Quasi-periodic solutions of a non-autonomous KdV-mKdV equation with periodic unbounded conditions

In this paper, the non-chiral boundary of the mixed kdv-mkdv equation is transformed into a chiral boundary by the construction of auxiliary functions, and a new linear difference format is constructed for the chiral boundary problem.Based on the traditional difference format, explicit and implicit differences are used alternately to construct a class of explicit-implicit (E-I) and implicit-explicit (I-E) alternating difference formats, and the unconditional stability of the numerical solutions is proved by taking advantage of the symmetric discrete numerical advantage of this class of alternating difference formats.The exact solution of the kdv-mkdv equation and its dynamical behavior are explored in the calculations using the semi-fixed separation of variables method combined with the phase diagram method for planar dynamical systems.Various types of exact solutions of the equations are obtained under special parametric conditions, and the existence problem of isolated wave solutions of the kdvmkdv equations is analyzed in conjunction with the exact solutions of the equations.Numerical examples verify the accuracy and feasibility of the constructed differential format, indicating the existence of isolated wave solutions for the KdV-mKdV equation.