The Application Of Inverse Scattering Transform On The (2+1) Dimensional Sawada–Kotera Equation

A burgeoning field which is focused on the study of solitons and the related issue of the construction of the solution to a wide class of the nonlinear equations has been discussed as an extremely active area of research investigation since 1950 s.The vitality of research on solitons stimulates a development of a systematic method named inverse scattering transform,which is mainly focusing on the nonlinear evolution equations.Only a certain important classes of evolution equations whose forms are capable to linearized can be solved by the method of inverse scattering transform.Those who have the capability of linearization often can be transformed into the Lax pair.In this paper,Sawada–Kotera equations are studied by the application of inverse scattering transform.To begin with the Lax pair of Sawada equations.after the observation of the one relied on spatial variables of the Lax pair,some assumptions for the regulations are constructed on the solution of Sawada equations,which leads to some bounded estimation on the corresponding eigenfunction.With the help of Fourier transform in the distribution sense,an asymptotical expansion of eigenfunction is obtained by Riemann-Lebesgue lemma,which also decides the candidate of the scattering data.Furthermore,what's from expression of eigenfunction in the integral form contributes to an estimation of the eigenfunction and scattering data.Moreover,with the application of formalistic differential operator z? and the Cauchy integral operator,we build connection between the solution of Sawada equations and the scattering data which is commonly named forward scattering transform.Finally,with asymptotical expansion and the temporal part of Lax pair,the evolution formula of the scattering data is obtained and the relation between the initial data of the solution to Sawada–Kotera equations and the one of scattering data is also constructed.