Pyramidal solitons in a generalized KdV equation

Abstract: The Korteweg – de Vries (KdV) equation is a classical model for describing long gravity waves on water and has among its solutions solitons – solitary waves propagating at constant speed and interacting elastically. The report discusses a generalized KdV equation in which the nonlinear term is given by an arbitrary smooth function, modeling a broader class of problems. The form of a solitary wave type solution in this equation may differ from the classical solitons of the KdV equation; in particular, depending on the nonlinearity, the number of inflection points may increase (we call such solutions pyramidal). Pyramidal solitons can arise in modeling internal waves and in problems of plasma physics. Their stability and interactions are studied using generalizations of the Gardner equation and other equations from the generalized KdV family.

This is joint work with E. Pelinovsky.

The main results of this work were published in the articles:

1. I. Melnikov, E. Pelinovsky, Pyramidal solitons: existence, instability, and interactions, Nonlinear Dyn. (2026), 114:934.

2. I. Melnikov, Soliton solution classification in the generalized Korteweg–de Vries equation, Nonlinear Dyn. (2025), 113:32641–32648.

Russianmathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods