Smooth multisoliton solutions and their peakon limit of Novikov's Camassa-Holm type equation with cubic nonlinearity

We consider Novikov's Camassa-Holm type equation with cubic nonlinearity. In particular, we present a compact parametric representation of the smooth bright multisolution solutions on a constant background and investigate their structure. We find that the tau-functions associated with the solutions are closely related to those of a model equation for shallow-water waves (SWW) introduced by Hirota and Satsuma. This novel feature is established by applying the reciprocal transformation to the Novikov equation. We also show by specifying a complex phase parameter that the smooth soliton is converted to a novel singular soliton with single cusp and double peaks. We demonstrate that both the smooth and singular solitons converge to a peakon as the background field tends to zero whereby we employ a method that has been developed for performing the similar limiting procedure for the multisoliton solutions of the Camassa-Holm equation. In the subsequent asymptotic analysis of the two- and $N$-soliton solutions, we confirm their solitonic behaviour. Remarkably, the formulas for the phase shifts of solitons as well as their peakon limits coincide formally with those of the Degasperis-Procesi equation. Last, we derive an infinite number of conservation laws of the Novikov equation by using a relation between solutions of the Novikov equation and those of the SWW equation. In appendix, we prove various bilinear identities associated with the tau-functions of the multisoliton solutions of the SWW equation.Comment: V2: final version, 39 pages, 5 figures, related paper: Y. Matsuno, J. Math. Phys. 54 (2013) 05150

The Lyapunov stability of the N-soliton solutions in the Lax hierarchy of the Benjamin-Ono equation

The Lyapunov stability is established for the N-soliton solutions in the Lax hierarchy of the Benjamin-Ono (BO) equation. We characterize the N-soliton profiles as critical points of certain Lyapunov functional. By using several results derived by the inverse scattering transform of the BO equation, we demonstarate the convexity of the Lyapunov functional when evaluated at the N-soliton profiles. From this fact, we deduce that the N-soliton solutions are energetically stable.Comment: To appear in Journa of Mathematical Physic

A novel multi-component generalization of the short pulse equation and its multisoliton solutions

We propose a novel multi-component system of nonlinear equations that generalizes the short pulse (SP) equation describing the propagation of ultra-short pulses in optical fibers. By means of the bilinear formalism combined with a hodograph transformation, we obtain its multi-soliton solutions in the form of a parametric representation. Notably, unlike the determinantal solutions of the SP equation, the proposed system is found to exhibit solutions expressed in terms of pfaffians. The proof of the solutions is performed within the framework of an elementary theory of determinants. The reduced 2-component system deserves a special consideration. In particular, we show by establishing a Lax pair that the system is completely integrable. The properties of solutions such as loop solitons and breathers are investigated in detail, confirming their solitonic behavior. A variant of the 2-component system is also discussed with its multisoliton solutions.Comment: Minor correction