Tommaso Brugarino

The integration of Burgers and Korteweg-de Vries equations with nonuniformities

Abstract In this letter we demonstrate that both Burgers and Korteweg-de Vries equations with nonuniformity terms can be reduced to a Burgers or Korteweg-de Vries equation with constant coefficients if these terms satisfy a compatibility condition.

Painlevé property, auto-Bäcklund transformation, Lax pairs, and reduction to the standard form for the Korteweg−De Vries equation with nonuniformities

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 Painleve 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‐Backlund 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.

Painlevé analysis and reducibility to the canonical form for the generalized Kadomtsev–Petviashvili equation

The most general Kadomtsev–Petviashvili (KP) type equation, [ut+a(t,x,y)u+b(t,x,y) ux+c(t,x,y)uux+d(t, x,y)uxxx]x+k(t,x,y) uyy=e(t,x,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.

Integrability of an inhomogeneous nonlinear Schrödinger equation in Bose–Einstein condensates and fiber optics

In this paper, we investigate the integrability of an inhomogeneous nonlinear Schrodinger equation, which has several applications in many branches of physics, as in Bose–Einstein condensates and fiber optics. The main issue deals with Painleve property (PP) and Liouville integrability for a nonlinear Schrodinger-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).

Two-dimensional solitons in shallow water of variable depth

Abstract In this letter we study the propagation of two-dimensional solitons in shallow water of variable depth, in case a relation between wave surfaces and variable depth exists. The existence of N -solitons and cnoidal waves is proved. Finally energetic considerations are presented.