Journal of East China Normal University (Natural Science) DGH. Stability of peakons for the DGH equation. CHEN Hui-ping

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1 ) Journal of East China Normal University Natural Science) No. 5 Sep : ) DGH, ) :,. DGH H 1.,,. : ; DGH ; : O29 : A Stability of peakons for the DGH equation CHEN Hui-ping Xinglin School, Nantong University, Nantong Jiangsu , China) Abstract: The peakons are peaked solitary wave solutions of a certain nolinear dispersive equation that is a model in shallow water theory. In this paper, the author showed that the peaked solitons to the DGH equation were orbital stable in H 1 norm by constructing a functional and conservation laws. The stability theorem indicates that, if a wave is close to the peakons at the beginning, it will remain close to some translate of it at any time later. Key words: stability; the DGH equation; peakons 0 KdV Camassa-Holm, Dullin [1], : { u t α 2 u xxt + c 0 u x + γu xxx + uu x = α 2 2u x u xx + uu xxx ), t > 0, x, ux, 0) = u 0 x), x, α, γ c 0, c 0 = gh > 0, h, g. Dullin, Gottwald Holm, DGH. 0.1) : : ); 08QH14006); 07SG29) :,,,. chp happy@sohu.com.

2 68 ) 2010 α = 0, DGH KDV., : ux, t), x, t. [2-4]). Bourgain [5] KDV, [6,7]., [8].,. α 0, : u t α 2 u xxt + c 0 u x c 0 α 2 u xxx + uu x = α 2 2u x u xx + uu xxx ). 0.2) DGH γ = c 0 α 2, α > 0). c > 0, ux, t) = ϕx ct) cϕ x + cα 2 ϕ xxx + c 0 ϕ x + ϕϕ x c 0 α 2 ϕ xxx = α 2 2ϕ x ϕ xx + ϕϕ xxx ). 0.) ϕ 0), 0.) cϕ + cα 2 ϕ xx + c 0 ϕ + 2 ϕ2 c 0 α 2 ϕ xx = α ϕ2 x + ϕϕ xx ). ϕ x, ϕ c + c 0 )ϕ 2 α 2 ϕ 2 x) = 0. ϕx) = Ce x α, x. ), peaked soliton =peakon). 0.2). Q = 1 α 2 2 x ) 1/2, Q 2 Q 2 f = G f = Gx y)fy)dy, f L 2 ), Gx) = 1 x e α, x., 0.2) u t α 2 u txx ) + c 0 u x α 2 u xxx ) + uu x α 2 u x u xx α 2 uu xxx ) = 2uu x α 2 u x u xx, Q 2 u t + c 0 u x + uu x ) = x u 2 + α2 2 u2 x). ) u t + u + c 0 )u x + x Q 2 u 2 + α2 2 u2 x u 2 H 1 α u2 + α 2 u 2 x)dx u C0, T, ); H 1 )) 0.2), 0 < ε < C 0, δ > 0, u0, ) cϕ H 1 α < δ, = 0. ut, ) cϕ ξt)) H 1 α < ε, t 0, T ), ξt) ut, ), C 0 = min1, α). DGH, [9].

3 5 : DGH u 0 H s ), s > 2 0.2) Kato [10,11]. Camassa-Holm. 1.1 [11] u 0 x) H s ), s > /2. T = T u 0 H s) > 0 u 0 0.2) u, u C [0, T ); H s )) C 1 [0, T ); H s 1 ) )., 0.2) Eu) = u 2 + α 2 u 2 x)dx, F u) = t.,,. 1.2 [12] 0.2). uu 2 + α 2 u 2 x)dx u 0 H 5 2 ) y 0 = u 0 u 0xx, x 0,. y 0 x) 0, x, x 0 ) y 0 x) 0, x x 0, + ), 1. [12] u 0 H ), s > 2, 1 2 u 0 H1 ). 0.2). 1.4 [12] 0.2). u 0 H ), s > 2, u 0xdx < 6 2 u 0 H1 ), u 0 H s ), s > 2., )..., ).., c = 1. c. [1],. [14],. DGH,, [15],. ϕx) = e x α H 1 ) x = 0., Eϕ) =, F ϕ) = 4 α u H 1 ) ξ, Eu) Eϕ) = u ϕ ξ) 2 H + 4αuξ) 4α. α 1

4 70 ) 2010 Eϕ) =, u ϕ ξ) 2 H = u ϕx ξ)) 2 + α 2 u α 1 x ϕ x x ξ)) 2 )dx = Eu) + Eϕ ξ)) 2 u x x)ϕ x x ξ)dx 2 ux)ϕx ξ)dx ξ = Eu) + Eϕ) u x x)ϕx ξ)dx + u x x)ϕx ξ)dx 2 ux)ϕx ξ)dx ξ = Eu) + Eϕ) 4αuξ) = Eu) Eϕ) 4αuξ) + 4α u H 1 ), Eu) [12]). 2.1 u H 1 ) Eu) M = max x ux), u. 2.1,., u H 1 ) C) Eu) = Eϕ) uξ) = M, uξ) u H 1 ), M = max x ux). F u) MEu) 2 αm. M x = ξ, { ux) αu x x), x < ξ, gx) := ux) + αu x x), x > ξ, ξ g 2 x)dx = ux) αu x x)) 2 dx + ux) + αu x x)) 2 dx ξ = u 2 x) + α 2 u 2 xx))dx αu 2 x) ξ + αu 2 x) ξ = Eu) u 2 ξ) = Eu) M ), ux)g 2 x)dx = = ξ ux)ux) αu x x)) 2 dx + ξ ux)ux) + αu x x)) 2 dx ux)u 2 x) + α 2 u 2 xx))dx 2 αu x) ξ + 2 αu x) ξ = F u) 4 αm. ux, t) M, u H 1 ). F u) 4 αm = ux)g 2 x)dx M g 2 x)dx = MEu) M.

5 5 : DGH 71 F u) MEu) 2 αm. 2. u H 1 ), u ϕ H 1 < δ, Eu) Eϕ) δ 2 ) + δ, F u) F ϕ) δ + δ + 1 ) δ ) u H 1 ),, 1 [12]. sup ux) 1 Eu)) 1 2 = 1 u H 1 x α. 2.2) Eu) Eϕ) = u H 1 α ϕ H 1 α ) u H 1 α + ϕ H 1 α ) δ2 ϕ H 1 α + δ) = δ2 + δ). 2.), F u) F ϕ) = uu 2 + α 2 u 2 x)dx ϕϕ 2 + α 2 ϕ 2 x)dx = u ϕ)u 2 + α 2 u 2 x)dx + ϕ u ϕ) 2 + α 2 u x ϕ x ) 2) dx + ϕ 2ϕu ϕ) + 2 ϕ x u x ϕ x ) ) dx u ϕ L Eu) + ϕ L u ϕ 2 H + ϕ α 1 L 2 u ϕ Hα 1 ϕ Hα 1 1 δ 2 ) δ + δ δ δ = δ + δ + 1 ) δ ) 2.). 2.4 u H 1 ), M = max x ux). δ < 1 4 C 0, Eu) Eϕ) δ2 + δ), F u) F ϕ) δ + δ + 1 ) δ 2, M 1 C 1 δ, C0 = min{1, α}, C 1 = C 1 α) = > 1, C2 1 > , F u) MEu) 2 αm. P, M MEu) + F u) ) P y) = y yeu) + F u).

6 72 ) 2010, Eϕ) = F ϕ) = 4 α, P 0 y) = y y + 2 = y 1) 2 y + 2). P 0 M) = P M) + MEu) Eϕ)) F u) F ϕ)). 2.4), P M) 0, M 1) 2 MEu) Eϕ)) F u) F ϕ)). 2.5) 4α 4α, 2.1), Eu) M 2 0, 0 < M < Eu)/ δ2 + δ) + )/ 1 + δ. 2.5), δ < 1 4 C 0 M δ ) Eu) Eϕ)) F u) F ϕ)) 4α 4α = δ δ2 + 9δ + 15 ) 4α ) = δ M 1 C 1 δ, C 2 1 α) = ) ), E F, Eut, )) = Eu 0 ) F ut, )) = F u 0 ), t ) u 0 δ = 1 9αC 1 ) 2 ε ε < C 0 δ < 1 4 C ) ut, ). 2.6) 2.1, ut, ξt)) 1 1 9α ε2, t 0, T ). 2.7) u ϕ ξ) 2 H 1 α = Eu 0) Eϕ) 4αuξ) + 4α, t 0, T ). 2.7) 2., 1 ) 2ε u ϕ ξ) 2 H ) 2ε ) α 1 9C 1 9C 1 9 ε2 ε 2, t 0, T ).. [ ] [ 1 ] DULLIN H, GOTTWALD G, HOLM D. An integrable shallow water equation with linear and nonlinear dispersion [J]. Phy ev Lett, 2001, 87: )

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