Lump solutions of the fractional Kadomtsev–Petviashvili equation

• Published in: Fractional calculus & applied analysis Vol. 27; no. 1; pp. 22 - 63
• Main Authors: Borluk, Handan, Bruell, Gabriele, Nilsson, Dag
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.02.2024
• Subjects: Abstract Harmonic Analysis; Analysis; Decay of lump solutions; Existence of lump solutions; Fractional Kadomtsev-Petviashvili equation (primary); Functional Analysis; Integral Transforms; Matematik; Matematisk analys; Mathematical Analysis; Mathematics; Mathematics and Statistics; Natural Sciences; Naturvetenskap; Operational Calculus; Original Paper; Petviashvili iteration; 35B06; Fractional Kadomtsev-Petviashvili equation (primary); Decay of lump solutions; 35S30; 35Q35 (primary); 35B65; Petviashvili iteration; 45K05; 35B40; Existence of lump solutions; 35C07
• ISSN: 1311-0454; 1314-2224
• DOI: 10.1007/s13540-023-00236-2

Of concern is the fractional Kadomtsev–Petviashvili (fKP) equation and its lump solution. As in the classical Kadomtsev–Petviashvili equation, the fKP equation comes in two versions: fKP-I (strong surface tension case) and fKP-II (weak surface tension case). We prove the existence of nontrivial lump solutions for the fKP-I equation in the energy subcritical case α > 4 5 by means of variational methods. It is already known that there exist neither nontrivial lump solutions belonging to the energy space for the fKP-II equation [ 9 ] nor for the fKP-I when α ≤ 4 5 [ 26 ]. Furthermore, we show that for any α > 4 5 lump solutions for the fKP-I equation are smooth and decay quadratically at infinity. Numerical experiments are performed for the existence of lump solutions and their decay. Moreover, numerically, we observe cross-sectional symmetry of lump solutions for the fKP-I equation.