Abstract
In this paper, the generalized unified method is used to construct multi-rational wave solutions of the (\(2 + 1\))-dimensional Kadomtsev–Petviashvili equation with variable coefficients. This is an extension of the previous work that was given by the same author in Osman and Abdel-Gawad (EPJ Plus 130(10):1–11, 2015). The (2 \(+\) 1)-dimensional Kadomtsev–Petviashvili equation with variable coefficients can be used to characterize many nonlinear phenomena in fluid dynamics, plasma physics and some other nonlinear science when the inhomogeneities of media and non-uniformities of boundaries are taken into consideration. To give more physical insight into the obtained solutions, we present graphically their representative structures by setting the arbitrary functions in the solutions as specific functions. Moreover, the influences of the variable coefficient functions and interaction properties of solitary waves are discussed for physical interests and possible applications.
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Osman, M.S., Abdel-Gawad, H.I.: Multi-wave solutions of the (2 \(+\) 1)-dimensional Nizhnik–Novikov–Veselov equations with variable coefficients. EPJ Plus. 130(10), 1–11 (2015)
Mirzazadeh, M., Eslami, M., Biswas, A.: 1-Soliton solution of KdV6 equation. Nonlinear Dyn. 80(1–2), 387–396 (2015)
Munro, S., Parkes, E.J.: Stability of solitary-wave solutions to a modified Zakharov–Kuznetsov equation. J. Plasma Phys. 64(4), 411–426 (2000)
Atre, R., Panigrahi, P.K.: Class of solitary wave solutions of the one-dimensional Gross–Pitaevskii equation. Phys. Rev. E. 73(5), 056611 (2006)
Zhou, Q., Ekici, M., Sonmezoglu, M., Mirzazadeh, M., Eslami, M.: Optical solitons with Biswas–Milovic equation by extended trial equation method. Nonlinear Dyn. 84(4), 1883–1900 (2016)
Wazwaz, A.M.: New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations. Appl. Math. Comput. 186(1), 130–141 (2007)
Kangalgil, F., Ayaz, F.: Solitary wave solutions for the KdV and mKdV equations by differential transform method. Chaos Soliton Fract. 41(1), 464–472 (2009)
Zhang, L., Khalique, C.M.: Exact solitary wave and quasi-periodic wave solutions of the KdV–Sawada–Kotera–Ramani equation. Adv. Differ. Equ-NY. 2015(1), 1–12 (2015)
Mohebbi, A.: Solitary wave solutions of the nonlinear generalized Pochhammer–Chree and regularized long wave equations. Nonlinear Dyn. 70(4), 2463–2474 (2012)
Ganji, D.D., Rafei, M.: Solitary wave solutions for a generalized Hirota–Satsuma coupled KdV equation by homotopy perturbation method. Phys. Lett. A. 356(2), 131–137 (2006)
Morris, R.M., Kara, A.H., Biswas, A.: An analysis of the Zhiber–Shabat equation including Lie point symmetries and conservation laws. Collect. Math. 67(1), 55–62 (2016)
Mirzazadeh, M., Eslami, M., Zerrad, E., Mahmood, M.F., Biswas, A., Belic, M.: Optical solitons in nonlinear directional couplers by sine–cosine function method and Bernoulli’s equation approach. Nonlinear Dyn. 81(4), 1933–1949 (2015)
Eslami, M., Vajargah, B.F., Mirzazadeh, M., Biswas, A.: Application of first integral method to fractional partial differential equations. Indian J. Phys. 88(2), 177–184 (2014)
Mirzazadeh, M., Arnous, A.H., Mahmood, M.F., Zerrad, E., Biswas, A.: Soliton solutions to resonant nonlinear Schrödinger’s equation with time-dependent coefficients by trial solution approach. Nonlinear Dyn. 81(1–2), 277–282 (2015)
Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg–deVries equation. Phys. Rev. E. 19(19), 1095 (1976)
Ablowitz, M.J., Clarkson, P.A.: Soliton, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)
Gu, C.: Soliton Theory and Its Application. NASA STI/Recon Technical Report A 1 (1995)
Li, Y., Zhang, J.E.: Darboux transformations of classical Boussinesq system and its multi-soliton solutions. Phys. Lett. A. 284(6), 253–258 (2001)
Guo, B., Ling, L., Liu, Q.P.: Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E. 85(2), 026607 (2012)
Hirota, R.: Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27(18), 1192–1194 (1971)
Hietarinta, J.: A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations. J. Math. Phys. 28(8), 1732–1742 (1987)
Wazwaz, A.M.: Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method. Appl. Math. Comput. 190(1), 633–640 (2007)
Wei, G.M., Gao, Y.T., Xu, T., Meng, X.H., Zhang, C.Y.: Painleve’ property and new analytic solutions for a variable-coefficient Kadomtsev–Petviashvili equation with symbolic computation. Chin. Phys. Lett. 25, 1599–1602 (2008)
Yomba, E.: Construction of new soliton-like solutions for the (2 \(+\) 1)-dimensional KdV equation with variable coefficients. Chaos Soliton Fract. 21, 75–79 (2004)
Ye, L.Y., Lv, Y.N., Zhang, Y., Jin, H.P.: Grammian solutions to a variable-coefficient KP equation. Chin. Phys. Lett. 25, 357–358 (2008)
Abdel-Gawad, H.I., Elazab, N.S., Osman, M.: Exact solutions of space dependent Korteweg–de Vries equation by the extended unified method. J. Phys. Soc. Jpn. 82, 044004 (2013)
Abdel-Gawad, H.I., Osman, M.: Exact solutions of the Korteweg–de Vries equation with space and time dependent coefficients by the extended unified method. Indian J. Pure Appl. Math. 45(1), 1–11 (2014)
Abdel-Gawad, H.I., Osman, M.: On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients. J. Adv. Res. 6(4), 593–599 (2015)
Abdel-Gawad, H.I., Tantawy, M., Osman, M.S.: Dynamic of DNA’s possible impact on its damage. Math. Methods Appl. Sci. 39(2), 168–176 (2016)
Wang, Y.Y., Zhang, J.F.: Variable-coefficient KP equation and solitonic solution for two-temperature ions in dusty plasma. Phys. Lett. A. 352(1), 155–162 (2006)
Yomba, E.: Abundant families of Jacobi elliptic function-like solutions for a generalized variable coefficients 2D KdV equation via the extended mapping method. Phys. Lett. A. 349(1), 212–219 (2006)
Li, L.L., Tian, B., Zhang, C.Y., Xu, T.: On a generalized Kadomtsev–Petviashvili equation with variable coefficients via symbolic computation. Phys. Scr. 76(5), 411 (2007)
Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)
Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)
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Osman, M.S. Nonlinear interaction of solitary waves described by multi-rational wave solutions of the (2 \(+\) 1)-dimensional Kadomtsev–Petviashvili equation with variable coefficients. Nonlinear Dyn 87, 1209–1216 (2017). https://doi.org/10.1007/s11071-016-3110-9
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DOI: https://doi.org/10.1007/s11071-016-3110-9