×

Investigation of interaction solutions for modified Korteweg-de Vries equation by consistent Riccati expansion method. (English) Zbl 1435.35337

Summary: A consistent Riccati expansion (CRE) method is proposed for obtaining interaction solutions to the modified Korteweg-de Vries (mKdV) equation. Using the CRE method, it is shown that interaction solutions such as the soliton-tangent (or soliton-cotangent) wave cannot be constructed for the mKdV equation. More importantly, exact soliton-cnoidal periodic wave interaction solutions are presented. While soliton-cnoidal interaction solutions were found to degenerate to special resonant soliton solutions for the values of modulus (\(n\)) closer to one (upper bound of modulus) in the Jacobi elliptic function, a normal kink-shaped soliton was observed for values of \(n\) closer to zero (lower bound).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C08 Soliton solutions
35C05 Solutions to PDEs in closed form
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abdusalam, H. A., On an improved complex tanh-function method, International Journal of Nonlinear Sciences and Numerical Simulation, 6, 2, 99-106 (2005) · Zbl 1401.35012
[2] Abdou, M. A.; Soliman, A. A., Modified extended tanh-function method and its application on nonlinear physical equations, Physics Letters A, 353, 6, 487-492 (2006)
[3] Sekulic, D. L.; Sataric, M. V.; Zivanov, M. B., Symbolic computation of some new nonlinear partial differential equations of nanobiosciences using modified extended tanh-function method, Applied Mathematics and Computation, 218, 7, 3499-3506 (2011) · Zbl 1238.92001
[4] Li, W. W.; Tian, Y.; Zhang, Z., F method and its application for finding new exact solutions to the sine-Gordon and sinh-Gordon equations, Applied Mathematics and Computation, 219, 3, 1135-1143 (2012) · Zbl 1288.35154
[5] Bhrawy, A. H.; Abdelkawy, M. A.; Biswas, A., Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi’s elliptic function method, Communications in Nonlinear Science and Numerical Simulation, 18, 4, 915-925 (2013) · Zbl 1261.35044
[6] Liu, H., Variational approach to nonlinear electrochemical system, International Journal of Nonlinear Sciences and Numerical Simulation, 5, 1, 95-96 (2004)
[7] Liang, J. F.; Gong, L. X., Complex wave solutions for (2+1)-dimensional modified dispersive water wave system, Communications in Theoretical Physics, 52, 1, 17-22 (2009) · Zbl 1170.76312
[8] Chen, S. F.; Guo, Q.; Xu, S.-L.; Belic’, M. .; Zhao, Y.; Zhao, D.; He, J.-R., Vortex solitons in bose-einstein condensates with spin-orbit coupling and gaussian optical lattices, Applied Mathematics Letters, 92, 15-21 (2019) · Zbl 1419.82010
[9] Milan, S. P.; Aleksandra, S.; Branislav, A.; Milivoj, R., Rotating solitons supported by a spiral waveguide, Physical Review A: Atomic, Molecular and Optical Physics, 98, 5 (2017)
[10] Cheng, J.; Xu, S.; Belić, M. R.; Li, H.; Zhao, Y.; Deng, W.; Sun, Y., Multipole solitons in a cold atomic gas with a parity-time symmetric potential, Nonlinear Dynamics, 95, 3, 2325-2332 (2019)
[11] Liu, X.; Triki, H.; Zhou, Q.; Mirzazadeh, M.; Liu, W.; Biswas, A.; Belic, M., Generation and control of multiple solitons under the influence of parameters, Nonlinear Dynamics, 95, 1, 143-150 (2019)
[12] Lou, S. Y., Consistent Riccati expansion for integrable systems, Studies in Applied Mathematics, 134, 3, 372-402 (2015) · Zbl 1314.35145
[13] Liu, X.; Yu, J.; Lou, Z., New interaction solutions from residual symmetry reduction and consistent Riccati expansion of the (2+1)-dimensional Boussinesq equation, Nonlinear Dynamics, 92, 4, 1469-1479 (2018) · Zbl 1398.37077
[14] Huang, L.; Qiao, Z.; Chen, Y., Soliton-cnoidal interactional wave solutions for the reduced Maxwell-Bloch equations, Chinese Physics B, 27, 2 (2018)
[15] Chen, J.; Wu, H.; Zhu, Q., Bäcklund transformation and soliton-cnoidal wave interaction solution for the coupled Klein-Gordon equations, Nonlinear Dynamics, 91, 3, 1949-1961 (2018) · Zbl 1390.37118
[16] Li, Y.; Hu, H., Nonlocal symmetries and interaction solutions of the Benjamin-Ono equation, Applied Mathematics Letters, 75, 18-23 (2018) · Zbl 1377.35010
[17] Liang, J.; Wang, X., Consistent Riccati expansion for finding interaction solutions of (2+1)‐dimensional modified dispersive water‐wave system, Mathematical Methods in the Applied Sciences (2019) · Zbl 1434.35142
[18] Liu, S.; Liu, S., Nonlinear Equation in Physics (2000), Beijing, China: Peking University Press, Beijing, China
[19] Leblond, H.; Mihalache, D., Few-optical-cycle solitons: modified korteweg-de vries sine-gordon equation versus other non-slowly-varying-envelope-approximation models, Physical Review A: Atomic, Molecular and Optical Physics, 79, 6 (2009)
[20] Leblond, H.; Mihalache, D., Few-optical-cycle dissipative solitons, Journal of Physics A: Mathematical and Theoretical, 43, 37 (2010) · Zbl 1203.78041
[21] Zabusky, N. J.; Kruskal, M. D., Interaction of solitons in acollisionless plasma and the recurrence of initial states, Physical Review Letters, 15, 6, 240-243 (1965) · Zbl 1201.35174
[22] Ono, H., Soliton fission in anharmonic lattices with reflectionless inhomogeneity, Journal of the Physical Society of Japan, 61, 12, 4336-4343 (1992)
[23] Ziegler, V.; Dinkel, J.; Setzer, C.; Lonngren, K. E., On the propagation of nonlinear solitary waves in a distributed Schottky barrier diode transmission line, Chaos Solitons and Fractals, 12, 9, 1719-1728 (2001) · Zbl 1022.35063
[24] Helal, M. A., Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics, Chaos Solitons and Fractals, 13, 9, 1917-1929 (2002) · Zbl 0997.35063
[25] Hirota, R., Exact solution of the korteweg—de vries equation for multiple Collisions of solitons, Physical Review Letters, 27, 18, 1192-1194 (1971) · Zbl 1168.35423
[26] Hirota, R., Exact envelope-soliton solutions of a nonlinear wave equation, Journal of Mathematical Physics, 14, 805-809 (1973) · Zbl 0257.35052
[27] Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H., Nonlinear-evolution equations of physical significance, Physical Review Letters, 31, 2, 125-127 (1973) · Zbl 1243.35143
[28] Akhmediev, N.; Korneev, V. I.; Mitskevich, N. V., N-modulation signals in a single-mode optical fiber with allowance for nonlinearity, Journal of Experimental and Theoretical Physics, 94, 159-170 (1988)
[29] Kevrekidis, P. G.; Khare, A.; Saxena, A.; Herring, G., On some classes of mKdV periodic solutions, Journal of Physics A: Mathematical and General, 37, 45, 10959-10965 (2004) · Zbl 1084.35073
[30] Jiao, X.; Lou, S., CRE method for solving mKdV equation and new interactions between solitons and cnoidal periodic waves, Communications in Theoretical Physics, 63, 1, 7-9 (2015) · Zbl 1305.35008
[31] Ma, S. H.; Fang, J. Y.; Zheng, C. L., New exact solutions for the (3+1)-dimensional Jimbo-Miwa system, Chaos Solitons and Fractals, 40, 3, 1352-1355 (2009)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.