Integrability of two coupled Kadomtsev–Petviashvili equations

The integrability of two coupled KP equations is studied. The simplified Hereman form of Hirota’s bilinear method is used to examine the integrability of each coupled equation. Multiple-soliton solutions and multiple singular soliton solutions are formally derived for each coupled KdV equation.     

Resonance and asymptotic solutions of a singular perturbation problem

A singular perturbation problem for a second-order ODE witha pair of singular boundary points and a pair of interior second-orderturning points is studied. Four leading-order uniform approximationsare formally constructed, each is restricted to a region includingone critical point. The neighbouring approximations are formallymatched independently on an overlap domain, yielding an asymptoticapproximation to leading order of the general solution. Twogeneralized formulae for the singular and turning point eigenvaluesthat exhibit the resonance conditions are derived. The resonancecriteria due to the influence of every possible combinationof the critical points are investigated.     

Breaking soliton equations and negative-order breaking soliton equations of typical and higher orders

We develop breaking soliton equations and negative-order breaking soliton equations of typical and higher orders. The recursion operator of the KdV equation is used to derive these models. We establish the distinct dispersion relation for each equation. We use the simplified Hirota’s method to obtain multiple soliton solutions for each developed breaking soliton equation. We also develop generalized dispersion relations for the typical breaking soliton equations and the generalized negative-order breaking soliton equations. The results provide useful information on the dynamics of the relevant nonlinear negative-order equations.