Classical-quantum simulation of single and multiple solitons generated from the KdV equation

Solitons are stable, localized solitary waves that can traverse a medium keeping their shape and speed without dissipating or diffusing. The soliton is a paramount concept in nonlinear science and it has been the subject of extensive research in physics. Experiments involving solitons have been cond...

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Bibliographic Details
Published in:Physica. D Vol. 484; p. 135012
Main Authors: Staino, Andrea, Gaitan, Frank, Basu, Biswajit
Format: Journal Article
Language:English
Published: Elsevier B.V 01.12.2025
Subjects:
Gradient-free integration
Korteweg-de Vries equation
Nonlinear waves
Quantum algorithm for PDEs
Quantum integration
Soliton collision
Solitons
Travelling waves
Korteweg-de Vries equation
Quantum algorithm for PDEs
Nonlinear waves
Travelling waves
Gradient-free integration
Solitons
Soliton collision
Quantum integration
ISSN:0167-2789
Online Access:Get full text
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Summary:Solitons are stable, localized solitary waves that can traverse a medium keeping their shape and speed without dissipating or diffusing. The soliton is a paramount concept in nonlinear science and it has been the subject of extensive research in physics. Experiments involving solitons have been conducted in multiple domains, including nonlinear optics, plasma physics, condensed matter, fluid mechanics, information coding and transmission. Designing experiments involving solitons often relies heavily on numerical simulations for predicting system behaviour and optimizing experimental parameters. These simulations require the resolution of nonlinear partial differential equations (PDEs), which are not solvable analytically in most realistic scenarios. On the other hand, numerical resolution of nonlinear PDEs can be computationally challenging, to accurately capture soliton dynamics, interaction effects and long-term stability regimes. Hence, constructing a quantum algorithm for soliton propagation that provides a computational speedup is of great interest. A recent quantum algorithm that solves nonlinear PDEs has been established in the literature. This algorithm has been proven to offer a quadratic speedup for Navier–Stokes and Burgers’ equations. In the present paper the capability of the gradient-free quantum solver to generate soliton solutions is studied. To verify the algorithm, single- and multi-soliton solutions of the well-known Korteweg–de Vries (KdV) equation are considered. First, the reliability of the quantum solver is investigated by comparing the solitary wave obtained from the numerical integration of the KdV with the corresponding analytical solution. Subsequently, the quantum-enabled emergence of solitons from different initial profiles as well as the recovery of known collision properties of classical solitons are examined. Results of numerical simulation of the quantum algorithm are compared with exact solutions and with a classical solver and excellent agreement is found. •A quantum algorithm for partial differential equations leading to soliton solutions.•Single- and multi-soliton solutions of the Korteweg–de Vries recovered.•Soliton properties such as spatio-temporal stability and collision effects preserved.
ISSN:0167-2789
DOI:10.1016/j.physd.2025.135012