Resumen de Qualitative Analysis and Analytical Wave Solutions for a Particular Scale-Invariant Analog of the Korteweg-de Vries Equation

Mohamed Atta, Lewa Alzaleq

• The Korteweg–de Vries (KdV) equation has long served as a foundational model in the study of nonlinear wave phenomena. Over time, researchers have developed extensions of the KdV equation to account for more complex physical settings, such as variable media, scaling effects, or external forces. One such extension is the scaleinvariant analog of the KdV (SIdV) equation, which retains some key features of the classical KdV equation–including the sech2 soliton solution–while exhibiting distinct conservation laws and nonlinear dynamics. In this study, we analyze a specific SIdV equation corresponding to the scaling parameter δ = 1 5 , representing a particular case within the broader KdV–SIdV family. This special value of δ is selected because it yields a mathematically tractable structure, enabling both a complete qualitative analysis (e.g., via Hamiltonian phase portraits) and the construction of exact traveling wave solutions—capabilities generally not possible for arbitrary non-zero δ. The analysis includes the derivation of conservation laws, a qualitative investigation through phase portraits, and a detailed classification of solution behaviors. We uncover a range of solution types, including smooth solitary waves, peakon solutions, singular waves, and periodic singular waves. Furthermore, we apply two direct analytical techniques–the tanh-method and the coth-method–to obtain both bounded and unbounded traveling wave solutions. These methods yield novel exact solutions that are not attainable through qualitative phase plane analysis alone. The results presented in this work contribute new findings to the theory of nonlinear dispersive equations and enhance the understanding of the rich solution structure of the SIdV family.