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LIU Yongqin WANG Weike 《武汉大学学报:自然科学英文版》2006,11(3):451-455
0 IntroductionInthis paper we consider the development of singularitiesfor strong solutionto the Cauchy problemut-utxx+3uux=2uxuxx+uuxxx,t >0, x∈Ru(0, x) =u0(x) , x∈R(1) This equation was physically derived in Refs .[1 ,2] todescribe the motion of solitary waves onshallow water ,whereu(t, x)describe thefree surface of the water above aflat bot-tom(or equivalently,thefluid velocityat ti metinthexdirec-tionin appropriate non-di mensional units ) .Equation (1) wasalso obtained and proved to b… 相似文献
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Lijun Zhang Jianming Zhang Yuzhen Bai Robert Hakl 《Journal of Applied Analysis & Computation》2019,9(5):1987-1998
The singular traveling wave solutions of a general 4-parameter family equation which unifies the Camass-Holm equation, the Degasperis-Procesi equation and the Novikov equation are investigated in this paper. At first, we obtain the explicit peakon solutions for one of its specific case that $a=(p+2)c$, $b=(p+1)c$ and $c=1$, which is referred to a generalized Camassa-Holm-Novikov (CHN) equation, by reducing it to a second-order ordinary differential equation (ODE) and solving its associated first-order integrable ODE. By observing the characteristics of peakon solutions to the CHN equation, we construct the peakon solutions for the general 4-parameter breaking wave equation. It reveals that singularities of the peakon solutions come up only when the solutions attain singular points of the equation, which might be a universal principal for all singular traveling wave solutions for wave breaking equations. 相似文献
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