# American Institute of Mathematical Sciences

October  2019, 39(10): 6131-6148. doi: 10.3934/dcds.2019267

## Accelerating dynamical peakons and their behaviour

 1 Department of Mathematics and Statistics, Brock University, St. Catharines, ON L2S3A1, Canada 2 Department of Mathematics, Faculty of Sciences, Universidad de Cádiz, Puerto Real, Cádiz, 11510, Spain

Received  March 2019 Revised  May 2019 Published  July 2019

A wide class of nonlinear dispersive wave equations are shown to possess a novel type of peakon solution in which the amplitude and speed of the peakon are time-dependent. These novel dynamical peakons exhibit a wide variety of different behaviours for their amplitude, speed, and acceleration, including an oscillatory amplitude and constant speed which describes a peakon breather. Examples are presented of families of nonlinear dispersive wave equations that illustrate various interesting behaviours, such as asymptotic travelling-wave peakons, dissipating/anti-dissipating peakons, direction-reversing peakons, runaway and blow up peakons, among others.

Citation: Stephen C. Anco, Elena Recio. Accelerating dynamical peakons and their behaviour. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6131-6148. doi: 10.3934/dcds.2019267
##### References:

show all references

##### References:
Asymptotic travelling-wave peakon
Direction-reversing peakon
Dissipating peakon
Blowing up peakon
Peakon breather
 [1] Stephen C. Anco, Elena Recio, María L. Gandarias, María S. Bruzón. A nonlinear generalization of the Camassa-Holm equation with peakon solutions. Conference Publications, 2015, 2015 (special) : 29-37. doi: 10.3934/proc.2015.0029 [2] Andrew N. W. Hone, Michael V. Irle. On the non-integrability of the Popowicz peakon system. Conference Publications, 2009, 2009 (Special) : 359-366. doi: 10.3934/proc.2009.2009.359 [3] Nakao Hayashi, Seishirou Kobayashi, Pavel I. Naumkin. Nonlinear dispersive wave equations in two space dimensions. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1377-1393. doi: 10.3934/cpaa.2015.14.1377 [4] Jibin Li, Yi Zhang. On the traveling wave solutions for a nonlinear diffusion-convection equation: Dynamical system approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1119-1138. doi: 10.3934/dcdsb.2010.14.1119 [5] Vural Bayrak, Emil Novruzov, Ibrahim Ozkol. Local-in-space blow-up criteria for two-component nonlinear dispersive wave system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6023-6037. doi: 10.3934/dcds.2019263 [6] Kai Yan, Zhijun Qiao, Yufeng Zhang. On a new two-component $b$-family peakon system with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5415-5442. doi: 10.3934/dcds.2018239 [7] Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391 [8] C. I. Christov, M. D. Todorov. Investigation of the long-time evolution of localized solutions of a dispersive wave system. Conference Publications, 2013, 2013 (special) : 139-148. doi: 10.3934/proc.2013.2013.139 [9] Jerry Bona, Hongqiu Chen. Well-posedness for regularized nonlinear dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1253-1275. doi: 10.3934/dcds.2009.23.1253 [10] Hongqiu Chen, Jerry L. Bona. Periodic traveling--wave solutions of nonlinear dispersive evolution equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4841-4873. doi: 10.3934/dcds.2013.33.4841 [11] Jerry L. Bona, Laihan Luo. More results on the decay of solutions to nonlinear, dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 151-193. doi: 10.3934/dcds.1995.1.151 [12] Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387 [13] Q-Heung Choi, Tacksun Jung. A nonlinear wave equation with jumping nonlinearity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 797-802. doi: 10.3934/dcds.2000.6.797 [14] Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099 [15] Jorge A. Esquivel-Avila. Qualitative analysis of a nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 787-804. doi: 10.3934/dcds.2004.10.787 [16] Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 [17] Qiaoyi Hu, Zhijun Qiao. Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103 [18] Qiaoyi Hu, Zhijun Qiao. Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2613-2625. doi: 10.3934/dcds.2016.36.2613 [19] Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171 [20] Xiuting Li, Lei Zhang. The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3301-3325. doi: 10.3934/dcds.2017140

2018 Impact Factor: 1.143