Scattering without Reflection, QM-SUSY, and the IST

Scattering without Reflection, QM-SUSY, and the Inverse Scattering Transform

In the context of solitons in integrable partial differential equations, transparent potentials emerge in two unrelated instances: first, as the Lax operators (whose eigenvalues are the integrals of motion of the PDE in question); and second, as the Bogoliubov-de-Genes (BdG) Hamiltonians (emerging from the linearization of the PDE). Transparent potentials, Lax operators, and BdG Hamiltonians form an entangled web the structure of which is yet to be understood.

While the transparency of the solitonic Lax operators is well understood, the similar assertion about solitonic BdG Hamiltonians is far from obvious. To this end, we show that the transparency of the BdG Hamiltonian is in fact a necessary condition that ensures the mutual transparency of the solitons (the sine-Gordon animation, above); the appearance of the reflection of waves in the BdG problem leads to a mutual destruction of the solitary waves upon collision (the “saw”-Gordon animation, above) ^[KHO15].

The transparency of both the Lax and the BdG Liouvillians can be analyzed using the methods of the quantum-mechanical supersymmetry (QM-SUSY)^{[KO11], [KHO15]}

We found that when one applies a nonintegrable perturbation to an integrable PDE, the notion of the “strength” of the perturbation depends dramatically on whether one works with the original equation or recasts the dynamics in Lax terms. For example, dramatic changes in the velocity of solitons can become small effects in the Lax formalism. We see this when we study the scattering of a two-soliton (a nonlinear superposition of two simple solitons that is itself an exact solution of the PDE) off a barrier. In the Lax context, the soliton velocity is a small real part of a complex number whose imaginary part, given by soliton's weight, is much larger in magnitude and, therefore, only negligibly changed in the process^[DO15].

We predicted theoretically and observed numerically that shock waves in the one-dimensional Bose gases are governed by a set of universal laws valid across the whole range of the interparticle interaction range^[DPPMO].