Free Particle to Complex KdV breathers through Isospectral Deformation

Abstract

The free particle in quantum mechanics in real space is endowed with supersymmetry, which enables a natural extension to complex spectra with a built-in parity (P) and time reversal (T) symmetry. It also explains the origin of unbroken and broken phases of the PT-symmetry and their relationship with the real and complex eigenvalues respectively, the latter further displaying zero-width resonances. This is possible as the extension of the eigenvalue problem to the complex plane enables the incorporation of bound and decaying states in the enlarged Hilbert space. The inherent freedom of modification of the potential without changing the spectra in supersymmetry naturally explains the connection of complex breather solutions of KdV with PT-symmetry and the free particle on the complex plane. Further, non-trivial zero-width resonances in the broken PT phase mandate a generalization that is directly connected to the sl(2, R) potential algebra.

Figures (4)

• Figure 1: Plots of $\mathbb{R}e(\tilde{V}_+(x))$ and $\mathbb{I}m(\tilde{V}_+(x))$ vs. $\xi$.
• Figure 2: $\mathbb{R}e(\tilde{V}_+(x))$ and $\mathbb{I}m(\tilde{V}_+(x))$ vs. $\eta$.
• Figure 3: KdV solutions of class I: (a) Complex breather for $\xi=aX+4a^3t$ with $\eta=2$ and (b) complex soliton for $\eta=-aX+4a^3t$ with $\xi=2$. In both cases $a=1$.
• Figure 4: Solutions of class II: (a) Complex KdV and (b) complex mKdV breathers for $\xi=aX-2a^3t$ with $\eta=2$. (c) Complex KdV and (d) complex mKdV solitons for $\eta=aX+2a^3t$ with $\xi=2$. In all cases $a=1$.