Tsunami Solitons Emerging from Superconducting Gap

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 $\mathit{\text{the}}\mathit{\text{Korteweg–de}}\mathit{\text{Vries}}\mathit{\text{(KdV)}}\mathit{\text{rocks}}$. 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 $\mathit{\text{isodispersive}}\mathit{\text{phases}}$ is introduced to characterize quasiperiodic multi-tsunami backgrounds and consider their classification.