Research Article | Open Access

Jing Wang, Na Xiong, Biao Li, "Peakon Solutions of Alice-Bob -Family Equation and Novikov Equation", *Advances in Mathematical Physics*, vol. 2019, Article ID 1519305, 8 pages, 2019. https://doi.org/10.1155/2019/1519305

# Peakon Solutions of Alice-Bob -Family Equation and Novikov Equation

**Academic Editor:**Zhijun Qiao

#### 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.

**(a)**

**(b)**

**(a)**

**(b)**

**(c)**

**(d)**

*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.

#### References

- B. Fuchssteiner and A. S. Fokas, “Symplectic structures, their Bäcklund transformations and hereditary symmetries,”
*Physica D: Nonlinear Phenomena*, vol. 4, no. 1, pp. 47–66, 1981/82. View at: Publisher Site | Google Scholar | MathSciNet - P. J. Olver and P. Rosenau, “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,”
*Physical Review E: Statistical, Nonlinear, and Soft Matter Physics*, vol. 53, no. 2, pp. 1900–1906, 1996. View at: Publisher Site | Google Scholar | MathSciNet - A. Constantin, “On the scattering problem for the Camassa-Holm equation,”
*Proceedings of the Royal Society A Mathematical, Physical and Engineering Sciences*, vol. 457, pp. 953–970, 2001. View at: Publisher Site | Google Scholar | MathSciNet - A. S. Fokas, “On a class of physically important integrable equations,”
*Physica D: Nonlinear Phenomena*, vol. 87, no. 1–4, pp. 145–150, 1995. View at: Publisher Site | Google Scholar | MathSciNet - C. Cotter, D. Holm, R. Ivanov, and J. Percival, “Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation,”
*Journal of Physics A Mathematical and Theoretical*, vol. 44, no. 26, pp. 2065–2088, 2011. View at: Google Scholar - R. Camassa, D. D. Holm, and J. M. Hyman, “A new integrable shallow water equation,”
*Advances in Applied Mechanics*, vol. 31, pp. 1–33, 1994. View at: Publisher Site | Google Scholar - R. Beals, D. H. Sattinger, and J. Szmigielski, “Multipeakons and the classical moment problem,”
*Advances in Mathematics*, vol. 154, no. 2, pp. 229–257, 2000. View at: Publisher Site | Google Scholar | MathSciNet - L. Luo, B.-Q. Xia, and Y.-F. Cao, “Peakon solutions to supersymmetric camassa-holm equation and degasperis-procesi equation,”
*Communications in Theoretical Physics*, vol. 59, no. 1, pp. 73–79, 2013. View at: Publisher Site | Google Scholar - D. D. Holm and R. I. Ivanov, “Smooth and peaked solitons of the CH equation,”
*Journal of Physics A: Mathematical and Theoretical*, vol. 43, no. 43, pp. 2741–2754, 2010. View at: Publisher Site | Google Scholar | MathSciNet - F. Gesztesy and H. Holden, “Real-valued algebro-geometric solutions of the Camassa-Holm hierarchy,”
*Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences*, vol. 366, no. 1867, pp. 1025–1054, 2008. View at: Publisher Site | Google Scholar | MathSciNet - Z. Qiao, “The Camassa-Holm hierarchy, -dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold,”
*Communications in Mathematical Physics*, vol. 239, no. 1-2, pp. 309–341, 2003. View at: Publisher Site | Google Scholar | MathSciNet - A. Constantin and W. A. Strauss, “Stability of peakons,”
*Communications on Pure and Applied Mathematics*, vol. 53, no. 5, pp. 603–610, 2000. View at: Publisher Site | Google Scholar | MathSciNet - A. Constantin and D. Lannes, “The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,”
*Archive for Rational Mechanics and Analysis*, vol. 192, no. 1, pp. 165–186, 2009. View at: Publisher Site | Google Scholar | MathSciNet - M. S. Alber, R. Camassa, Y. N. Fedorov, D. D. Holm, and J. E. Marsden, “The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type,”
*Communications in Mathematical Physics*, vol. 221, no. 1, pp. 197–227, 2001. View at: Publisher Site | Google Scholar | MathSciNet - R. S. Johnson, “On solutions of the Camassa-Holm equation,”
*Proceedings A*, vol. 459, no. 2035, pp. 1687–1708, 2003. View at: Publisher Site | Google Scholar | MathSciNet - B. Xia, Z. Qiao, and R. Zhou, “A synthetical two-component model with peakon solutions,”
*Studies in Applied Mathematics*, vol. 135, no. 3, pp. 248–276, 2015. View at: Publisher Site | Google Scholar | MathSciNet - A. Degasperis and M. Procesi, “Asymptotic integrability,”
*Symmetry and Perturbation Theory*, pp. 23–37, 1999. View at: Google Scholar - A. Degasperis, D. D. Kholm, and A. N. I. Khon, “A new integrable equation with peakon solutions,”
*Theoretical and Mathematical Physics*, vol. 133, no. 2, pp. 1463–1474, 2002. View at: Publisher Site | Google Scholar | MathSciNet - H. Lundmark and J. Szmigielski, “Degasperis-Procesi peakons and the discrete cubic string,”
*International Mathematics Research Papers*, vol. 2005, no. 2, pp. 53–116, 2005. View at: Google Scholar - V. Novikov, “Generalizations of the Camassa-Holm equation,”
*Journal of Physics A: Mathematical and Theoretical*, vol. 42, no. 34, Article ID 342002, 2009. View at: Publisher Site | Google Scholar | MathSciNet - Z. Qiao, “A new integrable equation with cuspons and W/M-shape-peaks solitons,”
*Journal of Mathematical Physics*, vol. 47, no. 11, pp. 1661–1664, 2006. View at: Google Scholar | MathSciNet - Z. J. Qiao, “New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solitons,”
*Journal of Mathematical Physics*, vol. 48, no. 8, pp. 249–315, 2007. View at: Publisher Site | Google Scholar | MathSciNet - D. P. Novikov, “Algebro-geometric solutions of the Krichever-Novikov equation,”
*Theoretical and Mathematical Physics*, vol. 121, no. 3, pp. 1567–1573, 1999. View at: Google Scholar - Z. Qiao and B. Xia, “Integrable peakon systems with weak kink and kink-peakon interactional solutions,”
*Frontiers of Mathematics in China*, vol. 8, no. 5, pp. 1185–1196, 2013. View at: Publisher Site | Google Scholar | MathSciNet - M. Kardell, “Lie symmetry analysis of the Novikov and Geng-Xue equations, and new peakon-like unbounded solutions to the Camassa-Holm, Degasperis-Procesi and Novikov equations,” https://arxiv.org/abs/1310.4927v2. View at: Google Scholar
- Y. Mi and C. Mu, “On the Cauchy problem for the modified Novikov equation with peakon solutions,”
*Journal of Differential Equations*, vol. 254, no. 3, pp. 961–982, 2013. View at: Publisher Site | Google Scholar | MathSciNet - B. L. Guo and L.-M. Ling, “Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations,”
*Chinese Physics Letters*, vol. 28, no. 11, pp. 110202–110205(4), 2011. View at: Google Scholar - M. J. Ablowitz and Z. H. Musslimani, “Integrable nonlocal nonlinear Schrödinger equation,”
*Physical Review Letters*, vol. 110, no. 6, Article ID 064105, 2013. View at: Publisher Site | Google Scholar - J. L. Ji and Z. N. Zhu, “Soliton solutions of an integrable nonlocal modified Korteweg--de Vries equation through inverse scattering transform,”
*Journal of Mathematical Analysis and Applications*, vol. 453, no. 2, pp. 973–984, 2017. View at: Publisher Site | Google Scholar | MathSciNet - S. Y. Lou and H. Y. Ruan, “Similarity reductions and nonlocal symmetry of the KdV equation,”
*Communications in Theoretical Physics*, vol. 25, no. 2, pp. 241–244, 1996. View at: Publisher Site | Google Scholar | MathSciNet - S. Y. Lou and X. B. Hu, “Infinitely many Lax pairs and symmetry constraints of the KP equation,”
*Journal of Mathematical Physics*, vol. 38, no. 12, pp. 6401–6427, 1997. View at: Publisher Site | Google Scholar | MathSciNet - Xian Li, Yao Wang, Meidan Chen, and Biao Li, “Lump solutions and resonance stripe solitons to the (2+1)-dimensional sawada-kotera equation,”
*Advances in Mathematical Physics*, vol. 2017, Article ID 1743789, 6 pages, 2017. View at: Publisher Site | Google Scholar | MathSciNet - J. Zhou, X. G. Li, and D. S. Wang, “
*N-*soliton solutions of the nonisospectral generalized Sawada-Kotera equation,”*Advances in Mathematical Physics*, vol. 2014, Article ID 547692, 5 pages, 2014. View at: Publisher Site | Google Scholar - L. L. Huang, Y. Chen, and Z. Y. Ma, “Nonlocal symmetry and interaction solutions of a generalized kadomtsev - petviashvili equation,”
*Communications in Theoretical Physics*, vol. 66, no. 2, pp. 189–195, 2016. View at: Publisher Site | Google Scholar - W. g. Cheng, B. Li, and Y. Chen, “Nonlocal symmetry and exact solutions of the (2+1)- dimensional breaking soliton equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 29, no. 1-3, pp. 198–207, 2015. View at: Publisher Site | Google Scholar | MathSciNet - Y.-k. Liu and B. Li, “Nonlocal symmetry and exact solutions of the -dimensional Gardner equation,”
*Chinese Journal of Physics*, vol. 54, no. 5, pp. 718–723, 2016. View at: Publisher Site | Google Scholar | MathSciNet - L. Huang and Y. Chen, “Nonlocal symmetry and similarity reductions for the Drinfeld-Sokolov-SATsuma-Hirota system,”
*Applied Mathematics Letters*, vol. 64, pp. 177–184, 2017. View at: Publisher Site | Google Scholar | MathSciNet - S. Y. Lou, “Alice-Bob systems, P-T-C symmetry invariant and symmetry breaking soliton solutions,”
*Journal of Mathematical Physics*, vol. 59, no. 8, Article ID 083507, 2018. View at: Publisher Site | Google Scholar | MathSciNet - S. Y. Lou and F. Huang, “Alice-bob physics: coherent solutions of nonlocal KdV systems,”
*Scientific Reports*, vol. 7, no. 1, p. 869, 2017. View at: Google Scholar - S. Y. Lou and Z. J. Qiao, “Alice-Bob peakon systems,”
*Chinese Physics Letters*, vol. 34, no. 10, pp. 1–4, 2017. View at: Google Scholar

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Copyright © 2019 Jing Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.