Morphological Evolution of Higher Order Nonlinear Kinetic Alfv\'en Waves in Structured Galactic Environments

Kinetic Alfven waves (KAWs) are fundamental to energy transport and small-scale structure formation in the turbulent, magnetized interstellar medium (ISM). While first-order Korteweg--de Vries (KdV) models describe weakly nonlinear KAW solitons, they fail in strongly inhomogeneous environments where higher-order effects become significant. We investigate higher-order "dressed" kinetic Alfven (KA) solitons in a structured ISM (warm ionized medium, H II regions, stellar-wind bubbles, supernova remnants). Using a multi-component fluid model with superthermal electrons, we derive an inhomogeneous KdV-type equation with cubic nonlinearity, nonlinear-dispersive cross terms, and fifth-order dispersion. The dressed soliton has a $\operatorname{sech}^2$ core decorated by higher-order corrections. We classify soliton morphologies across the Galactic plane as a function of electron suprathermality $\kappa_e$. Five classes ($\psi_{\rm I}$--$\psi_{\rm V}$) evolve non-monotonically with $\kappa_e$: strongly suprathermal ($\kappa_e=1.6$) favour negative double-hump ($\psi_{\rm III}$); intermediate $\kappa_e$ produce layered sequences of $\psi_{\rm II}$, $\psi_{\rm I}$, $\psi_{\rm IV}$, $\psi_{\rm V}$; near-Maxwellian ($\kappa_e=3.1$) revert to KdV-like $\psi_{\rm I}$. Localised $\psi_{\rm V}$ appear as a red ring around the SWB shell and a red core inside the SNR, showing embedded structures actively generate distinct morphologies. First-order KdV theory is insufficient; dressed solitons are the natural nonlinear states. The ISM morphology selects soliton class by modulating leading vs. higher-order terms. $\psi_{\rm V}$ features link macroscopic ISM structures to kinetic-scale fluctuations, offering candidates for extreme scattering events and pulsar scintillation. The non-monotonic $\kappa_e$ dependence can constrain electron suprathermality from observations.