Periodic peakons to a generalized μ-Camassa–Holm–Novikov equation

In this paper, we study the existence of peaked traveling wave solution of the generalized μ -Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ -version of a linear combination of the Novikov equation and Camassa-Hom equation. It is found that the equation admits single peaked traveling wave solutions.

We study the generalization of the dispersionless Kadomtsev–Petviashvili (dKP) equation in dimensions and with nonlinearity of degree , a model equation describing the propagation of weakly nonlinear, quasi one-dimensional waves in the absence of dispersion and dissipation, and arising in several physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear optics. In 2 + 1 dimensions and with quadratic nonlinearity, this equation is integrable through a novel inverse scattering transform, and it has been recently shown to be a prototype model equation in the description of the two-dimensional wave breaking of localized initial data. In higher dimensions and with higher nonlinearity, the generalized dKP equations are not integrable, but their invariance under motions on the paraboloid allows one to construct in this paper a family of exact solutions describing waves constant on their paraboloidal wave front and breaking simultaneously in all points of it, developing after breaking either multivaluedness or single-valued discontinuous profiles (shocks). Then such exact solutions are used to build the longtime behavior of the solutions of the Cauchy problem, for small and localized initial data, showing that wave breaking of small initial data takes place in the longtime regime if and only if . Lastly, the analytic aspects of such wave breaking are investigated in detail in terms of the small initial data, in both cases in which the solution becomes multivalued after breaking or it develops a shock. These results, contained in the 2012 master’s thesis of one of the authors (FS) [], generalize those obtained in [] for the dKP equation in dimensions with quadratic nonlinearity, and are obtained following the same strategy.

The Novikov equation (NE) has been discovered recently as a new integrable equation with cubic nonlinearities that is similar to the Camassa-Holm and Degasperis-Procesi equations, which have quadratic nonlinearities. NE is well-posed in Sobolev spaces Hs on both the line and the circle for s > 3/2, in the sense of Hadamard, and its data-to-solution map is continuous but not uniformly continuous. This work studies the continuity properties of NE further. For initial data in Hs, s > 3/2, it is shown that the solution map for NE is Hölder continuous in Hr-topology for all 0 ⩽ r < s with exponent α depending on s and r.

This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation (alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equations theory. Then, the Hölder continuity of the data-to-solution map to this equation are exhibited in some Sobolev spaces. After providing the blow-up criterion and the precise blow-up quantity in light of the Moser-type estimate in the Sobolev spaces, we then trace a portion and the whole of the precise blow-up quantity, respectively, along the characteristics associated with this equation, and obtain two kinds of sufficient conditions on the gradient of the initial data to guarantee the occurance of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon and multi-peakon solutions for this equation are also explored.

The Camassa-Holm equation, Degasperis—Procesi equation and Novikov equation are the three typical integrable evolution equations admitting peaked solitons. In this paper, a generalized Novikov equation with cubic and quadratic nonlinearities is studied, which is regarded as a generalization of these three well-known studied equations. It is shown that this equation admits single peaked traveling wave solutions, periodic peaked traveling wave solutions, and multi-peaked traveling wave solutions.

Peakons and periodic peakons are two kinds of special symmetric traveling wave solutions, which have important applications in physics, optical fiber communication, and other fields. In this paper, we study the orbital stability of peakons and periodic peakons for a generalized Camassa–Holm equation with quadratic and cubic nonlinearities (mCH–Novikov–CH equation). It is a generalization of some classical equations, such as the Camassa–Holm (CH) equation, the modified Camassa–Holm (mCH) equation, and the Novikov equation. By constructing an inequality related to the maximum and minimum of solutions with the conservation laws, we prove that the peakons and periodic peakons are orbitally stable under small perturbations in the energy space.

We prove the local existence of strong solutions of the periodic Hunter--Saxton equation, and we show that all strong solutions except space-independent solutions blow up in finite time.

Well-posedness and wave breaking of the degenerate Novikov equation

This paper mainly investigate the Cauchy problem for the generalised two-component Camassa–Holm type system, which includes the celebrated Camassa–Holm equation, Degasperis equation, Novikov equation, and the two-component cross-coupled Camassa–Holm system, Novikov system as special cases. Firstly, the local well-posedness of the system in nonhomogeneous Besov spaces \(B^{s}_{l,r}(\mathbb {R})\times B^{s}_{l,r}(\mathbb {R})\) with \(l,r\in [1,\infty ]\), \(s>\max \{2+1/l,5/2\}\) is established by using the Littlewood–Paley theory and transport equations theory. Moreover, we verify the blow-up occurs for this system only in the form of breaking waves. Finally, the waltzing peakons for the system and some numerical experiments to illustrate our results are performed.

A 4-parameter polynomial family of equations generalizing the Camassa-Holm and Novikov equations that describe breaking waves is introduced. A classification of low-order conservation laws, peaked travelling wave solutions, and Lie symmetries is presented for this family. These classifications pick out a 1-parameter equation that has several interesting features: it reduces to the Camassa-Holm and Novikov equations when the polynomial has degree two and three; it has a conserved H1 norm and it possesses N-peakon solutions when the polynomial has any degree; and it exhibits wave-breaking for certain solutions describing collisions between peakons and anti-peakons in the case N = 2.

This paper is concerned with two classes of cubic quasilinear equations, which can be derived as asymptotic models from shallow-water approximation to the 2D incompressible Euler equations. One class of the models has homogeneous cubic nonlinearity and includes the integrable modified Camassa–Holm (mCH) equation and Novikov equation, and the other class encompasses both quadratic and cubic nonlinearities. It is demonstrated here that both these models possess localized peaked solutions. By constructing a Lyapunov function, these peaked waves are shown to be dynamically stable under small perturbations in the natural energy space $H^1$, without restriction on the sign of the momentum density. In particular, for the homogeneous cubic nonlinear model, we are able to further incorporate a higher-order conservation law to conclude orbital stability in $H^1\cap W^{1,4}$. Our analysis is based on a strong use of the conservation laws, the introduction of certain auxiliary functions, and a refined continuity argument.

The bounded traveling wave solutions of a generalized CamassaHolm-Novikov equation with <i>p</i>=2 and <i>p</i>=3 are derived via the dynamical system approach. The singular wave solutions including peakons and cuspons are obtained by the bifurcation analysis of the corresponding singular dynamical system and the orbits intersecting with or approaching the singular lines. The results show that the generalized Camassa-Holm-Novikov equation with <i>p</i>=2 and <i>p</i>=3 both admit smooth solitary wave, smooth periodic wave solutions, solitary peakons, periodic peakons, solitary cuspons and periodic cuspons as well. It is worth pointing out that the Novikov equation has no bounded traveling wave solutions with negative wave speed, but has a family of new periodic cuspons which are distinguished with the normal periodic cuspons for their discontinuous first-order derivatives at both maximum and minimum.

Large-scale turbulence under a solitary wave: Part 2: Forms and evolution of coherent structures

Self-trapping and soliton-like propagation due to interplay of dispersion and nonlinearity-induced phase shift due to cascaded nonlinearities [1] has attracted a great deal of interest [2-6]. Soliton propagation may occur either in the form of strongly coupled symbiotic pairs for the fundamental-frequency (FF) and the second-harmonic (SII) fields [2-6], or in the limit of nonzero mismatch, for which the propagation is governed by an equivalent nonlinear Schrödinger (NLS) equation for the FF field, obtained by means of asymptotic techniques [7]. However, in χ(2) materials there always exist cubic nonlinearities which might become also important and even compete with quadratic nonlinearities. Here we analyze the effect of such competing nonlinearities on properties of solitary waves. As we will show, self-phase (SPM) and cross-phase (CPM) modulation induced by a cubic nonlinearity can strongly perturb solitary waves of a purely quadratic medium, and they may eventually destroy them. Nevertheless, we show that, even for such competing nonlinearities, stable solitary waves do exist.

New governing equations with combined quadratic and cubic nonlinearities are obtained to account for nonlinear strain waves in an elastic rod and in a plate. It is shown that strain solitary wave solutions of these equations arise as a result of balance between quadratic nonlinearity and dispersion and exists even in the absence of cubic nonlinearity. However, the amplitude, the width and the velocity of the wave are affected by the cubic nonlinearity causing, in particular, a narrowing of the longitudinal solitary wave. This allows to agree better with experiments on strain solitary wave generation in the rod and in the plate.