Abstract
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.
1 Introduction
We consider the following partial differential equation
where u(t, x) is a function of time t and a single spatial variable x, and
with 𝕊 = ℝ/ℤ which denotes the unit circle on ℝ2. Equation (1.1) can be reduced as μ-Novikov equation [39]
for k1 = 1 and k2 = 0, and the μ-Camassa-Holm equation [28]
for k1 = 0 and k2 = 1, respectively.
It is known that the Camassa-Holm equation of the following form [2, 20]
was proposed as a model for the unidirectional propagation of the shallow water waves over a flat bottom (see also [14, 25]), with u(x, t) representing the height of the water’s free surface in terms of non-dimensional variables. The Camassa-Holm equation (1.4) is completely integrable with a bi-Hamiltonian structure and an infinite number of conservation laws [2, 20], and can be solved by the inverse scattering method [5, 6, 30]. It is of interest to note that the Camassa-Holm equation (1.4) can also be derived by tri-Hamitonian duality from the Korteweg-de Vries equation (a number of additional examples of dual integrable systems derived applying the method of tri-Hamitonian duality can be found in [21, 42]). The Camassa-Holm equation (1.4) has two remarkable features: existence of peakon and multi-peakons [1, 2, 3] and breaking waves, i.e., the wave profile remains bounded while its slope becomes unbounded in finite time [7, 8, 10, 11, 12, 33]. Those peaked solitons were proved to be orbitally stable in the energy space [15, 16] and to be asymptotically stable under the Camassa-Holm flow [38] (see also [26, 27] for other equations). It is worth noting that solutions of this type are not mere abstractizations: the peakons replicate a feature that is characteristic for the waves of great height-waves of largest amplitude that are exact solutions of the governing equations for irrotational water waves [9, 13, 48]. Geometrically, the Camassa-Holm equation (1.4) describes the geodesic flows on the Bott-Virasoro group [37, 47] and on the diffeomorphism group of the unit circle under H1 metric [29], respectively. The Camassa-Holm equation (1.4) also arises from a non-stretching invariant planar curve flow in the centro-equiaffine geometry [4, 41]. Well-posedness and wave breaking of the Camassa-Holm equation (1.4) were studied extensively, and many interesting results have been obtained, see [7, 10, 11, 12, 33], for example. The μ-Camassa-Holm equation (1.3) was originally proposed as the model for the evolution of rotators in liquid crystals with an external magnetic field and self interatction [28]. It is interesting to note that this equation is integrable in the sense that it admits the Lax-pair and bi-Hamiltonian structure, and also describes a geodesic flow on the diffeomorphism group of 𝕊 with Hμ(𝕊) metric (which is equivalent to H1(𝕊) metric). Its integrability, well-posedness, blow-up and peakons were discussed in [19, 28].
It is observed that all nonlinear terms in the Camassa-Holm equation (1.4) are quadratic. In contrast to the integrable modified Korteweg-de Vries equation with a cubic nonlinearity, it is of great interest to find integrable Camassa-Holm type equations with cubic or higher-order nonlinearity admitting peakon solitons. Recently, two integrable Camassa-Holm type equtions with cubic nonlinearities have been appeared in literature. One was introduced by Olver and Rosenau [42](called the modified Camassa-Holm equation, see also [18, 21]) by using the tri-Hamiltonian duality approach, which takes the form
It was shown that the modified Camassa-Holm equation is integrable with the Lax-pair and the bi-Hamiltonian structure. It has single and multi-peaked traveling waves with a different character than of the Camassa-Holm equation (1.4) [22], and it also has new features of blow-up criterion and wave breaking mechanism. The issue of the stability of peakons for the modified Camassa-Holm equation were investigated in [46]. Like μ-Camassa-Holm equation (1.3), μ-version of the modified Camassa-Holm equation
was introduced in [44]. Its integrability, wave breaking, existence of peaked traveling waves and their stability were discussed in [34, 44]. The second one is the Novikov equation
which is integrable with the Lax pair [40]. A matrix Lax pair reprsentation to the Novikov equation was founded in [23]. It is also noticed that the Novikov equation admits a bi-Hamiltonian structure [23]. Existence of peaked solitons and multi-peakons for Novikov equation were obtained in [24, 40]. Orbital stability of the peaked solitons to the Novikov equation were discussed in [35]. The μ-Novikov equation (1.2), regarded as a μ-version of the Novikov equation, was introduced first in [39]. The existence of its single peakons was established in [39].
More recently, the following generalized μ-Camassa-Holm equation
was proposed in [45] as a μ-version of the generalized Camassa-Holm equation with quadradic and cubic nonlinearities
which was derived by Fokas [18] from the hydrodynamical wave, and can also obtained using the approach of tri-Hamiltonian duality [21, 42] to the bi-Hamiltonian Gardner equation
Note that the Lax pair of equation (1.9) was obtained in [43]. It was shown in [45] that a scale limit of equation (1.8) yields the following integrable equation
which describes asymptotic dynamics of a short capillarty-gravity wave [17], where v(t, x) denotes the fluid velocity on the surface. Notably, the generalized μ-Camassa-Holm equation (1.8) can be regarded as the integrable model that, in a sense, lies midway between equation (1.9) and its limiting version equation (1.11). It has been known that the generalized μ-Camassa-Holm equation (1.8) is formally integrable in the sense that it admits Lax formulation and bi-Hamiltonian form [45].
The existence of periodic peakons is of interest for nonlinear integrable equations because they are relatively new solitary waves (for most models the solitary waves are quite smooth). Applying the method of tri-Hamiltonian duality[21, 42] to the bi-Hamiltonian representation of the Korteweg-de Vries (KdV), modified Korteweg-de Vries (mKdV), and Gardner equation, the resulting dual systems, such as Camassa-Holm equation (1.4), the modified Camassa-Holm equation (1.5), and the generalized Camassa-Holm equation (1.9), exhibit nonlinear dispersion, and, in most cases, admit a remarkable variety of non-smooth soliton-like solutions, including peakons, compactons, tipons, rampons, mesaons, and so on [32]. It is known that Camassa-Holm equation (1.4), the modified Camassa-Holm equation (1.5), Novikov equation (1.7), and the generalized Camassa-Holm equation (1.9) [2, 22, 36, 40, 43] admit single peakons of the form
where the amplitude a is given by
for the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation, and the generalized Camassa-Holm equation, respectively. Their corresponding periodic peakons take the form
where the amplitude a is also given by
with
for the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation, and the generalized Camassa-Holm equation, respectively.
It is worth noting that the periodic peakons of the μ-integrable equation are of a manifestly different character. For example, in [28, 31, 39, 44], the authors showed that the μ-Camassa-Holm equatioin (1.3), μ-Novikov equation (1.2), modified μ-Camassa-Holm equation (1.6), and generalized μ-Camassa Holm equation (1.8) admit periodic peakons of the following form
where
and φ is extended periodically to the real line, the constant a takes value
respectively, for the μ-Camassa-Holm equation, μ-Novikov equation, modified μ-Camassa-Holm equation, and generalized μ-Camassa-Holm equation.
Motivated by the recent work [28, 44, 45], the aim of this paper is to investigate the existence of periodic peaked solution of the generalized μ-Novikov equation (1.1). Indeed, in Section 2, we give a short review on the notion of a strong and weak solution of the generalized μ-Novikov equation (1.1) and then show that equation (1.1) admits the periodic peakon, which is given by (1.15) with a replaced by
where the wave speed c satisfies
2 Peaked Traveling Waves
We first introduce the initial value problem of Equation (1.1) on the unit circle 𝕊, that is
We then formalize the notion of a strong (or classical) and weak solutions of the Equation (1.1) used throughout this paper.
Definition 2.1
If u ∈ C([0, T), Hs(𝕊)) ∩ C1([0, T), Hs–1(𝕊)) with s >
Note that the inverse operator
where g is given by [28]
Here [x] denote the greatest integer for
Plugging the formula for m := μ(u) – uxx in terms of u into Equation (1.1) results in the following fully nonlinear partial differential equation:
The formulation (2.5) allows us to define the notion of a weak solutions as follows.
Definition 2.2
Given the initial data u0 ∈ W1,3(𝕊), the function u ∈ L∞([0, T); W1,3(𝕊)) is said to be a weak solution to (2.1) if it satisfies the following identity:
for any smooth test function
Our main theorem is in the following.
Theorem 2.1
For any
where the amplitude
and ϕc(ξ) is extended periodically to the real line with period one.
Proof
Inspired by the forms of periodic peakons for the μ-CH equation [28](See also [44, 45]), we assume that the peaked periodic traveling wave of Equation (1.1) is given by
According to Definition 2.2 it is found that uc(t, x) satisfies the following equation
for some T > 0 and every test function
To evaluate Ij, j = 1, ⋯, 6, we need to consider two cases: (i) x > ct, and (ii) x ≤ ct.
For x > ct, we have
On the other hand,
and
It follows that
Plugging above expressions into (2.8), we deduce that for any
A similar computation yields for x ≤ ct that
and
This allows us to evaluate
Hence we arrive at
Since ψ(t, x) is an arbitrary, both cases imply that the parameter a fulfills the equation
Clearly, its solutions are given by which gives (2.7). Thus the theorem is proved.□
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (No. 2017R1C1B1002336).
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