The Darboux Transformation For The Integrable Couplings Of MKdV Equations And Its Solutions

Integrable couplings(IC),which can drive many classical soliton and integrable equations,is a popular subject in the field of soliton and integrable system.Darboux transformation(DT)is always regarded as one of the most effective methods to solve the soliton equations.By iterating,a family of solutions can be obtained.So,it is one of the important topics on the research of DT for IC.In this paper,we will respectively present the determinant representations of the N-fold DT and N-transformed solutions for the in-tegrable couplings of coupled mKdV equations and the bi-integrable couplings of coupled mKdV equations.When the reduction conditions are considered,we will obtain ones of the integrable couplings of mKdV equations(ICmKdV)and the bi-integrabel couplings of mKdV equations(BICmKdV),respectively.Some soliton solutions and smooth posi-ton solutions of ICmKdV and BICmKdV will be obtained.The detailed studies are listed as follows:In the first chapter,we briefly recall the development and some main results of soliton and integrable system,IC,DT and positon solutions.In the second chapter,we first recall the construction of integrable couplings for AKNS equations(ICAKNS).Then for the second non-trivial flows of ICAKNS,we obtain the determinant representations of N-fold DT and N-transformed solutions.Second,when the reduction conditions are considered,we obtain ones of ICmKdV.Finally,starting from the zero seed solutions,some soliton solutions and smooth positon solutions of ICmKdV are obtained.In the third chapter,we construct the bi-integrable couplings for AKNS equations(BICAKNS).By a similar way,we derive the determinant representations of N-fold DT and N-transformed solutions for the bi-integrable couplings of coupled mKdV equations.Considering the similar reduction conditions,we obtain ones of BICmKdV.Starting from the zero seed solutions,some soliton solutions and smooth positon solutions of BICmKdV are obtained.Some conclusions and discussions are listed in the last chapter.