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Singular solitons, shock waves, and other solutions to potential KdV equation. (English) Zbl 1306.35116

Summary: This paper addresses the potential Korteweg-de Vries equation. The singular 1-soliton solution is obtained by the aid of ansatz method. Subsequently, the \(G'/G\)-expansion method and the exp-function approach also gives a few more interesting solutions. Finally, the Lie symmetry analysis leads to another plethora of solution to the equation. These results are going to be extremely useful and applicable in applied mathematics and theoretical physics.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C08 Soliton solutions
35B06 Symmetries, invariants, etc. in context of PDEs
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[1] Abdelkawy, M.A., Bhrawy, A.\[H.: G^{\prime }/G\] G′/G-expansion method for two-dimensional force-free magnetic fields described by some nonlinear equations. Indian J. Phys. 87(6), 555-565 (2013) · doi:10.1007/s12648-013-0255-y
[2] Bhrawy, A., Tharwat, M.M., Abdelkawy, M.A.: Integrable system modelling shallow water waves: Kaup-Boussinesq shallow water system. Indian J. Phys. 87(7), 665-671 (2013) · doi:10.1007/s12648-013-0260-1
[3] Biswas, A., Kumar, S., Krishnan, E.V., Ahmed, B., Strong, A., Johnson, S., Yildirim, A.: Topological solitons and other solutions to potential KdV equation. Rom. Rep. Phys. 65(4), 1125-1137 (2013) · Zbl 1152.35482
[4] Biswas, A., Yildirim, A., Hayat, T., Aldossary, O.M., Sassaman, R.: Soliton perturbation theory for the generalized Klein-Gordon equation with full nonlinearity. Proc. Rom. Acad. 13(1), 32-41 (2012)
[5] Ebadi, G., Kara, A.H., Petkovic, M.D., Yildirim, A., Biswas, A.: Solitons and conserved quantities of the Ito equation. Proc. Rom. Acad. 13(3), 215-224 (2012)
[6] Guo, R., Tian, B., Wang, L.: Soliton solution for the reduced Maxwell-Bloch system in nonlinear optics via the \[NN\]-fold Darboux transformation. Nonlinear Dyn. 69, 2009-2020 (2012) · Zbl 1263.35200 · doi:10.1007/s11071-012-0403-5
[7] Guo, R., Hao, H.-Q.: Dynamic behaviors of the breather solutions for the \[AB\] AB system in fluid mechanics. Nonlinear Dyn. 74, 701-709 (2013) · doi:10.1007/s11071-013-0998-1
[8] Guo, R., Hao, H.-Q.: Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrödinger’s equation. Commun. Nonlinear Sci. Numer. Simul. 18(9), 2426-2435 (2013) · Zbl 1304.35641 · doi:10.1016/j.cnsns.2013.01.019
[9] Guo, R., Hao, H.-Q., Zhang, L.-L.: Bound solitons and breathers for the generalized generalized coupled nonlinear Schrödinger-Maxwell-Bloch system. Mod. Phys. Lett. 27(17), 1350130 (2013) · doi:10.1142/S0217984913501303
[10] Guo, Y., Wang, Y.: On Weierstrass elliptic function solutions for a \[(N+1)(N+1)\]-dimensional potential KdV equation. Appl. Math. Comput. 217(20), 8080-8092 (2011) · Zbl 1219.35231 · doi:10.1016/j.amc.2011.03.007
[11] Gupta, R.K., Bansal, A.: Similarity reductions and exact solutions of generalized Bretherton equation with time-dependent coefficients. Nonlinear Dyn. 71(1-2), 1-12 (2013) · Zbl 1268.35007 · doi:10.1007/s11071-012-0637-2
[12] Hirota, R., Hu, X.-B., Tang, X.-Y.: A vector potential KdV equation and vector Ito equation: soliton solutions, bilinear Bäcklund transformations and Lax pairs. J. Math. Anal. Appl. 288(1), 326-348 (2003) · Zbl 1055.35100 · doi:10.1016/j.jmaa.2003.08.046
[13] Liu, H., Li, J., Liu, L.: Lie symmetry analysis, optimal systems and exact solutions to the fifth-order KdV types of equations. J. Math. Anal. Appl. 368, 551-558 (2010) · Zbl 1192.35011 · doi:10.1016/j.jmaa.2010.03.026
[14] Liu, H., Li, J.: Lie symmetry analysis and exact solutions for the short pulse equation. Nonlinear Anal. 71, 2126-2133 (2009) · Zbl 1244.35003 · doi:10.1016/j.na.2009.01.075
[15] Lü, X., Peng, M.: Nonautonomous motion study on accelerated and decelerated solitons for the variable coefficient Lennels-Fokas model. Chaos 23, 013122 (2013) · Zbl 1319.37010 · doi:10.1063/1.4790827
[16] Lü, X., Peng, M.: Systematic construction of infinitely many conservation laws for certain nonlinear evolution equations in mathematical physics. Commun. Nonlinear Sci. Numer. Simul. 18, 2304-2312 (2013) · Zbl 1304.35030 · doi:10.1016/j.cnsns.2012.11.006
[17] Lü, X., Peng, M.: Painleve integrability and explicit solutions of the general two-coupled nonlinear Schrödinger system in the optical fiber communications. Nonlinear Dyn. 73(1-2), 405-410 (2013) · Zbl 1281.35078
[18] Lü, X.: Soliton behavior for a generalized mixed Schrödinger model with \[NN\]-fold Darboux transformation. Chaos 23, 033137 (2013) · Zbl 1323.35163
[19] Lü, X.: New bilinear Backlund transformation with multi-soliton solutions for the (2+1)-dimensional Sawada-Kotera-model. Nonlinear Dyn. doi:10.1007/s11071-013-1118-y · Zbl 1319.35222
[20] Triki, H., Wazwaz, A.M.: Dark solitons for a combined potential KdV equation and Schwarzian KdV equation with \[t\] t-dependent coefficients and forcing term. Appl. Math. Comput. 217(21), 8846-8851 (2011) · Zbl 1219.35261 · doi:10.1016/j.amc.2011.03.050
[21] Triki, H., Yildirim, A., Hayat, T., Aldossary, O.M., Biswas, A.: Topological and non-topological soliton solutions of the Bretherton equation. Proc. Rom. Acad. 13(2), 103-108 (2012)
[22] Triki, H., Yildirim, A., Hayat, T., Aldossary, O.M., Biswas, A.: Shock wave solution of the Benney-Luke equation. Rom. J. Phys. 57(7-8), 1029-1034 (2012)
[23] Wazwaz, A.M.: Analytic study on the one and two spatial dimensional potential KdV equations. Chaos, Solitons & Fractals. 36(1), 175-181 (2008) · Zbl 1152.35482 · doi:10.1016/j.chaos.2006.06.018
[24] Yang, Z.: New exact traveling wave solutions for two potential coupled KdV equations with symbolic computation. Chaos, Solitons & Fractals. 34(3), 932-939 (2007) · Zbl 1205.35285 · doi:10.1016/j.chaos.2006.04.030
[25] Xie, F., Zhang, Y., Lü, Z.: Symbolic computation in non-linear evolution equation: application to (3+1)-dimensional Kadomtsev Petviashvili equation. Chaos, Solitons & Fractals. 24(1), 257-263 (2005) · Zbl 1067.35095 · doi:10.1016/j.chaos.2004.09.019
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