Cauchy problem for modified Korteweg-de Vries equation with piecewise- type initial data
The modified Korteweg-de Vries (mKdV) equation \(u_ t-u^ 2u_ x+u_{xxx}=0\) with \(u(x,0)=f(x)\), \(x\in R\), \(t\geq 0\) is considered where the initial data f(x) belong to a certain Banach space \(X^ s\), \(s\geq 3\). By the use of the method of pseudoparabolic regularization the existence, uniqueness and continuous dependence on initial data of the solution of the Cauchy problem for the mKdV-equation is proved. Stability of the form of the solitary wave solution belonging to \(X^ s\) at any t is proved. The existence and uniqueness of the solution of the Cauchy problem for the Benjamin-Bona-Mahony equation \(u_ t+u_ x+u^ 2u_ x-u_{xxt}=0\) is also proved.
scientific article; zbMATH DE number 3905118