Well-Posedness of a Hirota–Satsuma System Posed on a Half Line

Abstract

This paper studies the initial boundary value problem of the Hirota–Satsuma system posed on the half line. $\left\{\begin{array}{cc}{u}_t-\alpha \left(u_{xxx}+6u{u}_x\right)=2\beta v{v}_x,& x>0,t>0,\\ v_t+v_{xxx}+3u{v}_x=0,& x>0,t>0,\\ u\left(x,0\right)=u_0\left(x\right),v\left(x,0\right)=v_0\left(x\right),& x>0,\\ u\left(0,t\right)=f\left(t\right),v\left(0,t\right)=g\left(t\right),& t>0.\end{array}\right)$ . For $-1/8<s<3/2,s\ne 1/2$ , we demonstrate that the abovementioned system is locally well-posed in $H^s\left(ℝ\right)\times H^{1+s}\left(ℝ\right)$ by utilizing several analytic boundary forcing operators.