Asymptotics for Large Time of Global Solutions to the Generalized Kadomtsev–Petviashvili Equation

Abstract:

We study the large time asymptotic behavior of solutions to the generalized Kadomtsev–Petviashvili (KP) equations

where σ= 1 or σ=− 1. When ρ= 2 and σ=− 1, (KP) is known as the KPI equation, while ρ= 2, σ=+ 1 corresponds to the KPII equation. The KP equation models the propagation along the x-axis of nonlinear dispersive long waves on the surface of a fluid, when the variation along the y-axis proceeds slowly [10]. The case ρ= 3, σ=− 1 has been found in the modeling of sound waves in antiferromagnetics [15]. We prove that if ρ≥ 3 is an integer and the initial data are sufficiently small, then the solution u of (KP) satisfies the following estimates:

for all t∈R, where κ= 1 if ρ= 3 and κ= 0 if ρ≥ 4. We also find the large time asymptotics for the solution.