【作者】 刘飞；

【导师】 丁丹平；

【作者基本信息】 江苏大学 ， 数学， 2020， 硕士

【摘要】 Fornberg-Whitham方程是浅水波模型,在这类模型中,方程是否具有孤立子解和波破碎现象具有很大的研究意义,在本文中我们首先研究带有弱耗散的For-nberg-Whitham方程的局部适定性和解的爆破;其次研究色散扰动的Fornberg-Whitham方程的孤立子解,探究扰动系数对Fornberg-Whitham方程解的影响。第三章我们研究弱耗散Fornberg-Whitham方程的Cauchy问题。利用先验估计和能量不等式,根据Banach不动点定理,得到了弱耗散Fornberg-Whitham方程初值问题解的存在唯一性结果。第四章我们研究弱耗散Fornberg-Whitham方程解的爆破性质。首先运用反证法给出弱耗散Fornberg-Whitham方程解的爆破准则,其后给出爆破时初值满足的条件。最后探究了爆破速率以及爆破点集的相关性质。研究结果表明FW方程解的爆破速率FW方程是否带弱耗散项无关,但是方程解的爆破条件却受到一定的影响。第五章我们研究带扰动色散项?u_xxx的Fornberg-Whitham方程的行波解。应用动力系统分叉理论,得到了色散扰动Fornberg-Whitham方程的三种行波解,并证明光滑孤立子解和周期尖角解的极限是尖峰孤立子解。当扰动系数?趋于零时,扰动色散Fornberg-Whitham方程趋于Fornberg-Whitham方程,此时我们得到的行波解与前人的研究结果是一致的。更多还原

【Abstract】 The Fornberg-Whitham equation is a shallow water wave model,in such models,whether the equation has soliton solution and wave breaking phenomenon is of great significance.In this paper,we first study the local well-posedness and the blow up of the solutions for the Fornberg-Whitham equation with weak dissipation;Secondly,we study the soliton solution of the dispersion perturbed Fornberg-Whitham equation and the influence of the perturbed coefficient on the solution of the Fornberg-Whitham equation.In Chapter 3,the Cauchy problem of the weakly dissipative Fornberg-Whitham equation is studied.By using the prior estimation and energy inequality,the existence and uniqueness of the solution of the initial value problem are obtained according to the Banach fixed point theorem.In Chapter 4,we study the blow up phenomena of the solutions of the weakly dissipative Fornberg-Whitham equation.The blow up criteria for the solution of the weakly dissipative Fornberg-Whitham equation are given by using the method of inverse proof,and then the conditions for the initial value to be satisfied at the time of blow up are given.Finally,the blow up rate and the properties of blow up point set are discussed.The results show that the blow up rate of FW equation has nothing to do with the weak dissipation term,but the blow up condition of FW equation is affected.In Chapter 5,we study the traveling wave solutions of Fornberg-Whitham equation with perturbed dispersion?u_xxx.By using the bifurcation theory of dynamical system,the implicit smooth soliton solution,the explicit peakon and the periodic cusp wave solution are obtained,and it proves that the limit of the smooth soliton solution and the periodic cusp wave solution is the peakon.When the perturbed coefficient?tends to zero,the perturbed dispersion Fornberg-Whitham equation tends to the Fornberg-Whitham equation,the traveling wave solution obtained is consistent with the Fornberg-Whitham equation.更多还原