Dullin-Gottwald-Holm方程尖峰孤立子附近解的稳定性

Orbital stability of solutions near the peakons for Dullin-Gottwald-Holm equation

研究Dullin-Gottwald-Holm(DGH)方程Cauchy问题在尖峰孤立子附近的解的轨道稳定性.运用伪共形变换方法,对DGH方程Cauchy问题在尖峰孤立波附近的解做如下分解:λ^(1/2)(t)u(t,y+x(t))=ε(t,y)+Q(y).通过对控制参数λ(t),x(t)的讨论,证明了余项ε(t,y)的稳定性;进一步得到了DGH方程Cauchy问题尖峰孤立波及其附近解的轨道稳定性.结果表明:若初值0与u 0在H 1意义下充分接近,则在时间T内初值对应的解仍任意接近,即(t,·+r 2)-u(t,·+r 1)H^(1)<ω,t∈[0,T).

The orbital stability of solutions around the peakons for Cauchy problem of the Dullin-Gottwald-Holm(DGH)equation is studied in this paper.Applying the method of pseudo-conformal transformation,the solution of DGH equation near the peakon is decomposed into following form:λ^(1/2)(t)u(t,y+x(t))=ε(t,y)+Q(y).By discussing the modulation parametersλ(t)and x(t),the stability of residual termε(t,y)is proved.Furthermore,the orbital stability of peakon solutions and near the peakons of Cauchy problem for DGH equation are obtained.The stability theorem indicates that,if the initial data 0 is sufficiently close to u 0 in H 1,then two corresponding solutions remain close within time T,that is(t,·+r 2)-u(t,·+r 1)H^(1)<ω,t∈[0,T).