Symmetry analysis, dynamical behavior, and conservation laws of the dual-mode nonlinear fluid model

The study aims to analyze conservation laws and dynamics of the dual-mode Gardner equation for ideal fluid models. Lie symmetry analysis is applied to find symmetry generators, which in turn describe translation symmetries and abelian algebra. Lie theory converts the equation into a nonlinear ordinary differential equation using similarity variables. The model is transformed into a planar dynamical system via Galilean transformation, with phase portraits generated using bifurcation parameters. Runge–Kutta method is utilized to compute both super nonlinear and nonlinear wave solutions, with all solutions illustrated in the phase plane. Sensitivity and multistability analysis are conducted to examine chaotic behavior, quasiperiodic dynamics, and time series. Lyapunov characteristic exponents are discussed for chaos assessment. Numerical simulations reveal significant dynamical changes with alterations in frequencies and amplitude values. Explicit solutions are constructed via the power series method. Exploration of phase velocity and dispersion effects on the equation is done through modulation instability criteria. The multiplier scheme characterizes conserved vectors.