Nonlinear Classical and Quantum Integrable Systems with PT-symmetries
- URL: http://arxiv.org/abs/2201.00089v1
- Date: Sat, 1 Jan 2022 01:50:53 GMT
- Title: Nonlinear Classical and Quantum Integrable Systems with PT-symmetries
- Authors: Julia Cen
- Abstract summary: Key feature of integrable systems is that they can be solved to obtain exact analytical solutions.
We show how new models can be constructed through generalisations of some well known nonlinear partial differential equations with PT-symmetries.
We develop new methods from well-known ones to obtain exact analytical soliton solutions for these new systems.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: A key feature of integrable systems is that they can be solved to obtain
exact analytical solutions. We show how new models can be constructed through
generalisations of some well known nonlinear partial differential equations
with PT-symmetries whilst preserving integrability. Subsequently, we develop
new methods from well-known ones to obtain exact analytical soliton solutions
for these new systems. The first PT-symmetric generalization we take are
extensions to the complex and multicomplex fields. In agreement with the
reality property present in PT-symmetric non-Hermitian quantum systems, we find
PT-symmetries also play a key role in the reality of conserved charges here. We
then extend our investigations to explore degenerate multi-soliton solutions
for the sine-Gordon and Hirota equations. In particular, we find the usual
time-delays from degenerate soliton solution scattering are time-dependent,
unlike the non-degenerate multi-soliton solutions, and provide a universal
formula to compute the exact time-delay values for the scattering of N-soliton
solutions. Other PT-symmetric extensions of integrable systems we take are of
nonlocal nature, with nonlocalities in space and/or in time, of time crystal
type. Whilst developing new methods for the construction of soliton solutions
for these systems, we find new types of solutions with different parameter
dependence and qualitative behaviour even in the one-soliton solution cases. We
exploit gauge equivalence between the Hirota system with continuous Heisenberg
and Landau-Lifschitz systems to see how nonlocality is inherited from one
system to another and vice versa. Extending investigations to the quantum
regime, we generalize the scheme of Darboux transformations for fully
time-dependent non-Hermitian quantum systems, which allows us to create an
infinite tower of solvable models.
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