2023/09/03 by Liu, Yong, Wei, Jun-cheng, Yang, Wen
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2309.01048
The KP-I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution. This is a traveling wave, and the KP-I equation in this case reduces to the Boussinesq equation. In this paper we classify the lump type solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman-Miller, we show that the lump type solutions are rational and their tau function has to be a polynomial of degree k(k+1). In particular, this implies that the lump solution is the unique ground state of the KP-I equation (as conjectured by Klein and Saut in \citeKlein0). Our result generalizes a theorem by Airault-McKean-Moser on the classification of rational solutions for the KdV equation.