We use an inverse scattering approach to study multi-peakon solutions of the Degasperis–Procesi (DP) equation, an integrable PDE similar to the Camassa–Holm shallow water equation. The spectral problem associated to the DP equation is equivalent under a change of variables to what we call the cubic string problem, which is a third order non-selfadjoint generalization of the well-known equation ...

Cubic Hermite interpolation ensures continuity of derivatives between elements, so as well as providing an excellent representation of a smooth function, it also accurately models its gradients. Another attractive feature of this interpolation is that because each node is shared by neighbouring elements, a mesh of cubic Hermite elements has no more unknowns than a mesh of quadratic Lagrange ele...

A quintic nonlinear Schrödinger (NLS) equation, with derivative cubic terms, that governs the propagation of nonlinear signals in a nonlinear transmission line (NLTL) is considered. By combining a special phase-imprint transformation with a modified lens transformation, we reduce the equation under consideration to a standard cubic NLS equation with a time-varying gain/loss term and obtain the ...