Geodesic
Super quasiperiodic wave solutions and asymptotic analysis for $\mathcal N=1$ supersymmetric $\text{KdV}$-type equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 166 (2011) no. 3, pp. 366-387 Cet article a éte moissonné depuis la source Math-Net.Ru

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Based on a general multidimensional Riemann theta function and the super Hirota bilinear form, we extend the Hirota method to construct explicit super quasiperiodic (multiperiodic) wave solutions of $\mathcal N=1$supersymmetric KdV-type equations in superspace. We show that the supersymmetric KdV equation does not have an $N$-periodic wave solution with arbitrary parameters for $N\ge2$. In addition, an interesting influencing band occurs among the super quasiperiodic waves under the presence of a Grassmann variable. We also observe that the super quasiperiodic waves are symmetric about this band but collapse along with it. We present a limit procedure for analyzing the asymptotic properties of the super quasiperiodic waves and rigorously show that the super periodic wave solutions tend to super soliton solutions under some “small amplitude” limits.
Keywords: supersymmetric KdV-type equation, super Hirota bilinear method, Riemann theta function, super quasiperiodic wave solution
Mots-clés : super soliton solution. 
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     author = {Y. C. Hon and Engui Fan},
     title = {Super quasiperiodic wave solutions and asymptotic analysis for $\mathcal N=1$ supersymmetric $\text{KdV}$-type equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {366--387},
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     volume = {166},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2011_166_3_a3/}
}
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Y. C. Hon; Engui Fan. Super quasiperiodic wave solutions and asymptotic analysis for $\mathcal N=1$ supersymmetric $\text{KdV}$-type equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 166 (2011) no. 3, pp. 366-387. http://geodesic.mathdoc.fr/item/TMF_2011_166_3_a3/

[1] R. Hirota, J. Satsuma, Progr. Theoret. Phys., 57:3 (1977), 797–807 | DOI | MR | Zbl

[2] R. Hirota, The Direct Methods in Soliton Theory, Cambridge Tracts Math., 155, Cambridge University Press, Cambridge | MR | Zbl

[3] X. B. Hu, P. A. Clarkson, J. Phys. A, 28:17 (1995), 5009–5016 | DOI | MR | Zbl

[4] X.-B. Hu, C.-X. Li, J. J. C. Nimmo, G.-F. Yu, J. Phys. A, 38:1 (2005), 195–204 | DOI | MR | Zbl

[5] R. Hirota, Y. Ohta, J. Phys. Soc. Japan, 60:3 (1991), 798–809 | DOI | MR | Zbl

[6] D.-j. Zhang, J. Phys. Soc. Japan, 71:11 (2002), 2649–2656 | DOI | MR | Zbl

[7] K. Sawada, T. Kotera, Progr. Theoret. Phys., 51:5 (1974), 1355–1367 | DOI | MR | Zbl

[8] A. A. Nakamura, J. Phys. Soc. Japan, 47:5 (1979), 1701–1705 | DOI | MR | Zbl

[9] A. Nakamura, J. Phys. Soc. Japan, 48:4 (1980), 1365–1370 | DOI | MR | Zbl

[10] H. H. Dai, E. G. Fan, X. G. Geng, Periodic wave solutions of nonlinear equations by Hirota's bilinear method, arXiv: nlin/0602015

[11] Y. C. Hon, E. G. Fan, Z. Qin, Modern Phys. Lett. B, 22:8 (2008), 547–553 | DOI | MR | Zbl

[12] E. G. Fan, Y. C. Hon, Phys. Rev. E, 78:3 (2008), 036607, 13 pp. | DOI | MR

[13] S. P. Novikov, Funkts. analiz i ego pril., 8:3 (1974), 54–66 | DOI | MR | Zbl

[14] B. A. Dubrovin, Funkts. analiz i ego pril., 9:3 (1975), 41–51 | DOI | MR | Zbl

[15] A. R. Its, V. B. Matveev, Funkts. analiz i ego pril., 9:1 (1975), 69–70 | DOI | MR | Zbl

[16] P. D. Lax, Comm. Pure Appl. Math., 28:1 (1975), 141–188 | DOI | MR | Zbl

[17] H. P. Mckean, P. van Moerbeke, Invent. Math., 30:3 (1975), 217–274 | DOI | MR | Zbl

[18] E. D. Belokolos, A. I. Bobenko, V. Z. Enol'skii, A. R. Its, V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer Ser. Nonlinear Dyn., Springer, Berlin, 1994 | Zbl

[19] F. Gesztesy, H. Holden, Soliton Equations and Their Algebro-Geometric Solutions, v. I, Cambridge Stud. Adv. Math., 79, $(1+1)$-Dimensional Continuous Models, Cambridge Univ. Press, Cambridge, 2003 | MR | Zbl

[20] F. Gesztesy, H. Holden, Phil. Trans. R Soc. A, 366:1867 (2008), 1025–1054 | DOI | MR | Zbl

[21] Z. J. Qiao, Comm. Math. Phys., 239:1–2 (2003), 309–341 | DOI | MR | Zbl

[22] L. Zampogni, Adv. Nonlinear Stud., 7:3 (2007), 345–380 | DOI | MR | Zbl

[23] R. G. Zhou, J. Math. Phys., 38:5 (1997), 2535–2546 | DOI | MR | Zbl

[24] C. Cao, Y. Wu, X. Geng, J. Math. Phys., 40:8 (1999), 3948–3970 | DOI | MR | Zbl

[25] X. G. Geng, Y. T. Wu, C. W. Cao, J. Phys. A, 32:20 (1999), 3733–3742 | DOI | MR | Zbl

[26] X. G. Geng, C. W. Cao, Nonlinearity, 14:6 (2001), 1433–1452 | DOI | MR | Zbl

[27] X. G. Geng, H. H. Dai, J. Y. Zhu, Stud. Appl. Math., 118:3 (2007), 281–312 | DOI | MR

[28] Y. C. Hon, E. G. Fan, J. Math. Phys., 46:3 (2005), 032701, 21 pp. | DOI | MR | Zbl

[29] Yu. I. Manin, A. O. Radul, Comm. Math. Phys., 98:1 (1985), 65–77 | DOI | MR | Zbl

[30] P. Mathieu, J. Math. Phys., 29:11 (1988), 2499–2506 | DOI | MR | Zbl

[31] W. Oevel, Z. Popowicz, Comm. Math. Phys., 139:3 (1991), 441–460 | DOI | MR | Zbl

[32] P. Mathieu, Phys. Lett. A, 128:3–4 (1988), 169–171 | DOI | MR

[33] Q. P. Liu, Lett. Math. Phys., 35:2 (1995), 115–122, arXiv: hep-th/9409008 | DOI | MR | Zbl

[34] Q. P. Liu, Y. F. Xie, Phys. Lett. A, 325:2 (2004), 139–143, arXiv: nlin/0212002 | DOI | MR | Zbl

[35] Q. P. Liu, M. Mañas, Phys. Lett. B, 436:3–4 (1998), 306–310, arXiv: solv-int/9806005 | DOI | MR

[36] A. S. Carstea, Nonlinearity, 13:5 (2000), 1645–1656 | DOI | MR | Zbl

[37] V. S. Vladimirov, I. V. Volovich, TMF, 59:1 (1984), 3–27 | DOI | MR | Zbl

[38] V. S. Vladimirov, I. V. Volovich, TMF, 60:2 (1984), 169–198 | DOI | MR | Zbl

[39] D. Mumford, Tata Lectures on Theta, II, Progress Math., 43, Jacobian Theta Functions and Differential Equations, Birkhäuser, Boston, MA, 1984 | MR | Zbl

[40] P. P. Kulish, A. M. Zeitlin, Phys. Lett. B, 581:1–2 (2004), 125–132, arXiv: hep-th/0312159 | DOI | MR | Zbl

[41] P. P. Kulish, A. M. Zeitlin, Phys. Lett. B, 597:2 (2004), 229–236, arXiv: hep-th/0407154 | DOI | MR | Zbl