Oscillating multidromion excitations in higher-dimensional nonlinear lattice with intersite and external on-site potentials using symbolic computation

doi:10.1088/1674-1056/23/1/010307

摘要/Abstract

摘要： We show by an extensive method of quasi-discrete multiple-scale approximation that nonlinear multi-dimensional lattice waves subjected to intersite and external on-site potentials are found to be governed by (N+1)-dimensional nonlinear Schrödinger (NLS) equation. In particular, the resonant mode interaction of (2+1)-dimensional NLS equation has been identified and the theory allows the inclusion of transverse effect. We apply the exponential function method to the (2+1)-dimensional NLS equation and obtain the class of soliton solutions with a purely algebraic computational method. Notably, we discuss in detail the effects of the external on-site potentials on the explicit form of the soliton solution generated recursively. Under the action of the external on-site potentials, the model presents a rich variety of oscillating multidromion patterns propagating in the system.

关键词: solitons, partial differential equations, lattice dynamics, exact solutions

Abstract: We show by an extensive method of quasi-discrete multiple-scale approximation that nonlinear multi-dimensional lattice waves subjected to intersite and external on-site potentials are found to be governed by (N+1)-dimensional nonlinear Schrödinger (NLS) equation. In particular, the resonant mode interaction of (2+1)-dimensional NLS equation has been identified and the theory allows the inclusion of transverse effect. We apply the exponential function method to the (2+1)-dimensional NLS equation and obtain the class of soliton solutions with a purely algebraic computational method. Notably, we discuss in detail the effects of the external on-site potentials on the explicit form of the soliton solution generated recursively. Under the action of the external on-site potentials, the model presents a rich variety of oscillating multidromion patterns propagating in the system.

Key words: solitons, partial differential equations, lattice dynamics, exact solutions

中图分类号: (Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations)

03.75.Lm

02.60.Lj (Ordinary and partial differential equations; boundary value problems) 04.20.Jb (Exact solutions)

B. Srividya, L. Kavitha, R. Ravichandran, D. Gopi. Oscillating multidromion excitations in higher-dimensional nonlinear lattice with intersite and external on-site potentials using symbolic computation[J]. 中国物理B, 2014, 23(1): 10307-010307.

参考文献 105

[1]	Fermi E, Pasta J and Ulam S 1955 Los Alamos National Laboratory Report (LA1940), also in Collected Papers of Enrico Fermi (Ed. Segré E) (Chicago: University of Chicago Press) 2 978
[2]	Zabusky N J and Kruskal M D 1965 Phys. Rev. Lett. 15 240
[3]	Zabusky N J 2005 Chaos 15 015102
[4]	Christov C I and Velarde M G 1994 Int. J. Bifurcat. Chaos 4 1095
[5]	Christov C I, Maugin G A and Velarde M G 1996 Phys. Rev. E 54 3621
[6]	Toda M 1989 Theory of Nonlinear Lattices (2nd edn.) (Berlin: Springer-Verlag)
[7]	Remoissenet M 1996 Waves Called Solitons (2nd edn.) (Berlin: Springer-Verlag)
[8]	Infeld E and Rowlands G 2000 Nonlinear Waves, Solitons and Chaos (2nd edn.) (England: Cambridge University Press)
[9]	Nekorkin V I and Velarde M G 2002 Patterns, Waves, Solitons, Chaos (Berlin: Springer-Verlag)
[10]	Dauxois T and Peyrard M 2006 Physics of Solitons (Cambridge: Cambridge University Press)
[11]	Denz C, Flach S and Kivshar Y S 2009 Nonlinearities in Periodic Structures and Metamaterials (Heidelberg: Springer-Verlag)
[12]	Huang G, Tong P and Hu B 1999 J. Phys: Condens. Matter 11 2759
[13]	Kivshar Y S, Haelterman M and Sheppard A P 1994 Phys. Rev. E 50 3161
[14]	Sievers A J and Takeno S 1988 Phys. Rev. Lett. 61 970
[15]	Takeno S and Sievers A J 1988 Solid State Commun. 67 1023
[16]	Huang G and Hu B 1998 Phys. Rev. B 57 5746
[17]	Hu B, Huang G and Velarde M G 2000 Phys. Rev. E 62 2827
[18]	Huang G 1995 Phys. Rev. B 51 12347
[19]	Huang G 2001 Chin. Phys. 6 0523
[20]	Brazhnyi V A, Konotop V V and Kuzmiak V 2006 Phys. Rev. Lett. 96 150402
[21]	Huang G, Deng L and Hang C 2005 Phys. Rev. E 72 036621
[22]	Huang G, Makarov V A and Velarde M G 2003 Phys. Rev. A 67 023604
[23]	Stegeman G I and Segev M 1999 Science 286 1518
[24]	Schultz C L, Ablowitz M J and Bar Y D 1987 Phys. Rev. Lett. 59 2825
[25]	Druzhinin O A and Ostrovskii L A 1991 Phys. Lett. A 160 357
[26]	Teixeira Rabelo J N and Maradudin A A 1995 Phys. Lett. A 205 349
[27]	Falo F, Bishop A R, Lomdahl P S and Horovitz B 1991 Phys. Rev. B 43 8081
[28]	Coullet P and Walgraef D 1989 Europhys. Lett. 10 525
[29]	Huss R, Mertens F G and Gaididei Y B 1995 in Fluctuation Phenomena: Disorder and Nonlinearity (Ed. Bishop A R, Jiménez S and Vázquez L) (Singapore: World Scientific) p. 244
[30]	Flach S, Kladko K and Mackay R S 1997 Phys. Rev. Lett. 78 1207
[31]	Pouget J, Remoissenet M and Tamga J M 1993 Phy. Rev. B 47 14866
[32]	Goncalves A L and Konotop V V 2000 Phys. Rev. B 62 14105
[33]	Kiselev S A and Sievers A J 1997 Phys. Rev. B 55 5755
[34]	Bonart D, Mayer A P and Schröder U 1995 Phys. Rev. Lett. 75 870
[35]	Zolotaryuk Y, Savin A V and Christiansen D L 1998 Phys. Rev. B 57 14213
[36]	Suzuki T, Shimono M and Takeno S 1999 Phys. Rev. Lett. 82 1474
[37]	Konotop V V and Takeno S 1997 Phys. Rev. B 55 11342
[38]	Duncan D B, Eilbeck J C, Walshaw C H and Zakharov V E 1991 Phys. Lett. A 158 107
[39]	Gaididei Y, Flytzanis N and Mertens F G 1995 Physica D 82 229
[40]	Gaididei Y B, Christiansen P L and Mingaleev S F 1995 Phys. Scr. 51 289
[41]	Gaididei Y, Huss R and Mertens F G 1998 Eur. Phys. J. B 6 257
[42]	Ostrovskaya E A, Mingaleev S F, Kivshar Y S, Gaididei Y B and Christiansen P L 2001 Phys. Lett. A 282 157
[43]	Kevrekidis P G, Malomed B A and Gaididei Y B 2002 Phys. Rev. E 66 016609
[44]	Burlakov V M, Kiselev S A and Pyrkov V N 1990 Phys. Rev. B 42 4921
[45]	Butt I A and Wattis J A D 2006 J. Phys. A: Math. Gen. 39 4955
[46]	Butt I A and Wattis J A D 2007 J. Phys. A: Math. Theoret. 40 1239
[47]	Astakhova T Y and Vinogradov G A 2006 J. Phys. A: Math. Gen. 39 3593
[48]	Hoq Q E, Carretero-González R, Kevrekidis P G, Malomed B A, Frantzeskakis D J, Bludov Y V and Konotop V V 2008 Phys. Rev. E 78 036605
[49]	Marín J L, Eilbeck J C and Russell F M 1998 Phys. Lett. A 248 225
[50]	Marín J L, Russell F M and Eilbeck J C 2001 Phys. Lett. A 281 21
[51]	Russell F M and Eilbeck J C 2007 Eur. Phys. Lett. 78 10004
[52]	Huang G, Konotop V V, Tam H W and Hu B 2001 Phys. Rev. E 64 056619
[53]	Sun Z Y, Gao Y T, Yu X, Meng X H and Liu Y 2009 Wave Motion 46 511
[54]	Chow K W 2002 Wave Motion 35 71
[55]	Borhanifar A, Kabir M M and Maryam Vahdat L 2009 Chaos Soliton. Fract. 42 1646
[56]	Huang W H, Zhang J F and Sheng Z M 2002 Chin. Phys. 11 1101
[57]	Bai C L 2001 Chin. Phys. 10 1091
[58]	Fokas A S and Ablowitz M J 1983 in Nonlinear Phenomena: Lecture Notes in Physics (Ed. Wolf K B) (Berlin: Springer-Verlag) 189 137
[59]	Hietarinta J and Hirota R 1990 Phys. Lett. A 145 237
[60]	Matveev V B and Salle M A 1991 Darboux Transformations and Solitons (Berlin: Springer-Verlag)
[61]	Boiti M, Leon J J P, Martina L and Pempinelli F 1988 Phys. Lett. A 132 432
[62]	Wazwaz A M 2005 Comput. Math. Appl. 49 1101
[63]	Kavitha L, Sathishkumar P, Nathiyaa T and Gopi D 2009 Phys. Scr. 79 035403
[64]	Fan E and Zhang H 1998 Phys. Lett. A 246 403
[65]	Abdelhalim E and Emad H A 2012 Wave Motion 49 296
[66]	Kavitha L, Saravanan M, Srividya B and Gopi D 2011 Phys. Rev. E 84 066608
[67]	He J H and Wu X H 2007 Comput. Math. Appl. 54 881
[68]	Wang C, Wu Y Y and Wu W 2005 Wave Motion 41 329
[69]	Li H M and Wu F M 2004 Chin. Phys. Lett. 21 1425
[70]	Zheng L X, Zhang J L, Wang Y M and Wang M L 2004 Acta Phys. Sin. 53 4045 (in Chinese)
[71]	Kavitha L, Sathiskumar P and Gopi D 2009 Phys. Scr. 79 015402
[72]	Kavitha L, Prabhu A and Gopi D 2009 Chaos Soliton. Fract. 42 2322
[73]	Kavitha L, Srividya B, Akila N and Gopi D 2010 Nonl. Sci. Lett. A 1 95
[74]	Kavitha L, Srividya B and Gopi D 2010 J. Magn. Magn. Mater. 322 1793
[75]	Wang D L, Yan X H and Wang F J 2007 Chin. Phys. Lett. 24 1817
[76]	Wang F J, Yan X H and Wang F L 2008 Chin. Phys. Lett. 25 4
[77]	Zhu S H, Huang G and Cui W N 2002 Chin. Phys. 11 0919
[78]	Wang D L and Feng P C 2003 Acta Phys. Sin. 52 1332 (in Chinese)
[79]	Tsurui A 1972 Prog. Theor. Phys. 48 1196
[80]	Huang G, Shi Z P and Xu Z X 1993 Phys. Rev. B 47 14561
[81]	Yoshimura K and Watanabe S 1991 J. Phys. Soc. Jpn. 60 82
[82]	Campa A, Giansanti A, Tenenbaum A, Levi D and Ragnisco O 1993 Phys. Rev. B 48 10168
[83]	Jeffrey A and Kawahawa T 1982 Asymptotic Methods in Nonlinear Wave Theory (London: Pitman)
[84]	Konotop V V 1996 Phys. Rev. E 54 4266
[85]	Ruan H Y and Chen Y X 2003 Chaos Soliton. Fract. 15 947
[86]	Nistazakis H E, Frantzeskakis D J, Balourdos P S, Tsigopoulos A and Malomed B A 2000 Phys. Lett. A 278 68
[87]	Rosanov N N, Vladimirov A G, Skryabin D V and Firth W J 2002 Phys. Lett. A 293 45
[88]	Lakshmanan M and Radha R 1997 Pramana J. Phys. 48 163
[89]	Radha R and Lakshmanan M 1997 Chaos Soliton. Fract. 8 17
[90]	Zhong W P, Belić M R, Mihalache D, Malomed B A and Huang T 2012 Romanian Reports in Physics 64 1399
[91]	Hongyan Z and Hongqing Z 2009 Chaos Soliton. Fract. 39 120
[92]	Aly R S 2012 Appl. Math. Lett. 25 687
[93]	Kates R E and Kaup D J 1988 Astron. Astrophys. 206 9
[94]	Fibich G 2001 SIAM J. Appl. Math. 61 1680
[95]	Gagnon L and Bélanger P A 1991 Phys. Rev. A 43 6187
[96]	Agrawal G P 1991 Phys. Rev. A 44 7493
[97]	He J H and Wu X H 2006 Chaos Soliton. Fract. 30 700
[98]	He J H 2008 Int. J. Mod. Phys. B 22 3487
[99]	He J H 2006 Non-Perturbative Methods for Strongly Nonlinear Problems (Ph.D. Dissertation) (Berlin: de-verlag im Internet GmbH)
[100]	Wu X H and He J H 2007 Comput. Math. Appl. 54 966
[101]	Bekir A and Boz A 2007 Int. J. Nonlinear Sci. Numer. Simul. 8 505
[102]	Marinakis V 2008 Z. Naturforsch 63a 653
[103]	Zhang S and Zhang H Q 2009 Phys. Lett. A 373 2501
[104]	Zhu S D 2007 Int. J. Nonlinear Sci. Numer. Simul. 8 461
[105]	Zhu S D 2007 Int. J. Nonlinear Sci. Numer. Simul. 8 465