On the Cauchy problem for the modified Novikov equation with peakon solutions

Abstract

A new nonlinear dispersive partial differential equation with cubic nonlinearity, which includes the famous Novikov equation as special case, is investigated. We first establish the local wellposedness in a range of the Besov spaces B s p,r , p, r ∈ [1, ∞], s > max{ 3 2 , 1 + 1 p } but s = 2 + 1 p (which generalize the Sobolev spaces H s ), well-posedness in H s with s > 3 2 , is also established by applying Kato's semigroup theory. Then we give the precise blow-up scenario. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we prove that peakon solutions to the equation are global weak solutions.

Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

“…Most of our results can be carried out to the periodic case. It is well known that the H 1 × L 2norm conserved quantity plays a key role in studying the blow-up phenomenon and global existence [34,56,64,62]. Unfortunately, one cannot find this similar conservation law for Equ.…”

Accelerating dynamical peakons and their behaviour

“…Note that local existence of X(t) holds whenever A(t) exists and the function g 0 (u)/(uf 0 (u)) is locally integrable with respect to u. The following Theorem is an immediate consequence of the ODEs (32) foṙ X andẌ, as well as use of the quadrature (27) for X similarly to the proof of Theorem 1. Finally, combining part (i) of both Theorems, we obtain necessary and sufficient conditions on the nonlinearities f (u, u x ) and g(u, u x ) so that the asymptotic behaviour of dynamical peaked waves is a travelling-wave peakon.…”

“…where τ 1 = τ (A 1 ) and c 1 = c(A 1 ). These two expressions (26)- (27), or equivalently the integral expressions…”

“…Much of the interest in these equations is that, firstly, they are integrable systems having a Lax pair, bi-Hamiltonian structure, hierarchies of symmetries and conservation laws; secondly, the peakon solutions are orbitally stable [15,16,17], which implies the shape of the peakon is unchanged under small perturbations; thirdly, they possess N -peakon weak solutions given by a linear superposition of single peakons with time-dependent amplitudes and speeds; and fourthly, they exhibit wave breaking in which certain smooth initial data yields solutions whose gradient u x blows up in a finite time while u stays bounded [9,10,11,12,13,14]. Moreover, the CH and DP equations arise as models for water waves [19,20,21], and it is known that the travelling wave solutions of greatest height for the Euler equations governing water waves have a peak at their crest (see [22,23,24,25] All of these equations, and their various modified versions and nonlinear generalizations [26,27,28,29,30,31,32], belong to the general family of nonlinear dispersive wave equations m t + f (u, u x )m + (g(u, u x )m) x = 0, m = u − u xx (1) where f and g are arbitrary non-singular functions of u and u x . Remarkably, as shown in recent work [33], every equation in this family (1) possesses N -peakon weak solutions…”

Well-Posedness and Analyticity for an Integrable Two-Component System With Cubic Nonlinearity

“…The integrability, peaked solitons, well-posedness and blow-up phenomena to the Novikov equation have been studied extensively [31,32,42,43,37]. An alternative modified Camassa-Holm equation was introduced in [22,35]. Multicomponent versions of the Camassa-Holm equation have been introduced and studied in [19,23,24,[28][29][30]13].…”