Coherent structure of Alice-Bob modified Korteweg de-Vries equation. (English) Zbl 1398.37076
Summary: To describe two-place events, Alice-Bob systems have been established by means of the shifted parity and delayed time reversal in [the second author, “Alice-Bob systems, \(P_{s}\)-\(T_{d}\)-\(C\) principles and multi-soliton solutions”, Preprint, arXiv:1603.03975]. In this paper, we mainly study exact solutions of the integrable Alice-Bob modified Korteweg de-Vries (AB-mKdV) system. The general \(N\)th Darboux transformation for the AB-mKdV equation is constructed. By using the Darboux transformation, some types of shifted parity and time reversal symmetry breaking solutions including one-soliton, two-soliton, and rogue wave solutions are explicitly obtained. In addition to the similar solutions of the mKdV equation (group invariant solutions), there are abundant new localized structures for the AB-mKdV systems.
MSC:
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35C08 | Soliton solutions |
Keywords:
nonlinear nonlocal partial differential equations; Darboux transformations; exact solutions; solitons; rogue wavesReferences:
| [1] | Ablowitz, MJ; Musslimani, ZH, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., 110, 064105, (2013) · doi:10.1103/PhysRevLett.110.064105 |
| [2] | Bender, CM, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys., 70, 947-1018, (2007) · doi:10.1088/0034-4885/70/6/R03 |
| [3] | Markum, H; Pullirsch, R; Wettig, T, Non-Hermitian random matrix theory and lattice QCD with chemical potential, Phys. Rev. Lett., 83, 484-487, (1999) · Zbl 1051.81602 · doi:10.1103/PhysRevLett.83.484 |
| [4] | Lin, Z; Schindler, J; Ellis, FM; Kottos, T, Experimental observation of the dual behavior of PT-symmetric scattering, Phys. Rev. A, 85, 050101, (2012) · doi:10.1103/PhysRevA.85.050101 |
| [5] | Ruter, CE; Makris, KG; EI-Ganainy, R; Christodoulides, DN; Segev, M; Kip, D, Observation of paritytime symmetry in optics, Nat. Phys., 6, 192-195, (2010) · doi:10.1038/nphys1515 |
| [6] | Musslimani, ZH; Makris, KG; EI-Ganainy, R; Christodoulides, DN, Optical solitons in PT periodic potentials, Phys. Rev. Lett., 100, 030402, (2008) · doi:10.1103/PhysRevLett.100.030402 |
| [7] | Dalfovo, F; Giorgini, S; Pitaevskii, LP; Stringari, S, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71, 463-512, (1999) · doi:10.1103/RevModPhys.71.463 |
| [8] | Lou, S.Y.: Alice-Bob systems, \(P_{s}\)-\(T_{d}\)-\(C\) principles and multi-soliton solutions, arXiv:1603.03975v2 [nlin.SI], (2016) |
| [9] | Jia, M., Lou, S.Y.: An AB-KdV equation: exact solutions, symmetry reductions and Bäcklund transformations, arXiv:1612.00546v1 [nlin.SI], (2016) |
| [10] | Ablowitz, MJ; Musslimani, ZH, Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation, Nonlinearity, 29, 915-946, (2016) · Zbl 1338.37099 · doi:10.1088/0951-7715/29/3/915 |
| [11] | Ji, JL; Zhu, ZN, Soliton solutions of an integrable nonlocal modified Korteweg-de Vries equation through inverse scattering transform, J. Math. Anal. Appl., 453, 973, (2017) · Zbl 1369.35079 · doi:10.1016/j.jmaa.2017.04.042 |
| [12] | Ablowitz, MJ; Musslimani, ZH, Integrable discrete PT symmetric model, Phys. Rev. E, 90, 032912, (2014) · doi:10.1103/PhysRevE.90.032912 |
| [13] | Song, CQ; Xiao, DM; Zhu, ZN, Solitons and dynamics for a general integrable nonlocal coupled nonlinear Schrödinger equation, Commun. Nonlin. Sci. Numer. Simulat., 45, 13-28, (2017) · Zbl 1485.35346 · doi:10.1016/j.cnsns.2016.09.013 |
| [14] | Dimakos, M; Fokas, AS, Davey-Stewartson type equations in 4+2 and 3+1 possessing soliton solutions, J. Math. Phys., 54, 081504, (2013) · Zbl 1290.35199 · doi:10.1063/1.4817345 |
| [15] | Fokas, AS, Integrable nonlinear evolution partial differential equations in 4+2 and 3+1 dimensions, Phys. Rev. Lett., 96, 190201, (2006) · Zbl 1228.35199 · doi:10.1103/PhysRevLett.96.190201 |
| [16] | Fokas, AS, Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation, Nonlinearity, 29, 319-324, (2016) · Zbl 1339.37066 · doi:10.1088/0951-7715/29/2/319 |
| [17] | Yan, ZY, Integrable PT-symmetric local and nonlocal vector nonlinear schrodinger equations: a unified two-parameter model, Appl. Math. Lett., 47, 61-68, (2015) · Zbl 1322.35132 · doi:10.1016/j.aml.2015.02.025 |
| [18] | Yan, ZY, Nonlocal general vector nonlinear schrodinger equations: PT symmetribility, and solutions, Appl. Math. Lett., 62, 101-109, (2016) · Zbl 1351.35197 · doi:10.1016/j.aml.2016.07.010 |
| [19] | Lou, SY; Qiao, ZJ, Alice-bob peakon systems, Chin. Phys. Lett., 34, 100201, (2007) · doi:10.1088/0256-307X/34/10/100201 |
| [20] | Lou, S.Y., Huang, F.: Alice-Bob Physics, coherent solutions of nonlocal KdV systems, arXiv:1606.03154v1 [nlin.SI], (2016); Sci. Rep. 7, 869 (2017) |
| [21] | Lou, SY, Consistent Riccati expansion for integrable systems, Stud. Appl. Math., 134, 372-402, (2015) · Zbl 1314.35145 · doi:10.1111/sapm.12072 |
| [22] | Gao, Y; Tang, XY, A coupled variable coefficient modified KdV equation arising from a two-layer fluid system, Commun. Theor. Phys., 48, 961-970, (2007) · Zbl 1267.35194 · doi:10.1088/0253-6102/48/6/001 |
| [23] | Ji, JL; Zhu, ZN, On a nonlocal modified Korteweg-de Vries equation: integrability, Darboux transformation and soliton solutions, Commun. Nonlin. Sci. Numer. Simulat., 42, 699-708, (2017) · Zbl 1473.37081 · doi:10.1016/j.cnsns.2016.06.015 |
| [24] | Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991) · Zbl 0744.35045 · doi:10.1007/978-3-662-00922-2 |
| [25] | Gu, C.H., Hu, H.S., Zhou, Z.X.: Darboux Transformation in Soliton Theory and its Geometric Applications. Shanghai Sci-Tech Publishing House, Shanghai (2005) |
| [26] | Ma, W.X.: Wronskian solutions to integrable equations. Discret. Contin. Dyn. Syst, Suppl 506-515 (2009) · Zbl 1184.37057 |
| [27] | Wazwaz, A-M, Multiple soliton solutions and multiple complex solitons for two distinct Boussinesq equations, Nonlin. Dyn., 85, 731-737, (2016) · doi:10.1007/s11071-016-2718-0 |
| [28] | Wazwaz, A-M; El-Tantawy, SA, New (3+1)-dimensional equations of Burgers type and Sharma Tasso Olver type: multiple soliton solutions, Nonlin. Dyn., 87, 2457-2461, (2017) · Zbl 1373.37161 · doi:10.1007/s11071-016-3203-5 |
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.
