-
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.
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Keywords:
- rogue wave /
- line rogue wave /
- two-dimensional self-similar transformation /
- nonlinear Schrödinger equation
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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
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[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
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[16] Yan Z Y 2011 Phys. Lett. A 375 4274
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[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
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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
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[43] Solli D R Ropers C, Koonath P, Jalali B 2007 Nature 450 1054
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
Zhang J F, Jin M Z, Hu W C 2020 Acta Phys. Sin. 69 244205
Google Scholar
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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
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Google Scholar
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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
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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
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Google Scholar
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Google Scholar
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Google Scholar
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