TY - JOUR

T1 - Asymptotics for large time of global solutions to the generalized Kadomtsev-Petviashvili equation

AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

AU - Saut, Jean Claude

PY - 1999/4

Y1 - 1999/4

N2 - We study the large time asymptotic behavior of solutions to the generalized Kadomtsev-Petviashvili (KP) equations { ut + uxxx + σ∂-1xuyy = -(uρ)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, ∥ux(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.

AB - We study the large time asymptotic behavior of solutions to the generalized Kadomtsev-Petviashvili (KP) equations { ut + uxxx + σ∂-1xuyy = -(uρ)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, ∥ux(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.

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U2 - 10.1007/s002200050569

DO - 10.1007/s002200050569

M3 - Article

AN - SCOPUS:0033243765

SN - 0010-3616

VL - 201

SP - 577

EP - 590

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -