Abstract
By requiring and substituting into the -family equation and Novikov equation, we can obtain Alice-Bob peakon systems, where and are the arbitrary shifted parity transformation and delayed time reversal transformation, respectively. The nonlocal integrable Camassa-Holm equation and Degasperis-Procesi equation can be derived from the Alice-Bob -family equations by choosing different parameters. Some new types of interesting solutions are solved including explicit one-peakons, two-peakons, and -peakons solutions.
In shallow water theory, a lot of integrable equations are derived to model wave propagation in water of variable depth. We consider the -family equations where is an real constant.
When , (1) reduces to Camassa-Holm (CH) equationwhich was first reported by Fuchssteiner and Fokas in the context of hereditary symmetries [1]. It was rediscovered by Camassa and Holm as a model for shallow water waves. This equation shares most of the important properties of an integrable system of a Korteweg-de Vries type, for example, the existence of the bi-Hamiltonian structure [1, 2], Lax pair formalism [1], integrable by the inverse scattering transform [3], and other remarkable properties of integrable equations [4, 5]. Also, this equation admits peaked solitary wave solutions [3, 6], peaked soliton solutions [1, 7–9], and algebrogeometric solutions [10, 11]. The most interesting feature of the CH equation is that it admits peaked peakon solutions [1, 7]. A peakon is a weak solution in some Sobolev space with corner at its crest. The stability and analysis study of peakons were discussed in several references [7, 12–16].
When , (1) simplifies to Degasperis-Procesi (DP) equationwhich was proposed firstly by Degasperis and Procesi. This equation can also be considered as a model for shallow water wave and satisfied the asymptotic integrability to third order [17]; then in the complete integrability of the system it was proved because of the existence of a Lax pair and a bi-Hamiltonian structure based on a third order spectral problem [18]. The -peakon solutions of the DP solution have been obtained [19]. In fact, every member of the -family equation has peakon solutions for each . However, it is not the case that every member is also integrable. In fact, the DP and CH equations are the only integrable members of the -family equation. As we all know, the CH equation is widely used. Many new integrable equations can be regarded as a generalization of the well-known CH equation [6, 20]including FORQ (or MCH) system [21, 22] and Novikov equation [23]. In 2013, the integrable peakon systems with weak kink and kink-peakon interactional solutions of the FORQ system are first introduced by Qiao and Xia [24]. Next, we will study the Novikov equation as follows:Like the CH equation, the Novikov equation has the Lax pair, bi-Hamiltonian structure, Lie symmetry, algebraic-geometric solutions [23, 25] and infinitely many conserved quantities. The Novikov equation also enjoys two other important properties of the CH equation; it admits peakon solutions and the Cauchy problem [26].
We all know that nonlocal integrable systems have attracted much attention in different nonlocal nonlinear equations, for example, the nonlocal nonlinear Schrödinger equation [27, 28], the nonlocal modified KdV systems [29], the (2+1)-dimensional KdV equation [30], KP equation [31], (2+1)-dimensional Sawada-Kotera equation [32, 33], nonlocal symmetry for the gKP equation [34], nonlocal symmetry of the (2+1)- dimensional breaking soliton equation [35], (2+1)-dimensional Gardner equation [36], and Drinfeld-Sokolov-Satsuma-Hirota system [37]. Recently, Lou introduced Alice-Bob (AB) models to study two-place physical problems [38]. In [38, 39], many possible models named AB systems are proposed; some special types of group invariant multisoliton solutions for the KdV-KP-Toda type, mKdV-sG type, discrete type, and NLS type AB systems are explicitly constructed. In [40], nonlocal integrable peakon equations are obtained and shown to have peakon solutions for some AB peakon systems. Now, we will investigate peakon solutions for nonlocal integrable -family equation and Novikov equation.
Based on the AB approach [40], take to (1) and (4); the following AB -family equation and AB Novikov equation can be generated:where is an arbitrary shifted parity () and delayed time reversal () invariant functionand the definitions of and are ; and are constants.
The rest of the paper is organized as follows. Firstly, peakon solutions of Alice-Bob (AB) -family equation are introduced; the dynamic properties of these peakon solutions are displayed vividly by some figures. The peakon solutions of the AB Camassa-Holm equation and the AB Degasperis-Procesi equation can be derived. Secondly, we are devoted to peakon solutions of AB Novikov equation. Finally, short conclusion remarks are made in the paper.
Next, let us solve the peakon solutions to (5) for some special functions .
Example 1. Setting in (5) yields the following system:One-peakon solution of (8) iswhere is real parameters to be determined which are nontraveling solitary waves with a fast decayed standing peak, as shown in Figure 1.
Furthermore, we can obtain the -peakon dynamical system of (5)where , , requiring that , () be positive or negative at the same time. We obtain the following explicit two-peakon solutions:where , are real function with respect to variable , and :wherewhich require that , , , and be positive or negative at the same time which are nontraveling solitary waves with a fast decayed standing peak, as shown in Figure 2.
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Example 2. Substituting into (5), we can get the following AB CH equation:We can derive the one-peakon solutions of the systemHere we discuss the two-peakon solutions to CH equationwhere , are real function with respect to variable which needs to satisfy , , , and being positive or negative at the same time. The -peakon dynamical system of (15) is as follows:wherewhere , , , and , , are positive or negative at the same time.
Example 3. Setting to (5), we can get the AB system of the DP equationwhich admits the following one-peakon solution of (21):The two-peakon solutions are given in the following form:wherewhich require that , , , and be positive or negative at the same time. The -peakon dynamical system of (21) can be obtained as follows:and requires that and () have the same positive or negative qualities.
Example 4. When or of (5), we can obtain AB -peakon solution as (8).
Example 5. When of (5), this system has only one-peakon solutionsNo multipeakon solution is found for this special example.
Just like -family equation, taking to AB Novikov (6),We can obtain the following one-peakon solution:where is arbitrary constant. The two-peakon solutions can be derivedwhere which require that , , , and be positive or negative at the same time. Furthermore, we can obtain the -peakon dynamical system of (28):where where and () are positive or negative at the same time. Similar to the -family equation when the takes the different functions of and , we can get the same single peakon and two-peakon forms.
In this study, we have obtained the Alice-Bob (AB) peakon system for -family equation and Novikov equation, from which the nonlocal integrable Camassa-Holm equation and Degasperis-Procesi equation can be derived by choosing different parameters. Then the peakon solutions of these AB peakon systems are concretely established, and the dynamic properties of these peakon solutions are displayed vividly by some figures. For every peakon systems, there may exist different versions of integrable AB peakon systems, such as Novikov’s cubic nonlinear equation, generalized peakon system, and Li-Liu-Popowicz’s system. We believe that these systems deserve further investigation.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
We are very grateful to Professors S Y Lou and E G Fan for valuable discussions. This work is supported by the National Natural Science Foundation of China under Grants Nos. 11775121, 11805106, and 11435005 and K.C. Wong Magna Fund in Ningbo University.