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

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.