Analytical solutions and dynamical behaviors of the extended Bogoyavlensky-Konopelchenko equation in deep water dynamics

Abstract In this study, we delve into the mathematical intricacies of the novel Bogoyavlensky-Konopelchenko equation, which finds practical applications in understanding the dynamics of internal waves in deep water. This equation holds significance across scientific fields such as plasma physics, nonlinear optics, and fluid dynamics. The equation extends the (2+1)-dimensional Bogoyavlensky-Konopelchenko equation by adding the second-order derivative terms <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> </mml:math> due to second-order dissipative elements. The generalized exponential rational function method, crucial in mechanical engineering, analyzes analytical solutions featuring symmetric waveform representations. The planar dynamical system, derived via Galilean transformation with mathematical models and parameter values, enhances problem comprehension. Sensitivity analysis and phase portraits of equilibrium points highlight symmetrical properties. The global analysis identifies periodic, quasi-periodic, and chaotic behaviors, corroborated by Poincaré maps, attractor, power spectrum, return map, and a symmetric basin of the largest Lyapunov exponent.