Abstract
We consider the negative order KdV (NKdV) hierarchy which generates nonlinear integrable equations. Selecting different seed functions produces different evolution equations. We apply the traveling wave setting to study one of these equations. Assuming a particular type of solution leads us to solve a cubic equation. New solutions are found, but none of these are classical solitary traveling wave solutions.
Highlights
We show all the possible seed functions for the negative order Korteweg-de Vries (KdV) (NKdV) hierarchy, as well as the equations these produce with the setting k = −1
Starting from the KdV spectral problem, we showed how to produce the KdV hierarchy
We gave the different evolution equations that arise after using the different possible seed functions
Summary
The study of negative order models from an integrable hierarchy becomes more and more important in seeking peakon and weak solutions. Those CH peakon solutions develop singularities in finite time, capturing the features of breaking waves [15]. Lax representations for the positive and negative KdV hierarchies [17] They studied the first member in the NKdV hierarchy and provided all possible traveling wave solutions, including soliton, kink wave, and periodic solutions. We know that the NKdV equation contains soliton solutions, including the interesting peakons It remains to be seen whether other equations in the NKdV hierarchy possess solitary traveling wave solutions.
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