The goal of this paper was to provide a general analysis of the solutions to a higher-order p-Laplacian operator with nonlinear advection. Generally speaking, it is well known that any solution to a higher-order operator exhibits oscillations. In the present study, an advection term is introduced. This will allow us to analyze smoothing conditions in the solutions. The study of existence and uniqueness is based on a variational approach. Solutions are analyzed with an energy formulation initially discussed by Saint-Venant and extended in the works by Tikhonov and Täklind. This variational principle is supported by the definition of generalized norms under Hilbert-Sobolev spaces, enabling focus on the oscillating properties of solutions. Afterward, the paper introduces an analysis to characterize the traveling wave kind of solutions together with their characterization to understand the oscillations. Finally, a numerical exploration focuses on the smoothing conditions by the action of the nonlinear advection term. As a main finding to report: There exist a traveling wave speed ( $\lambda$) and an advection coefficient ( $c^{\ast }$) for which the profile's first minimum is almost positive, and such positivity holds beyond the first minimum.

The higher order diffusion can be understood as a generalization to the classical fickian diffusion. To account for such generalization, the Landau-Ginzburg free energy concept is applied leading to a fourth order spatial operator. This kind of diffusion induces a set of instabilities in the proximity of the critical points raising difficulties to study the convergence of Travelling Waves (TW) solutions. This paper aims at introducing a system of two species driven by a mutual interaction towards prospering and with a logistic term in their respective reactions. Previous to any analytical finding of TW solutions, the instabilities of such solutions are studied. Afterwards, the Geometric Perturbation Theory is applied to provide means to search for a linearized hyperbolic manifold in the proximity of the equilibrium points. The homotopy graphs for each of the flows to the hyperbolic manifolds are provided, so that analytical solutions can be obtained in the proximity of the critical points. Additionally, the set of eigenvalues in the homotopy graphs tend to cluster and synchronize for increasing values of the TW-speed.