Asymptotics for large time of global solutions to the generalized Kadomtsev-Petviashvili equation

We study the large time asymptotic behavior of solutions to the generalized Kadomtsev-Petviashvili (KP) equations { u_t + u_xxx + σ∂^-1_xu_yy = -(u^ρ)_x, (t,x,y) ∈ R × R², u(0,x,y) = u₀(x,y), (x,y) ∈ R², (KP) 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: ∥u(t)∥∞ ≤ C(1 + |t|)^-1(log(2 + |t|))^κ, ∥u_x(t)∞ ≤ C(1 + |t|)^-1 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.