Multi-lump solutions of mKP-1,2 equations with integrable boundary condition via $\overline{\partial}$-dressing

Abstract

We constructed new classes of exact multi-lump solutions of mKP-1,2 equations with integrable boundary condition $u(x,y,t)\big|_{y=0}=0$ by the use of $\overline\partial$-dressing method of Zakharov and Manakov. We exactly satisfied reality and boundary conditions for the field $u(x,y,t)$ using general determinant formula for multi-lump solutions. We illustrated new calculated classes by simple examples of two-lump solutions and demonstrated how fulfilment of integrable boundary condition $u\big|_{y=0}=0$ via special nonlinear superposition of several single lumps leads to formation of certain eigenmodes for the field $u(x,y,t)$ in semiplane $y\geq0$, the analogs of standing waves on the string arising from corresponding boundary conditions at endpoints of string.