It is demonstrated that the KdV equation with nonuniformities, ut+a(t)u+(b(x,t)u)x +c(t)uux+d(t)uxxx +e(x,t)=0, has the Painlevé property if the compatibility condition among the coefficients of it holds: bt+(a−Lc)b+bbx +dbxxx =2ah+hL(d/c2)+(dh/dt)+ce +x[2a2+aL(d3/c4)+(da/dt) +L(d/c)L(d/c2)+(d/dt)L(d/c)], where L=(d/dt)lg and h(t) is an arbitrary function of t. The auto-Bäcklund transformation and Lax pairs for this equation are obtained by truncating the Laurent expansion. Furthermore, assuming the compatibility condition, then the KdV equation with nonuniformities is transformable, via suitable variable transformations, to the standard KdV.

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In this paper, we investigate the integrability of an inhomogeneous nonlinear Schrödinger equation, which has several applications in many branches of physics, as in Bose–Einstein condensates and fiber optics. The main issue deals with Painlevé property (PP) and Liouville integrability for a nonlinear Schrödinger-type equation. Solutions of the integrable equation are obtained by means of the Darboux transformation. Finally, some applications on fiber optics and Bose–Einstein condensates are proposed (including Bose–Einstein condensates in three-dimensional in cylindrical symmetry).

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The most general Kadomtsev-Petviashvili (KP) type equation, [ tt, + ii(t,x,y)u + b(t,xa)u, + c(~,x,Y)uu~ + d(t,x,y)u,]. + k(w,y)u, = d&y), is studied and the conditions for the coefficients, in order that it owns complete integrability, are determined via a Painleve test. Finally, it is proved that the above conditions are the same as those requested for reducing the equation to the canonical form via suitable transformations.