Exact solutions and dynamic properties of Ito-type coupled nonlinear wave equations. (English) Zbl 07427918
Summary: In this paper, we conduct qualitative and quantitative analysis to Ito system by the complete discrimination system for polynomial method (CDSPM). Through the traveling wave transformation, the original equations could be transformed into a dynamic system. A conserved quantity, namely the Hamiltonian, is established, and then the qualitative properties such as equilibrium points are gotten. Especially, a typical phase portrait is emphasized to present prior estimate of existences of smooth soliton, cusped soliton and periodic solutions. Concrete examples of these solutions are all constructed to verify our conclusion directly.
Keywords:
complete discrimination system for polynomial method; periodic solution; soliton solution; qualitative analysis; quantitative analysisReferences:
| [1] | Liu, H.; Bai, C. L.; Xin, X. P., A novel Lie group classification method for generalized cylindrical KdV type of equation: exact solutions and conservation laws, J. Math. Fluid Mech., 21, 4, 1-7 (2019) · Zbl 1428.37068 |
| [2] | Fan, E. G., Travelling wave solutions for two generalized Hirota-Satsuma coupled KdV systems, Z. Naturforsch. A, 56, 3-4, 312-318 (2001) |
| [3] | Kawamoto, S., Cusp soliton solutions of the Ito-Type coupled nonlinear wave equation, J. Phys. Soc. Jpn., 53, 4, 1203-1205 (1984) |
| [4] | Ito, M., Symmetries and conservation laws of a coupled nonlinear wave equation, Phys. Lett. A, 91, 7, 335-338 (1982) |
| [5] | Baskonus, H. M.; Kayan, M., Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations, Appl. Math. Nonlinear Sci. (2021) |
| [6] | Hirota, R.; Ohta, Y., Hierarchies of coupled soliton equations I, J. Phys. Soc. Jpn., 60, 3, 798-809 (1991) · Zbl 1160.37395 |
| [7] | Rogers, C.; William, F., Bäcklund Transformations and Their Applications (1982), Academic Press · Zbl 0492.58002 |
| [8] | Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of soliton, Phys. Rev. Lett., 27, 1456-1458 (1971) |
| [9] | Roshid, H. O.; Noor, N.; Khatun, M. S., Breather, multi-shock waves and localized excitation structure solutions to the extended BKP-Boussinesq equation, Commun. Nonlinear Sci., 8, Article 105867 pp. (2021) · Zbl 1469.35080 |
| [10] | Ciancio, A.; Yel, G.; Kumar, A., On the complex mixed dark-bright wave distributions to some conformable nonlinear integrable models, Fractals, 13, 7, Article 2240018 pp. (2021) |
| [11] | Ma, W. X., N-soliton solutions and the Hirota conditions in (2+1)-dimensions, Opt. Quantum Electron., 52, 12, 511-522 (2020) |
| [12] | Ma, W. X., N-soliton solution and the Hirota condition of a (2+1)-dimensional combined equation, Math. Comput. Simul., 190, 270-279 (2021) · Zbl 1540.35103 |
| [13] | Ma, W. X., N-soliton solutions and the Hirota conditions in (1+1)-dimensions, Int. J. Nonlinear Sci. Numer. Simul. (2021) |
| [14] | Liu, C. S., Classification of all single travelling wave solutions to Calogero-Degasperis-Focas equation, Commun. Theor. Phys., 48, 4, 601-604 (2007) · Zbl 1267.35065 |
| [15] | Liu, C. S., All single traveling wave solutions to (3+1)-dimensional Nizhnok-Novikov-Veselov equation, Commun. Theor. Phys., 45, 991-992 (2006) |
| [16] | Liu, C. S., The classification of travelling wave solutions and superposition of multi-solutions to Camassa-Holm equation with dispersion, Chin. Phys. B, 16, 7, 1832-1837 (2007) |
| [17] | Liu, C. S., Representations and classification of traveling wave solutions to Sinh-Gördon equation, Commun. Theor. Phys., 49, 1, 153-158 (2008) · Zbl 1392.35063 |
| [18] | Liu, C. S., Solution of ODE \(u'' + p(u) ( u^\prime )^2 + q(u) = 0\) and applications to classifications of all single travelling wave solutions to some nonlinear mathematical physics equations, Commun. Theor. Phys., 49, 2, 291-296 (2008) · Zbl 1392.35064 |
| [19] | Liu, C. S., Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations, Comput. Phys. Commun., 181, 2, 317-324 (2010) · Zbl 1205.35262 |
| [20] | Kai, Y., The classification of the single travelling wave solutions to the variant Boussinesq equations, Pramana, 87, 4, 1-5 (2016) |
| [21] | Dai, D. Y.; Yuan, Y. P., The classification and representation of single traveling wave solutions to the generalized Fornberg-Whitham equation, Appl. Math. Comput., 242, 729-735 (2014) · Zbl 1334.35010 |
| [22] | Fan, H. L., The classification of the single traveling wave solutions to the generalized equal width equation, Appl. Math. Comput., 219, 2, 748-754 (2012) · Zbl 1288.35162 |
| [23] | Kai, Y.; Chen, S. Q.; Zhang, K., A study of the shallow water waves with some Boussinesq-type equations, Waves Random Complex Media (2021) |
| [24] | Kai, Y., Qualitative and quantitative analysis of nonlinear dynamics by the complete discrimination system for polynomial method, Chaos Solitons Fractals, 141, Article 110314 pp. (2020) · Zbl 1496.35146 |
| [25] | Cao, C. W., A qualitative test for single soliton solution, J. Zhengzhou Univ., 2, 3-7 (1984) |
| [26] | Zheng, X.; Xiao, Q.; Ouyang, Z., A smooth soliton solution and a periodic cuspon solution of the Novikov equation, Appl. Math. Lett., 112, 34, Article 106786 pp. (2021) · Zbl 1453.35042 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
