TY - JOUR

T1 - Solitons in a Box-Shaped Wave Field with Noise

T2 - Perturbation Theory and Statistics

AU - Mullyadzhanov, Rustam

AU - Gelash, Andrey

N1 - Funding Information:
The first part of the Letter was supported by the Russian Science Foundation Grant No. 19-79-10225 (R. M. for derivation of the perturbation framework). The second part was supported by the Russian Science Foundation Grant No. 20-71-00022 (A. G. for the work on the noise-induced effects). The numerical code was developed under the state contract with the Institute of Thermophysics SB RAS. The authors thank Dr. D. Agafontsev for fruitful discussions on virtual soliton eigenvalues. Statistical simulations were performed at the Novosibirsk Supercomputer Center (NSU).
Publisher Copyright:
© 2021 American Physical Society.

PY - 2021/6/11

Y1 - 2021/6/11

N2 - We investigate the fundamental problem of the nonlinear wave field scattering data corrections in response to a perturbation of initial condition using inverse scattering transform theory. We present a complete theoretical linear perturbation framework to evaluate first-order corrections of the full set of the scattering data within the integrable one-dimensional focusing nonlinear Schrödinger equation (NLSE). The general scattering data portrait reveals nonlinear coherent structures - solitons - playing the key role in the wave field evolution. Applying the developed theory to a classic box-shaped wave field, we solve the derived equations analytically for a single Fourier mode acting as a perturbation to the initial condition, thus, leading to the sensitivity closed-form expressions for basic soliton characteristics, i.e., the amplitude, velocity, phase, and its position. With the appropriate statistical averaging, we model the soliton noise-induced effects resulting in compact relations for standard deviations of soliton parameters. Relying on a concept of a virtual soliton eigenvalue, we derive the probability of a soliton emergence or the opposite due to noise and illustrate these theoretical predictions with direct numerical simulations of the NLSE evolution. The presented framework can be generalized to other integrable systems and wave field patterns.

AB - We investigate the fundamental problem of the nonlinear wave field scattering data corrections in response to a perturbation of initial condition using inverse scattering transform theory. We present a complete theoretical linear perturbation framework to evaluate first-order corrections of the full set of the scattering data within the integrable one-dimensional focusing nonlinear Schrödinger equation (NLSE). The general scattering data portrait reveals nonlinear coherent structures - solitons - playing the key role in the wave field evolution. Applying the developed theory to a classic box-shaped wave field, we solve the derived equations analytically for a single Fourier mode acting as a perturbation to the initial condition, thus, leading to the sensitivity closed-form expressions for basic soliton characteristics, i.e., the amplitude, velocity, phase, and its position. With the appropriate statistical averaging, we model the soliton noise-induced effects resulting in compact relations for standard deviations of soliton parameters. Relying on a concept of a virtual soliton eigenvalue, we derive the probability of a soliton emergence or the opposite due to noise and illustrate these theoretical predictions with direct numerical simulations of the NLSE evolution. The presented framework can be generalized to other integrable systems and wave field patterns.

UR - http://www.scopus.com/inward/record.url?scp=85108151712&partnerID=8YFLogxK

U2 - 10.1103/PhysRevLett.126.234101

DO - 10.1103/PhysRevLett.126.234101

M3 - Article

AN - SCOPUS:85108151712

VL - 126

JO - Physical Review Letters

JF - Physical Review Letters

SN - 0031-9007

IS - 23

M1 - 234101

ER -