SOLUTIONS OF LOCAL AND NONLOCAL DISCRETE COMPLEX MODIFIED KORTEWEG-DE
VRIES EQUATIONS AND CONTINUUM LIMITS
Abstract
Cauchy matrix approach for the discrete Ablowitz-Kaup-Newell-Segur
equations is reconsidered, where two ‘proper’ discrete
Ablowitz-Kaup-Newell-Segur equations and two ‘unproper’ discrete
Ablowitz-Kaup-Newell-Segur equations are derived. The ‘proper’ equations
admit local reduction, while the ‘unproper’ equations admit nonlocal
reduction. By imposing the local and nonlocal complex reductions on the
obtained discrete Ablowitz-Kaup-Newell-Segur equations, two local and
nonlocal discrete complex modified Korteweg-de Vries equations are
constructed. For the obtained local and nonlocal discrete complex
modified Korteweg-de Vries equations, soliton solutions and Jordan-block
solutions are presented by solving the determining equation set. The
dynamical behaviors of 1-soliton solution are analyzed and illustrated.
Continuum limits of the resulting local and nonlocal discrete complex
modified Korteweg-de Vries equations are discussed.