Well-posedness and dynamics of solutions to the generalized KdV with low power nonlinearity

TL;DR

This paper establishes local well-posedness for generalized KdV equations with low power nonlinearities, including the Schamel equation, and explores their solution dynamics, soliton interactions, and decay properties through numerical analysis.

Contribution

It extends well-posedness results to fractional weights and exponential decay, and provides a detailed numerical study of solution behaviors and soliton interactions in low power nonlinearities.

Findings

Well-posedness in weighted $H^1$ spaces with polynomial and exponential decay.

Numerical confirmation of well-posedness and solution behaviors.

Observation of soliton formation, radiation, and breathers in low power nonlinearities.

Abstract

We consider two types of the generalized Korteweg - de Vries equation, where the nonlinearity is given with or without absolute values, and, in particular, including the low powers of nonlinearity, an example of which is the Schamel equation. We first prove the local well-posedness of both equations in a weighted subspace of $H^1$ that includes functions with polynomial decay, extending the result of Linares et al [39] to fractional weights. We then investigate solutions numerically, confirming the well-posedness and extending it to a wider class of functions that includes exponential decay. We include a comparison of solutions to both types of equations, in particular, we investigate soliton resolution for the positive and negative data with different decay rates. Finally, we study the interaction of various solitary waves in both models, showing the formation of solitons, dispersive radiation and even breathers, all of which are easier to track in nonlinearities with lower power.