On integrability of a new dynamical system associated with the BBM-type hydrodynamic flow

D. Dutykh; Ya. Prykarpatsky

doi:10.1002/mma.10317

2025

On integrability of a new dynamical system associated with the BBM-type hydrodynamic flow

MMath. Methods Appl. Sci.

D. Dutykh and Ya. Prykarpatsky

Math. Methods Appl. Sci. 48(1): 107-121 (2025)

Open Access

arXiv Version DOI

Abstract

This article explores the exceptional integrability property of a family of higher-order Benjamin-Bona-Mahony (BBM)-type nonlinear dispersive equations. Here, we highlight its deep relationship with a generalized infinite hierarchy of the integrable Riemann-type hydrodynamic equations. A previous Lie symmetry analysis revealed a particular case which was conjectured to be integrable. Namely, a Lie-Baecklund symmetry exists, thus highlighting another associated dynamical system. Here, we investigate these two equations using the gradient-holonomic integrability scheme. Moreover, we construct their infinite hierarchy of conservation laws analytically, using three compatible Poisson structures to prove the complete integrability of both dynamical systems. We investigate these two equations using the so-called gradient-holonomic integrability scheme. Based on this scheme, applied to the equation under consideration, we have analytically constructed its infinite hierarchy of conservation laws, derived two compatible Poisson structures and proved its complete integrability.

Keywords

math.APpartial differential equationsgradient-holonomic integrability schemenlin.PSRiemann-type hydrodynamic equationsfluid dynamicsLie symmetry analysisPoisson structuresmath-phcomplete integrabilityLie-Baecklund symmetryBBM-type hydrodynamic flow58J70, 37J35 (primary), 76B15, 35A30, 37K35 (secondary)conservation lawsnlin.SIMathematicsnonlinear dynamicsmath.MPmathematical physicsnonlinear dispersive equations

Bibliographic record

Journal: Math. Methods Appl. Sci.

Volume: 48

Issue: 1

Pages: 107-121

DOI: 10.1002/mma.10317

arXiv: 2407.07900

BibTeX Citation

@article{Dutykh2025integrabilitydynamical,
  author = {Dutykh, D. and Prykarpatsky, Ya.},
  title = {On integrability of a new dynamical system associated with the BBM-type hydrodynamic flow},
  journal = {Math. Methods Appl. Sci.},
  year = {2025},
  volume = {48},
  number = {1},
  pages = {107--121},
  doi = {10.1002/mma.10317},
  abstract = {This article explores the exceptional integrability property of a family of higher-order Benjamin-Bona-Mahony (BBM)-type nonlinear dispersive equations. Here, we highlight its deep relationship with a generalized infinite hierarchy of the integrable Riemann-type hydrodynamic equations. A previous Lie symmetry analysis revealed a particular case which was conjectured to be integrable. Namely, a Lie-Baecklund symmetry exists, thus highlighting another associated dynamical system. Here, we investigate these two equations using the gradient-holonomic integrability scheme. Moreover, we construct their infinite hierarchy of conservation laws analytically, using three compatible Poisson structures to prove the complete integrability of both dynamical systems. We investigate these two equations using the so-called gradient-holonomic integrability scheme. Based on this scheme, applied to the equation under consideration, we have analytically constructed its infinite hierarchy of conservation laws, derived two compatible Poisson structures and proved its complete integrability.},
  keywords = {math.AP, partial differential equations, gradient-holonomic integrability scheme, nlin.PS, Riemann-type hydrodynamic equations, fluid dynamics, Lie symmetry analysis, Poisson structures, math-ph, complete integrability, Lie-Baecklund symmetry, BBM-type hydrodynamic flow, 58J70, 37J35 (primary), 76B15, 35A30, 37K35 (secondary), conservation laws, nlin.SI, Mathematics, nonlinear dynamics, math.MP, mathematical physics, nonlinear dispersive equations}
}