M Kdv Solitons on the Background of Quasi-Periodic Finite-Gap Solutions

Using commutation methods, the authors present a general formalism to construct Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) $N$-soliton solutions relative to arbitrary (m)KdV background solutions. As an illustration of these techniques, the authors combine them with algebro-geometric methods and Hirota's $\tau$-function approach to systematically derive the (m)KdV $N$-soliton solutions on quasi-periodic finite-gap backgrounds.