Qualitative behavior and impact of fractional parameter on soliton structures of the time fractional (2+1)-dimensional generalized Boussinesq equation via multi-reduction techniques

Nurzhan Serikbayev; Rajesh Kumar Gupta; Gaukhar Shaikhova; Sharmila; Nurzhan Serikbayev; Rajesh Kumar Gupta; Gaukhar Shaikhova; Sharmila

doi:10.3934/math.20261287

AIMS Mathematics

2026, Volume 11, Issue 10: 32806-32838. doi: 10.3934/math.20261287

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Research article

Qualitative behavior and impact of fractional parameter on soliton structures of the time fractional (2+1)-dimensional generalized Boussinesq equation via multi-reduction techniques

1.
Department of General and Theoretical Physics, Institute of Physical and Technical Sciences, L.N. Gumilyov Eurasian National University, Astana 010000, Kazakhstan
2.
Department of Mathematics, School of Basic Sciences, Central University of Haryana, Mahendergarh, Haryana 123031, India

Received: 14 July 2026 Revised: 27 August 2026 Accepted: 08 September 2026 Published: 10 October 2026
MSC : 39A33, 35C10, 35Q51

The fractional (2+1)-dimensional generalized Boussinesq equation, which incorporates the classical (1+1)- and (2+1)-dimensional Boussinesq and Benjamin-Ono equations as special cases, is investigated symbolically to find the infinitesimal generators and multi-reduction via invariance. The reduced equations are solved using the power series and analytical methods, and at the same time, we acquire a family of the soliton solutions, including singular, combined-singular, breather solitons, bright solitons, and two families of the analytic solutions via two different schemes. With respect to the wave field for the wave dynamics in atmospheric sciences and oceanography, our breather, kink, singular, and bright soliton solutions are closely linked to the nonlinear wave dynamics in the Boussinesq equation, including oscillating wave packets, step-like transitions, and stable localized pulses balanced by nonlinearity and dispersion. The analysis of the fractional parameter $ \beta $'s influence on soliton dynamics through 3D and 2D plots for particular values of parameters shows how $ \beta $ affects the dispersion, amplitude, and intensity of waves. Furthermore, the qualitative behavior of the previously unstudied integer-order equation is investigated through chaotic analysis using a planar dynamical system approach, which has not been explored previously. Key novelties of this work are the derivation of new symmetries, the construction of diverse analytic solutions, and the analysis of the system's dynamical behavior.
Citation: Nurzhan Serikbayev, Rajesh Kumar Gupta, Gaukhar Shaikhova, Sharmila. Qualitative behavior and impact of fractional parameter on soliton structures of the time fractional (2+1)-dimensional generalized Boussinesq equation via multi-reduction techniques[J]. AIMS Mathematics, 2026, 11(10): 32806-32838. doi: 10.3934/math.20261287

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Abstract

The fractional (2+1)-dimensional generalized Boussinesq equation, which incorporates the classical (1+1)- and (2+1)-dimensional Boussinesq and Benjamin-Ono equations as special cases, is investigated symbolically to find the infinitesimal generators and multi-reduction via invariance. The reduced equations are solved using the power series and analytical methods, and at the same time, we acquire a family of the soliton solutions, including singular, combined-singular, breather solitons, bright solitons, and two families of the analytic solutions via two different schemes. With respect to the wave field for the wave dynamics in atmospheric sciences and oceanography, our breather, kink, singular, and bright soliton solutions are closely linked to the nonlinear wave dynamics in the Boussinesq equation, including oscillating wave packets, step-like transitions, and stable localized pulses balanced by nonlinearity and dispersion. The analysis of the fractional parameter $ \beta $'s influence on soliton dynamics through 3D and 2D plots for particular values of parameters shows how $ \beta $ affects the dispersion, amplitude, and intensity of waves. Furthermore, the qualitative behavior of the previously unstudied integer-order equation is investigated through chaotic analysis using a planar dynamical system approach, which has not been explored previously. Key novelties of this work are the derivation of new symmetries, the construction of diverse analytic solutions, and the analysis of the system's dynamical behavior.

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