Dynamics of a quartic Korteweg–de Vries equation with multiple dissipations via an Abelian integral approach

We analyze both solitary and periodic wave solutions of a quartic Korteweg–de Vries (KdV) equation that incorporates multiple dissipative effects. The investigation primarily focuses on the dynamical behavior within a two-dimensional invariant manifold. To establish the existence of solitary waves, we employ the evaluation of the associated Abelian integral along a homoclinic loop, a method that offers significant insights into both their existence and stability. Furthermore, we rigorously derive periodic traveling waves by analyzing the dynamics induced by degenerate Hopf bifurcations, homoclinic bifurcations, and Poincaré bifurcations. These bifurcations play a pivotal role in identifying the conditions under which a unique periodic traveling wave arises and scenarios where two distinct periodic waves coexist. Notably, we also examine the intriguing coexistence of a solitary wave and a periodic wave. This comprehensive analysis sheds light on the intricate dynamics of the KdV equation under the influence of multiple dissipative mechanisms, enriching our understanding of its complex wave phenomena.