-
我们提出了二维自相似变换理论, 以聚焦的(2 + 1)维NLS方程(数学称为抛物型的非线性微分方程)为模型, 构建了它被转变为聚焦的(1 + 1)维NLS方程的二维自相似变换, 深入研究了它的空间怪波激发, 发现除了(1 + 1)维NLS方程的Peregrine孤子、高阶怪波和多怪波诱导的线怪波所具有的短寿命特征外, 由Akhmediev呼吸子(AB)和Kuznetsov-Ma孤子(KMS)诱导的线怪波也具有这种短寿命特征. 这与由亮孤子(包括多孤子)诱导的空间相干结构保持形状和幅值不变的演化特征完全不同. 通过图示展现了本文例举的各类线怪波的演化规律. 本文揭示的线怪波激发新机制, 有助于提升对高维非线性波动模型的相干结构的新认识.A two-dimensional self-similarity transformation theory is established, and the focusing (parabolic) (2 + 1)-dimensional NLS equation is taken as the model. The two-dimensional self-similarity transformation is proposed for converting the focusing (2 + 1)-dimensional NLS equation into the focusing (1 + 1) dimensional NLS equations, and the excitation of its novel line-rogue waves is further investigated. It is found that the spatial coherent structures induced by the Akhmediev breathers (AB) and Kuznetsov-Ma solitons (KMS) also have the short-lived characteristics which are possessed by the line-rogue waves induced by the Peregrine solitons, and the other higher-order rogue waves and the multi-rogue waves of the (1 + 1) dimensional NLS equations. This is completely different from the evolution characteristics of spatially coherent structures induced by bright solitons (including multi-solitons and lump solutions), with their shapes and amplitudes kept unchanged. The diagram shows the evolution characteristics of all kinds of resulting line rogue waves. The new excitation mechanism of line rogue waves revealed contributes to the new understanding of the coherent structure of high-dimensional nonlinear wave models.
-
Keywords:
- rogue wave /
- line rogue wave /
- two-dimensional self-similar transformation /
- nonlinear Schrödinger equation
[1] Onorato M, Osborne A R, Serio M, Bertone S 2001 Phys. Rev. Lett. 86 5831
Google Scholar
[2] Kharif C, Pelinovsky E, Slunyaev A 2009 Rogue Waves in the Ocean. In: Advances in Geophysical and Environmental Mechanics and Mathematics (Berlin: Springer-Verlag)
[3] Adcock T A A, Taylor P H 2014 Rep. Prog. Phys. 77 105901
Google Scholar
[4] Dudley J M, Genty G, Eggleton B J 2008 Opt. Express 16 3644
Google Scholar
[5] Kasparian J, Bejot P, Wolf J P, Dudley J M 2009 Opt. Express 17 12070
Google Scholar
[6] Shukla P K, Moslem W M 2012 Phys. Lett. A 376 1125
Google Scholar
[7] Tsai Y Y, Tsai Y, Lin I 2016 Nature Phys. 12 573
Google Scholar
[8] Bludov Y V, Konotop V V, Akhmediev N 2009 Phys. Rev. A 80 033610
Google Scholar
[9] Vinayagam P S, Radha R, Porsezian K 2013 Phys. Rev. E 88 042906
Google Scholar
[10] Ginzburg N S, Rozental R M, Sergeev A S, Fedotov A E, Zotova I V, Tarakanov V P 2017 Phys. Rev. Lett. 119 034801
Google Scholar
[11] Hohmann R, Kuhl U, Stockmann H J, Kaplan L, Heller E J 2010 Phys. Rev. Lett. 104 093901
Google Scholar
[12] Ganshin A N, Efimov V B, Kolmakov G V, Mezhov-Deglin L P, McClintock P V E 2008 Phys. Rev. Lett. 101 065303
Google Scholar
[13] Stenflo L, Marklund M 2010 J. Plasma Phys. 76 293
Google Scholar
[14] Hammani K, Kibler B, Finot C, Morin P, Millot G P 2011 Opt. Lett. 36 112
Google Scholar
[15] Shats M, Punzmann H, Xia H 2010 Phys. Rev. Lett. 104 104503
Google Scholar
[16] Yan Z Y 2011 Phys. Lett. A 375 4274
[17] Han J F, Liang T, Duan W S 2019 Euro. Phys. J. E 42 5
Google Scholar
[18] Copus M G, Camley R E 2020 Phys. Rev. B 102 220410
Google Scholar
[19] Dysthe K, Krogstad H, Muller P 2008 Ann. Rev. Flu. Mech. 40 287
Google Scholar
[20] Benetazzo A, Ardhuin F, Bergamasco F, Cavaleri L, Guimarães P V, Schwendeman M, Sclavo M, Thomson J, Torsello A 2017 Sci. Rep. 7 8276
[21] Peregrine D H 1983 J. Aust. Math. Soc. B 25 16
Google Scholar
[22] Akhmediev N, Korneev V I 1986 Theor. Math. Phys. 69 1089
Google Scholar
[23] Kuznetsov E 1977 Sov. Phys. Dokl. 22 507
Google Scholar
[24] Ma Y C 1979 Stud. Appl. Math. 60 43
[25] Akhmediev N, Ankiewicz A, Soto-Crespo J M 2009 Phys. Rev. E 80 026601
Google Scholar
[26] He J S, Zhang H R, Wang L H, Porsezian K, Fokas A S 2013 Phys. Rev. E 87 052914
Google Scholar
[27] Dubard P, Matveev V B 2011 Nat. Hazards Earth Syst. Sci. 11 667
Google Scholar
[28] Ankiewicz A, Clarkson P A, Akhmediev N 2010 J. Phys. A 43 122002
Google Scholar
[29] Kedziora D J, Ankiewicz A, Akhmediev N 2011 Phys. Rev. E 84 056611
Google Scholar
[30] Zhao L C, Guo B, Ling L 2016 J. Math. Phys. 57 043508
Google Scholar
[31] Onorato M, Residori S, Bortolozzo U, Montina A, Arecchi F 2013 Phys. Rep. 528 47
Google Scholar
[32] Soto-Crespo J M, Devine N, Akhmediev N 2016 Phys. Rev. Lett. 116 103901
Google Scholar
[33] Akhmediev N, Dudley J M, Solli D R, Turitsyn S K 2013 J. Opt. 15 060201
Google Scholar
[34] Ohta Y, Yang J K 2012 Proc. R. Soc. Lond. Ser. A 468 1716
Google Scholar
[35] Christian K, Efim P 2003 Euro. J. Mech. B Fluids 22 603
[36] Wang L H, He J S, Xu H, Wang J, Porsezian K 2017 Phys. Rev. E 95 042217
Google Scholar
[37] Dudley J M, Dias F, Erkintalo M, Genty G 2014 Nature Photo. 8 755
Google Scholar
[38] Zhang X E, Chen Y, Tang X Y 2018 Comput. Math. Appl. 76 1938
Google Scholar
[39] Chen J C, Chen Y, Feng B F, Maruno K, Ohta Y 2018 J. Phys. Soc. Jpn. 87 094007
Google Scholar
[40] Wang M M, Chen Y 2021 Nonlinear Dyn. 104 2621
Google Scholar
[41] Wang M M, Chen Y 2021 Nonlinear Dyn. 98 1781
Google Scholar
[42] Birkholz S, Nibbering E T J 2013 Phys. Rev. Lett. 111 243903
Google Scholar
[43] Solli D R Ropers C, Koonath P, Jalali B 2007 Nature 450 1054
Google Scholar
[44] Solli D R, Ropers C, Jalali B 2008 Phys. Rev. Lett. 101 233902
Google Scholar
[45] Kibler B, Fatome J, Finot C, Millot G, Genty G, Wetzel B, Akhmediev N, Dias F, Dudley J M 2012 Sci. Rep. 2 463
Google Scholar
[46] Chabchoub A, Hoffmannn N P, Akhmediev N 2011 Phys. Rev. Lett. 106 204502
Google Scholar
[47] Bailung H, Sharma S K, Nakamura Y 2011 Phys. Rev. Lett. 107 255005
Google Scholar
[48] Xiong H, Gan J H, Wu Y 2017 Phys. Rev. Lett. 119 153901
Google Scholar
[49] 张解放, 金美贞, 胡文成 2020 物理学报 69 244205
Google Scholar
Zhang J F, Jin M Z, Hu W C 2020 Acta Phys. Sin. 69 244205
Google Scholar
[50] 张解放, 金美贞 2020 物理学报 69 214203
Google Scholar
Zhang J F, Jin M Z 2020 Acta Phys. Sin. 69 214203
Google Scholar
[51] Martin D U, Yuen H C 1980 Wave Motion 2 215
Google Scholar
[52] Carter J D, Segur H 2003 Phys. Rev. E 68 045601
Google Scholar
[53] Manakov S V 1974 Sov. Phys. JETP 38 248
[54] Pecseli H L 1985 IEEE Trans. Plasma Sci. 13 53
Google Scholar
[55] Gross E P 1963 J. Math Phys. 4 195
Google Scholar
[56] Pitaevskii L P 1961 Sov. Phys. JETP 13 451
Google Scholar
[57] Benney D J, Roskes G J 1969 Stud. Appl. Math. 48 377
Google Scholar
[58] Myra J R and Liu C S 1980 Phys. Fluids 23 2258
Google Scholar
[59] Gorza S P, Roig N 2004 Phys. Rev. Lett. 92 084101
Google Scholar
[60] Gorza S P, Kockaert P, Emplit P, Haelterman M 2009 Phys. Rev. Lett. 102 13410
Google Scholar
[61] Fermann M E 2000 Phys. Rev. Lett. 84 6010
Google Scholar
[62] Ankiewicz A, Kedziora D J, Akhmediev N 2011 Phys. Lett. A 375 2782
Google Scholar
[63] Kedziora D J, Ankiewicz A, Akhmediev N 2012 Phys. Rev. E 86 056602
Google Scholar
[64] Ohta Y, Yang J K 2012 Proc. R. Soc. A 468 1716
Google Scholar
[65] Kimmoun O, Hsu H C, Branger H, Li M S, Chen Y Y, Kharif C, Onorato M, Kelleher E J R, Kibler B, Akhmediev N, Chabchoub A 2016 Sci. Rep. 6 8516
Google Scholar
-
图 1 一阶单线怪波(28)在
$ (x, y) $ -平面上随$ t $ 的演化图: (a)$ t = - 3 $ ; (b)$ t = 0 $ ; (c)$ t = 5 $ , 自由参数选取为$\alpha = \beta = \lambda = 1,$ $\gamma = 2, {\kappa _0} = {\iota _0} = - {a_0} = {{ {b}}_0} = 1$ Fig. 1. Evolution plots of the first-order single line-rogue wave (28) with the
$ t $ on the$ (x, y) $ -plane. (a)$ t = - 3 $ ; (b)$ t = 0 $ ; (c)$ t = 5 $ . The free parameters are chosen as$ \alpha = \beta = 1, $ $\gamma = 2, {\kappa _0} = {\iota _0} = -{a_0} = {{ {b}}_0} = 1$ 
图 2 二阶单线怪波(29)在
$ (x, y) $ -平面上随$ t $ 的演化图: (a)$ t = - 3 $ ; (b)$ t = 0 $ ; (c)$ t = 5 $ , 自由参数选取为$ \alpha = \beta = \lambda = 1, $ $\gamma = 2, {\kappa _0} = {\iota _0} = -{a_0} = {{{b}}_0} = 1$ ,$ {\xi _d} = {\tau _d} = 0 $ Fig. 2. Evolution plots of the second-order single line-rogue wave (29) with the
$ t $ on the$ (x, y) $ -plane. (a)$ t = - 3 $ ; (b)$ t = $ $ 0 $ ; (c)$ t = 5 $ . The free parameters are chosen as$ \alpha = \beta = 1, $ $\gamma = 2, {\kappa _0} = {\iota _0} = -{a_0} = {{{b}}_0} = 1, {\xi _d} = {\tau _d} = 0$ 
图 3 双线怪波(37)在
$ (x, y) $ -平面上随$ t $ 的演化图: (a)$ t = - 3 $ ; (b)$ t = 0 $ ; (c)$ t = 5 $ , 自由参数选取为$\alpha = $ $ \beta = \lambda = 1,$ $\gamma = 2, {\kappa _0} = {\iota _0} = -{a_0} = {{ {b}}_0} = 1$ ,$\delta = 25, \mu = 100$ Fig. 3. Evolution plots of the double line-rogue wave (37) with
$ t $ on the$ (x, y) $ -plane. (a)$ t = - 3 $ ; (b)$ t = 0 $ ; (c)$ t = 5 $ .The free parameters are chosen as$ \alpha = \beta = 1, $ $\gamma = 2,~ {\kappa _0} = {\iota _0} = -{a_0} = {{ {b}}_0} = 1, ~\delta = 25, \mu = 100$ 
图 4 多线怪波(38)在
$ (x, y) $ -平面上随$ t $ 的演化图: (a)$ t = - 3 $ ; (b)$ t = 0 $ ; (c)$ t = 5 $ , 自由参数选取为$ \alpha = \beta = \lambda = 1, $ $\gamma = 2, {\kappa _0} = {\iota _0} = -{a_0} = {{ {b}}_0} = 1$ ,$a = 0.45$ Fig. 4. Evolution plots of the multi-line rogue wave (38) with
$ t $ on the$ (x, y) $ -plane. (a)$ t = - 3 $ ; (b)$ t = 0 $ ; (c)$ t = 5 $ . The free parameters are chosen as$ \alpha = \beta = 1, $ $\gamma = 2, {\kappa _0} = {\iota _0} = -{a_0} = {{ {b}}_0} = 1, a = 0.45$ 
图 5 线怪波(39)在
$ (x, y) $ -平面上随$ t $ 的演化图 (a)$ t = - 3 $ , (b)$ t = 0 $ , (c)$ t = 5 $ , 自由参数选取为$ \alpha = \beta = \lambda = 1, $ $\gamma = 2, $ $ {\kappa _0} = {\iota _0} = -{a_0} = {{ {b}}_0} = 1$ ,$a = 0.45$ Fig. 5. Evolution plots of the line rogue waves (39) with
$ t $ on the$ (x, y) $ -plane. (a)$ t = - 3 $ ; (b)$ t = 0 $ ; (c)$ t = 5 $ . The free parameters are chosen as$ \alpha = \beta = 1, $ $\gamma = 2, {\kappa _0} = {\iota _0} = -{a_0} = {{ {b}}_0} = 1, a = 0.45$ -
[1] Onorato M, Osborne A R, Serio M, Bertone S 2001 Phys. Rev. Lett. 86 5831
Google Scholar
[2] Kharif C, Pelinovsky E, Slunyaev A 2009 Rogue Waves in the Ocean. In: Advances in Geophysical and Environmental Mechanics and Mathematics (Berlin: Springer-Verlag)
[3] Adcock T A A, Taylor P H 2014 Rep. Prog. Phys. 77 105901
Google Scholar
[4] Dudley J M, Genty G, Eggleton B J 2008 Opt. Express 16 3644
Google Scholar
[5] Kasparian J, Bejot P, Wolf J P, Dudley J M 2009 Opt. Express 17 12070
Google Scholar
[6] Shukla P K, Moslem W M 2012 Phys. Lett. A 376 1125
Google Scholar
[7] Tsai Y Y, Tsai Y, Lin I 2016 Nature Phys. 12 573
Google Scholar
[8] Bludov Y V, Konotop V V, Akhmediev N 2009 Phys. Rev. A 80 033610
Google Scholar
[9] Vinayagam P S, Radha R, Porsezian K 2013 Phys. Rev. E 88 042906
Google Scholar
[10] Ginzburg N S, Rozental R M, Sergeev A S, Fedotov A E, Zotova I V, Tarakanov V P 2017 Phys. Rev. Lett. 119 034801
Google Scholar
[11] Hohmann R, Kuhl U, Stockmann H J, Kaplan L, Heller E J 2010 Phys. Rev. Lett. 104 093901
Google Scholar
[12] Ganshin A N, Efimov V B, Kolmakov G V, Mezhov-Deglin L P, McClintock P V E 2008 Phys. Rev. Lett. 101 065303
Google Scholar
[13] Stenflo L, Marklund M 2010 J. Plasma Phys. 76 293
Google Scholar
[14] Hammani K, Kibler B, Finot C, Morin P, Millot G P 2011 Opt. Lett. 36 112
Google Scholar
[15] Shats M, Punzmann H, Xia H 2010 Phys. Rev. Lett. 104 104503
Google Scholar
[16] Yan Z Y 2011 Phys. Lett. A 375 4274
[17] Han J F, Liang T, Duan W S 2019 Euro. Phys. J. E 42 5
Google Scholar
[18] Copus M G, Camley R E 2020 Phys. Rev. B 102 220410
Google Scholar
[19] Dysthe K, Krogstad H, Muller P 2008 Ann. Rev. Flu. Mech. 40 287
Google Scholar
[20] Benetazzo A, Ardhuin F, Bergamasco F, Cavaleri L, Guimarães P V, Schwendeman M, Sclavo M, Thomson J, Torsello A 2017 Sci. Rep. 7 8276
[21] Peregrine D H 1983 J. Aust. Math. Soc. B 25 16
Google Scholar
[22] Akhmediev N, Korneev V I 1986 Theor. Math. Phys. 69 1089
Google Scholar
[23] Kuznetsov E 1977 Sov. Phys. Dokl. 22 507
Google Scholar
[24] Ma Y C 1979 Stud. Appl. Math. 60 43
[25] Akhmediev N, Ankiewicz A, Soto-Crespo J M 2009 Phys. Rev. E 80 026601
Google Scholar
[26] He J S, Zhang H R, Wang L H, Porsezian K, Fokas A S 2013 Phys. Rev. E 87 052914
Google Scholar
[27] Dubard P, Matveev V B 2011 Nat. Hazards Earth Syst. Sci. 11 667
Google Scholar
[28] Ankiewicz A, Clarkson P A, Akhmediev N 2010 J. Phys. A 43 122002
Google Scholar
[29] Kedziora D J, Ankiewicz A, Akhmediev N 2011 Phys. Rev. E 84 056611
Google Scholar
[30] Zhao L C, Guo B, Ling L 2016 J. Math. Phys. 57 043508
Google Scholar
[31] Onorato M, Residori S, Bortolozzo U, Montina A, Arecchi F 2013 Phys. Rep. 528 47
Google Scholar
[32] Soto-Crespo J M, Devine N, Akhmediev N 2016 Phys. Rev. Lett. 116 103901
Google Scholar
[33] Akhmediev N, Dudley J M, Solli D R, Turitsyn S K 2013 J. Opt. 15 060201
Google Scholar
[34] Ohta Y, Yang J K 2012 Proc. R. Soc. Lond. Ser. A 468 1716
Google Scholar
[35] Christian K, Efim P 2003 Euro. J. Mech. B Fluids 22 603
[36] Wang L H, He J S, Xu H, Wang J, Porsezian K 2017 Phys. Rev. E 95 042217
Google Scholar
[37] Dudley J M, Dias F, Erkintalo M, Genty G 2014 Nature Photo. 8 755
Google Scholar
[38] Zhang X E, Chen Y, Tang X Y 2018 Comput. Math. Appl. 76 1938
Google Scholar
[39] Chen J C, Chen Y, Feng B F, Maruno K, Ohta Y 2018 J. Phys. Soc. Jpn. 87 094007
Google Scholar
[40] Wang M M, Chen Y 2021 Nonlinear Dyn. 104 2621
Google Scholar
[41] Wang M M, Chen Y 2021 Nonlinear Dyn. 98 1781
Google Scholar
[42] Birkholz S, Nibbering E T J 2013 Phys. Rev. Lett. 111 243903
Google Scholar
[43] Solli D R Ropers C, Koonath P, Jalali B 2007 Nature 450 1054
Google Scholar
[44] Solli D R, Ropers C, Jalali B 2008 Phys. Rev. Lett. 101 233902
Google Scholar
[45] Kibler B, Fatome J, Finot C, Millot G, Genty G, Wetzel B, Akhmediev N, Dias F, Dudley J M 2012 Sci. Rep. 2 463
Google Scholar
[46] Chabchoub A, Hoffmannn N P, Akhmediev N 2011 Phys. Rev. Lett. 106 204502
Google Scholar
[47] Bailung H, Sharma S K, Nakamura Y 2011 Phys. Rev. Lett. 107 255005
Google Scholar
[48] Xiong H, Gan J H, Wu Y 2017 Phys. Rev. Lett. 119 153901
Google Scholar
[49] 张解放, 金美贞, 胡文成 2020 物理学报 69 244205
Google Scholar
Zhang J F, Jin M Z, Hu W C 2020 Acta Phys. Sin. 69 244205
Google Scholar
[50] 张解放, 金美贞 2020 物理学报 69 214203
Google Scholar
Zhang J F, Jin M Z 2020 Acta Phys. Sin. 69 214203
Google Scholar
[51] Martin D U, Yuen H C 1980 Wave Motion 2 215
Google Scholar
[52] Carter J D, Segur H 2003 Phys. Rev. E 68 045601
Google Scholar
[53] Manakov S V 1974 Sov. Phys. JETP 38 248
[54] Pecseli H L 1985 IEEE Trans. Plasma Sci. 13 53
Google Scholar
[55] Gross E P 1963 J. Math Phys. 4 195
Google Scholar
[56] Pitaevskii L P 1961 Sov. Phys. JETP 13 451
Google Scholar
[57] Benney D J, Roskes G J 1969 Stud. Appl. Math. 48 377
Google Scholar
[58] Myra J R and Liu C S 1980 Phys. Fluids 23 2258
Google Scholar
[59] Gorza S P, Roig N 2004 Phys. Rev. Lett. 92 084101
Google Scholar
[60] Gorza S P, Kockaert P, Emplit P, Haelterman M 2009 Phys. Rev. Lett. 102 13410
Google Scholar
[61] Fermann M E 2000 Phys. Rev. Lett. 84 6010
Google Scholar
[62] Ankiewicz A, Kedziora D J, Akhmediev N 2011 Phys. Lett. A 375 2782
Google Scholar
[63] Kedziora D J, Ankiewicz A, Akhmediev N 2012 Phys. Rev. E 86 056602
Google Scholar
[64] Ohta Y, Yang J K 2012 Proc. R. Soc. A 468 1716
Google Scholar
[65] Kimmoun O, Hsu H C, Branger H, Li M S, Chen Y Y, Kharif C, Onorato M, Kelleher E J R, Kibler B, Akhmediev N, Chabchoub A 2016 Sci. Rep. 6 8516
Google Scholar
-
[1] 饶继光, 陈生安, 吴昭君, 贺劲松. 空间位移 $\mathcal{PT}$ 对称非局域非线性薛定谔方程的高阶怪波解. 物理学报, 2023, 72(10): 104204. doi: 10.7498/aps.72.20222298[2] 张解放, 俞定国, 金美贞. (2+1)维Zakharov方程的自相似变换和线怪波簇激发. 物理学报, 2022, 71(8): 084204. doi: 10.7498/aps.71.20211181 [3] 王艳, 李禄. 基于怪波实现光脉冲串的全光放大. 物理学报, 2021, 70(22): 224213. doi: 10.7498/aps.70.20210959 [4] 张解放, 金美贞, 胡文成. 非自治Kadomtsev-Petviashvili方程的自相似变换和二维怪波构造. 物理学报, 2020, 69(24): 244205. doi: 10.7498/aps.69.20200981 [5] 陈智敏, 段文山. 弹性管中的怪波. 物理学报, 2020, 69(1): 014701. doi: 10.7498/aps.69.20191308 [6] 闻小永, 王昊天. 高阶Ablowitz-Ladik方程的局域波解及稳定性分析. 物理学报, 2020, 69(1): 010205. doi: 10.7498/aps.69.20191235 [7] 李敏, 王博婷, 许韬, 水涓涓. 四阶色散非线性薛定谔方程的明暗孤立波和怪波的形成机制. 物理学报, 2020, 69(1): 010502. doi: 10.7498/aps.69.20191384 [8] 蒋涛, 黄金晶, 陆林广, 任金莲. 非线性薛定谔方程的高阶分裂改进光滑粒子动力学算法. 物理学报, 2019, 68(9): 090203. doi: 10.7498/aps.68.20190169 [9] 崔少燕, 吕欣欣, 辛杰. 广义非线性薛定谔方程描述的波坍缩及其演变. 物理学报, 2016, 65(4): 040201. doi: 10.7498/aps.65.040201 [10] 李淑青, 杨光晔, 李禄. Hirota方程的怪波解及其传输特性研究. 物理学报, 2014, 63(10): 104215. doi: 10.7498/aps.63.104215 [11] 胡文成, 张解放, 赵辟, 楼吉辉. 光纤放大器中非自治光畸波的传播控制研究. 物理学报, 2013, 62(2): 024216. doi: 10.7498/aps.62.024216 [12] 王晶, 马瑞玲, 王龙, 孟俊敏. 采用混合模型数值模拟从深海到浅海内波的传播. 物理学报, 2012, 61(6): 064701. doi: 10.7498/aps.61.064701 [13] 宋诗艳, 王晶, 孟俊敏, 王建步, 扈培信. 深海内波非线性薛定谔方程的研究. 物理学报, 2010, 59(2): 1123-1129. doi: 10.7498/aps.59.1123 [14] 宋诗艳, 王晶, 王建步, 宋莎莎, 孟俊敏. 应用非线性薛定谔方程模拟深海内波的传播. 物理学报, 2010, 59(9): 6339-6344. doi: 10.7498/aps.59.6339 [15] 余超凡, 梁国栋, 曹锡金. 一维分子晶体系统的极化子-孤子压缩态、系统基态性质和量子涨落. 物理学报, 2008, 57(7): 4402-4411. doi: 10.7498/aps.57.4402 [16] 雷 霆, 涂成厚, 李恩邦, 李勇男, 郭文刚, 魏 岱, 朱 辉, 吕福云. 高能量无波分裂超短脉冲自相似传输的理论研究和数值模拟. 物理学报, 2007, 56(5): 2769-2775. doi: 10.7498/aps.56.2769 [17] 龚伦训. 非线性薛定谔方程的Jacobi椭圆函数解. 物理学报, 2006, 55(9): 4414-4419. doi: 10.7498/aps.55.4414 [18] 阮航宇, 李慧军. 用推广的李群约化法求解非线性薛定谔方程. 物理学报, 2005, 54(3): 996-1001. doi: 10.7498/aps.54.996 [19] 陈宏平, 王 箭, 何国光. 光脉冲传输数值模拟的分步小波方法. 物理学报, 2005, 54(6): 2779-2783. doi: 10.7498/aps.54.2779 [20] 阮航宇, 陈一新. (2+1)维非线性薛定谔方程的环孤子,dromion,呼吸子和瞬子. 物理学报, 2001, 50(4): 586-592. doi: 10.7498/aps.50.586
下载:
计量
- 文章访问数: 4226
- PDF下载量: 73
- 被引次数: 0
