=(1,\frac{2}{3})$, and the soliton parameters are $(\epsilon_1;x_1,t_1,\varphi_1)=(\frac{1}{3};0,0,0)$. (a) and (b) are contour plots for $|q|^2$ and $u-2|\xi|^2$. (c) and (d) show snapshots at special $t$ for $|q|^2,|\xi|^2, u-2|\xi|^2-u_0,$ and $v$. The gif animation for (c) and (d) is available suppltakahashi_2025_16936981.
• Figure 3: (Color online) Five-soliton solution with two tsunami solitons and three KdV rocks suppltakahashi_2025_16936981. The plotted objects are the same as those in Fig. \ref{['fig:onesol']}. The parameters are $(u_0,q_0)=(1,\frac{2}{3})$ for background, $(\epsilon_1;x_1,t_1,\varphi_1)=(\frac{1}{3};0,-5,0),\ (\epsilon_2;x_2,t_2,\varphi_2)=(\frac{3}{5};-3,13,0)$ for tsunami solitons, and $(\epsilon_3;x_3,t_3,\varphi_3)=(-\frac{9}{5};-5,0,\frac{\pi}{6}),\ (\epsilon_4;x_4,t_4,\varphi_4)=(\frac{5}{2};10,0,\frac{\pi}{6}),\ (\epsilon_5;x_5,t_5,\varphi_5)=(\frac{5}{4};19,0,\frac{\pi}{6})$ for KdV rocks. The gif animation for (c) and (d) is available suppltakahashi_2025_16936981.