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

Abstract

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.

Nsoliton solutions to the Bogoyavlenskii-Schiff equation and a quest for the soliton solution in (3 1) dimensions

“…The last term is the curvature term. The Painlevé test indicates that d = 1 and 2 cases are integrable but that d = 3 case has the movable branch point [18,19,20]. This is the same situation as the KdV (d = 1), the BS equation (d = 2) and new equations (d = 3).…”

Generation of Secondary Solitary Waves in the Variable‐Coefficient Korteweg–de Vries Equation

“…Generally, Equation (1) is not integrable by the inverse scattering transform method, with some exceptions when the coefficients α(s) and λ(s) satisfy some special relations [5,6]. The variable-coefficient KdV Equation (1) has been extensively studied by a combination of asymptotic methods and numerical computations.…”

A new symmetry approach to solve space–time‐dependent KdV systems

“…It is surprising to discover how much work has been invested in the study of KdV equations with explicit space-time dependence; these equations originate from shallow water problems in water with variable depth and from investigations considering non-homogeneous media [1][2][3].…”

“…However, because most of the nonlinear equations arising in physics possess coe cients depending also on the independent space and time variable, this type of equations has attracted recently much attention [1][2][3].…”