Research On Traveling Wave Solutions Of Constantin-Lannes Equation

Constantin-Lannes equation,also known as moderate amplitude shallow water wave model,which describes the corresponding evolution of the free surface water waves of moderate amplitude in the shallow water regime.Constantin-Lannes equation has not only solitary waves phenomenon but also wave breaking phenomenon.Constantin-Lannes equation has very effective significance for the development of shallow water wave theory.This thesis is devoted to studying the traveling wave solutions of the Constantin-Lannes equation,including peakon solutions,classification and symmetry of traveling wave solutions.This thesis is organized as follows.(1)We study peakon solutions of the Constantin-Lannes equation.We give the definition of peakon solutions and a formulation to the weak solution of the ConstantinLannes equation.Moreover,we utilize the formulation of weak solution to prove the non-existence of peakon solutions of Constantin-Lannes equation.(2)We study the classification of traveling wave solutions of the ConstantinLannes equation.Firstly,we give an approach to the classification of all traveling waves and the results of traveling waves to the Camassa-Holm equation.Secondly,we calculate the weak form of the Constantin-Lannes equation.Finally,by applying the weak form of the Constantin-Lannes equation,according to the approach of classification of all traveling waves,we classify all traveling wave solutions of the Constantin-Lannes equation.(3)We study the symmetries of the Constantin-Lannes equation,by establishing the connection between the symmetric structure and the steady structure solutions,we prove that symmetric waves are traveling waves.