On solitons in media modelled by the hierarchical KdV equation

On a hierarchy of nonlinearly dispersive generalized Korteg - de Vries evolution equations

“…dominates the behavior of U ′′ as U → U − max (ξ → 0) and, since the compact solutions derived hold only for m > 1 and n > 0, |U ′′ | → ∞ as U → U − max (ξ → 0). The presence of an infinite second derivative at ξ = 0 means that the compact waveforms constructed are peaked like the (non-compact) peakon solutions of the Camassa-Holm equation [9], hence we refer to the family of solutions given by Eqs ( 17)- (18) by the portmanteau peakompactons. The possibility of an infinite second derivative as ξ → {0, ±ξ 0 } implies that we have constructed pseudo-classical solutions of Eq.…”

“…Additionally, in modeling various physical phenomena beyond long waves over shallow water, many generalizations and modifications of the KdV equation have been derived over the years, including but not limited to the modified KdV equation [4,6], the KdV-Burgers equation [5], the cylindrical KdV equation [7], the extended KdV equation [8], the Camassa-Holm equation [9], the K(n, m) equations [10], the K * (l, p) equations [11], the KdV-Kuramoto-Sivashinsky-Velarde equation [12], the generalized integrable KdV equation [13, eq. ( 4)], KdV-based higher-order equations [14,15], hierarchical KdV equations [16][17][18], the Destrade-Saccomandi equation [19, eq. (17)], the Jordan-Saccomandi equation [20, eq.…”

On a hierarchy of nonlinearly dispersive generalized KdV equations

“…As in [19,20], it is now straightforward to piece together a compact traveling wave solution such that U = 0 for ξ ∈ (−∞, −ξ 0 ] ∪ [+ξ 0 , +∞), U is given implicitly by eqs. ( 17)- (18) for ξ ∈ (−ξ 0 , +ξ 0 )\{0} and U (0) = U max .…”

“…Additionally, in modeling various physical phenomena beyond long waves over shallow water, many generalizations and modifications of the KdV equation have been derived over the years, including but not limited to: the modified KdV equation [6,4], the KdV-Burgers equation [5], the cylindrical KdV equation [7], the extended KdV equation [8], the Camassa-Holm equation [9], the K(n, m) equations [10], the K * (l, p) equations [11], the KdV-Kuramoto-Sivashinsky-Velarde equation [12], the generalized integrable KdV equation [13, eq. ( 4)], KdV-based higher-order equations [14,15], hierarchical KdV equations [16,17,18], the Destrade-Saccomandi equation [19, eq. (17)], the Jordan-Saccomandi equation [20, eq.…”

Simulation of solitary wave propagation in carbon fibre reinforced polymer

“…The solitary waves can experience a phenomenon called 'selection' where the amplitude and velocity of a solitary wave tend to finite values, which depend on the nonlinearity and dispersion [10][11][12]. In some microstructured models, the solitary wave propagation can also be sensitive to the ratio of macro-and microstructural dispersions and a general 'shape' of the initial profile [13], which could likewise be used for diagnostic purposes.…”