搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

(2+1)维Zakharov方程的自相似变换和线怪波簇激发

张解放 俞定国 金美贞

引用本文:
Citation:

(2+1)维Zakharov方程的自相似变换和线怪波簇激发

张解放, 俞定国, 金美贞

Self-similar transformation and excitation of rogue waves for (2+1)-dimensional Zakharov equation

Zhang Jie-Fang, Yu Ding-Guo, Jin Mei-Zhen
PDF
HTML
导出引用
  • 首先建立(2+1)维(二维空间和一维时间)Zakharov方程的自相似变换, 并将该系统转换为(1+1)维非线性薛定谔(nonlinear Schrödinger, NLS)方程; 然后基于该相似变换和已知的(1+1)维NLS方程有理形式解, 通过选择合适参数得到了(2+1)维Zakharov方程在x-y平面上丰富的线怪波簇激发, 发现产生线怪波簇最大辐值时的传播距离$ z $值完全不同, 而且形状和幅度可以得到有效调控; 最后借助图示展现了二维怪波的传播特征. 此外, 发现在x-y平面上, 当参数$ \gamma = 1 $时, 呈现线怪波; 而当参数$ \gamma \ne 1 $时, 线怪波转变为离散的局域怪波. 随参数$ \gamma $的增大, 可以在x-y平面限定区域获得时空局域的怪波, 这与Peregrine在(1+1)维NLS方程中发现的“Kuznetsov-Ma孤子”(Kuznetsov-Ma soliton, KMS)或“Akhmediev呼吸子”(Akhmediev breather, AB)极限情形的“Peregrine 孤子”(Peregrine soliton, PS)类似. 本文提出的(2+1)维Zakharov方程怪波方法可以作为获得高维怪波激发的有效途径, 并推广应用于其他(2+1)维非线性系统.
    The search for the excitation of two-dimensional rogue wave in a (2+1)-dimensional nonlinear evolution model is a research hotspot. In this paper, the self-similar transformation of the (2+1)-dimensional Zakharov equation is established, and this equation is transformed into the (1+1)-dimensional nonlinear Schrödinger equation. Based on the similarity transformation and the rational formal solution of the (1+1)-dimensional nonlinear Schrödinger equation, the rogue wave excitation of the (2+1)-dimensional Zakharov equation is obtained by selecting appropriate parameters. We can see that the shape and amplitude of the rogue waves can be effectively controlled. Finally, the propagation characteristics of line rogue waves are diagrammed visually. We also find that the line-type characteristics of two-dimensional rogue wave are present in the x-y plane when the parameter $ \gamma = 1 $. The line rogue wave is converted into discrete localized rogue wave in the x-y plane when the parameter $ \gamma \ne 1 $. The spatial localized rogue waves with short-life can be obtained in the required x-y plane region. This is similar to the Peregrine soliton (PS) first discovered by Peregrine in the (1+1)-dimensional NLS equation, which is the limit case of the “Kuznetsov-Ma soliton” (KMS) or “Akhmediev breather” (AB). The proposed approach to constructing the line rogue waves of the (2+1) dimensional Zakharov equation can serve as a potential physical mechanism to excite two-dimensional rogue waves, and can be extended to other (2+1)-dimensional nonlinear systems.
      通信作者: 张解放, Zhangjief@cuz.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 61877053)资助的课题.
      Corresponding author: Zhang Jie-Fang, Zhangjief@cuz.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61877053).
    [1]

    Divinsky B V, Levin B V, Lopatukhin L I, Pelinovsky E N, Slyunyaev A V 2004 Dokl. Earth Sci. 395 438

    [2]

    Onorato M, Osborne A R, Serio M, Bertone S 2001 Phys. Rev. Lett. 86 5831Google Scholar

    [3]

    Ginzburg N S, Rozental R M, Sergeev A S, Fedotov A E, Zotova I V, Tarakanov V P 2017 Phys. Rev. Lett. 119 034801Google Scholar

    [4]

    Akhmediev N, Dudley J M, Solli D R, Turitsyn S K 2013 J. Opt. 15 060201Google Scholar

    [5]

    Bludov Y V, Konotop V V, Akhmediev N 2009 Phys. Rev. A 80 033610Google Scholar

    [6]

    Moslem W M 2011 Phys. Plasm. 18 032301Google Scholar

    [7]

    Stenfl L, Marklund M 2010 J. Plasm. Phys. 76 293Google Scholar

    [8]

    Onorato M, Residori S, Bortolozzo U, Montina A, Arecchi F 2013 Phys. Rep. 528 47Google Scholar

    [9]

    Müller P, Garrett C, Osborne A 2005 Oceanography 18 66Google Scholar

    [10]

    Hohmann R, Kuhl U, Stockmann H J, Kaplan L, Heller E J 2010 Phys. Rev. Lett. 104 093901Google Scholar

    [11]

    Ganshin A N, Efimov V B, Kolmakov G V, Mezhov Deglin P V, McClintock E 2008 Phys. Rev. Lett. 101 065303Google Scholar

    [12]

    Yan Z Y 2010 Commun. Theor. Phys. 54 947Google Scholar

    [13]

    Shats M, Punzmann H, Xia H 2010 Phys. Rev. Lett. 104 104503Google Scholar

    [14]

    Xia H, Maimbourg T, Punzmann H, Shats M 2012 Phys. Rev. Lett. 109 114502Google Scholar

    [15]

    Pu J C, Li J, Chen Y 2021 Chin. Phys. B 30 060202Google Scholar

    [16]

    Zhou H J, Chen Y 2021 Nonlinear. Dynam. 106 3437Google Scholar

    [17]

    Solli R, Ropers C, Koonath P, Jalali B 2007 Nature 450 1054Google Scholar

    [18]

    Chabchoub A, Hoffmann N P, Akhmediev N 2011 Phys. Rev. Lett. 106 204502Google Scholar

    [19]

    Peregrine D H 1983 J. Aust. Math. Soc. Ser. B:Appl. Math. 25 16Google Scholar

    [20]

    Akhmediev N, Ankiewicz A, Soto Crespo J M 2009 Phys. Rev. E 80 026601Google Scholar

    [21]

    Kedziora D J, Ankiewicz A, Akhmediev N 2012 Phys. Rev. E 86 056602Google Scholar

    [22]

    张解放, 金美贞, 胡文成 2020 物理学报 69 244205Google Scholar

    Zhang J F, Jin M Z, Hu W C 2020 Acta Phys. Sin. 69 244205Google Scholar

    [23]

    张解放, 金美贞 2020 物理学报 69 214203Google Scholar

    Zhang J F, Jin M Z 2020 Acta Phys. Sin. 69 214203Google Scholar

    [24]

    Zakharov V E (edited by Bullough R, Caudrey P) 1980 The Inverse Scattering Method (Vol. 17) (Berlin: Springer) pp243–285 DOI: 10.1007/978-3-642-81448-8_7

    [25]

    Radha R, Lakshmanan M 1994 Inverse Problems 10 L29Google Scholar

    [26]

    Strachan I A B 1992 Inverse Problems 8 L21Google Scholar

    [27]

    Radha R, Lakshmanan M 1997 J. Phys. A:Math. Gen. 30 3229Google Scholar

    [28]

    沈守枫, 张隽 2008 应用数学和力学 29 1254Google Scholar

    Shen S F, Zhang J 2008 Appl. Math. Mech. 29 1254Google Scholar

    [29]

    Wang J, Chen L W, Liu C F 2014 Appl. Math. Comput. 249 76

    [30]

    程丽, 张翼 2016 长江大学学报(自科版) 13 35

    Chen L, Zhang Y 2016 J. Yangtze Univ. (Nat. Sci. Ed.) 13 35 (in Chinese)

    [31]

    Wang X B, Tian S F, Zhang T T 2018 Proceedings of Roceedings of the American Society 146 3353Google Scholar

    [32]

    Chen M D, Li B 2017 Modern Phys. Lett. B 31 1750298

    [33]

    Fokas A S 1994 Inverse Problems 10 19Google Scholar

    [34]

    Chen J C, Chen Y 2014 J. Nonlinear Math. Phys. 21 454

    [35]

    Radha R, Lakshmanan M 1997 Chaos, Solitons and Fractals 8 17Google Scholar

    [36]

    Villarroel J, Prada J, Estévez P G 2009 Stud. Appl. Math. 122 395Google Scholar

    [37]

    Estévez P G 1999 J. Math. Phys. 40 1406Google Scholar

    [38]

    Rao J G, Wang L H, Zhang Y, He J S 2015 Commun. Theor. Phys. 64 605Google Scholar

    [39]

    Chen T T, Hu P Y, He J S 2019 Commun. Theor. Phys. 71 496Google Scholar

    [40]

    Rao J G, Mihalacheb D, Cheng Y, He J S 2019 Phys. Lett. A 383 1138Google Scholar

  • 图 1  传播距离$ z = - 3, 0, 1 $时一阶线怪波$ \left| u \right| $ (a), (b), (c) 三维图; (d), (e), (f) 对应x-y面上的投影图; (a), (d) A = 1.45; (b), (e) A = 4.24; (c), (f) A = 1.71

    Fig. 1.  One-order line rogue wave$ \left| u \right| $respectively at the propagation distance $ z = - 3, 0, 1 $: (a), (b), (c) Three-dimensional plots; (d), (e), (f) the corresponding contour plots in the x-y plane; (a), (d) A = 1.45; (b), (e) A = 4.24; (c), (f) A = 1.71.

    图 2  传播距离$ z = - 3, 0, 1 $时二阶线怪波$ \left| u \right| $ (a), (b), (c)三维图; (d), (e), (f) 对应x-y面上的投影图; (a), (d) A = 1.58; (b), (e) A = 7.32; (c), (f) A=1.93

    Fig. 2.  Two-order line rogue wave$ \left| u \right| $respectively at the propagation distance $ z = - 3, 0, 1 $: (a), (b), (c) Three-dimensional plots; (d), (e), (f) the corresponding contour plots in the x-y plane; (a), (d) A = 1.58; (b), (e) A = 7.32; (c), (f) A = 1.93.

    图 3  传播距离$ z = - 3, 0, {\text{ 2}} $时二阶线怪波$ \left| u \right| $ (a), (b), (c) 三维图; (d), (e), (f) 对应x-y面上的投影图; (a), (d) A = 1.52; (b), (e) A = 3.99; (c), (f) A = 1.69; 参数$ {w_0} = \gamma = {\beta _0} = 1, {\varsigma _0} = {\zeta _0} = {\varphi _0} = \delta = 0, \mu = 100 $

    Fig. 3.  Two-order line rogue wave$ \left| u \right| $respectively at the propagation distance$ z = - 3, 0, {\text{ 2}} $: (a), (b), (c) Three-dimensional plots; (d), (e), (f) the corresponding contour plots in the x-y plane; (a), (d) A = 1.52; (b), (e) A = 3.99; (c), (f) A = 1.69; the parameters are$ {w_0} = \gamma = {\beta _0} = 1, {\varsigma _0} = {\zeta _0} = {\varphi _0} = \delta = {\text{0, }} $$ \mu = 100 $.

    图 4  传播距离$ z = - {\text{1}}{\text{.5, }} - 0.{\text{635}}, {\text{ 0}}{\text{.5}} $时二阶线怪波簇$ \left| u \right| $ (a), (b), (c) 三维图; (d), (e), (f) 对应x-y面上的投影图; (a), (d) A = 1.80; (b), (e) A = 5.81; (c), (f) A = 2.23; 参数$ {w_0} = \gamma = {\beta _0} = 1, {\varsigma _0} = {\zeta _0} = {\varphi _0} = \mu = 0, \delta = 100 $

    Fig. 4.  Two-order line rogue wave$ \left| u \right| $respectively at the propagation distance$ z = - {\text{1}}{\text{.5, }} - 0.{\text{635}}, {\text{ 0}}{\text{.5}} $: (a), (b), (c) Three-dimensional plots; (d), (e), (f) the corresponding contour plots in the x-y plane; (a), (d) A = 1.80; (b), (e) A = 5.81; (c), (f) A=2.23; the parameters are$ {w_0} = \gamma = {\beta _0} = 1, {\varsigma _0} = {\zeta _0} = {\varphi _0} = \mu = 0, \delta = 100 $.

    图 5  传播距离$ z = 0{, }0.{\text{5, 1}}{\text{.0}} $时二阶线怪波簇$ \left| u \right| $ (a), (b), (c)三维图; (d), (e), (f) 对应x-y面上的投影图; (a), (d) A = 2.76; (b), (e) A = 4.36; (c), (f) A = 2.78; 参数$ {w_0} = \gamma = {\beta _0} = 1, {\varsigma _0} = {\zeta _0} = {\varphi _0} = 0, \mu = 40, \delta = 20 $

    Fig. 5.  Two-order line rogue wave$ \left| u \right| $respectively at the propagation distance$ z = 0{, }0.{\text{5, 1}}{\text{.0}} $: (a), (b), (c) Three-dimensional plots; (d), (e), (f) the corresponding contour plots in the x-y plane; (a), (d) A = 2.76; (b), (e) A = 4.36; (c), (f) A = 2.78; the parameters are$ {w_0} = \gamma = {\beta _0} = 1, {\varsigma _0} = {\zeta _0} = {\varphi _0} = 0, \mu = 40, \delta = 20 $.

    图 6  (a), (b), (c)传播距离$ z = - {\text{1, }}0.{\text{29}}, 1{\text{.5}} $时二线怪波簇$ \left| u \right| $x-y平面上的三维图; (d), (e), (f)对应的投影图; (a), (d) A = 2.76; (b), (e) A = 4.53; (c), (f) A = 1.72; 参数$ {w_0} = \gamma = {\beta _0} = 1, {\varsigma _0} = {\zeta _0} = {\varphi _0} = 0, \mu = 20, \delta = - 20 $

    Fig. 6.  Two-order line rogue wave$ \left| u \right| $respectively at the propagation distance$ z = - {\text{1, }}0.{\text{29}}, 1{\text{.5}} $: (a), (b), (c) Three-dimensional plots; (d), (e), (f) the corresponding contour plots in the x-y plane; (a), (d) A=2.76; (b), (e) A = 4.53; (c), (f) A = 1.72; the parameters are$ {w}_{0}=\gamma ={\beta }_{0}=1, {\varsigma }_{0}={\zeta }_{0}={\phi }_{0}=\text{0, } $$ \mu = 20, \delta = - 20 $.

    图 7  传播距离$ z = - {\text{1}}{\text{.5, }} - 0.{\text{295, 1}} $时二阶线怪波簇$ \left| u \right| $ (a), (b), (c) 三维图; (d), (e), (f)对应x-y面上的投影图; (a), (d) A = 1.72; (b), (e) A = 4.39; (c), (f) A = 2.39; 参数$ {w_0} = \gamma = {\beta _0} = 1, {\varsigma _0} = {\zeta _0} = {\varphi _0} = {\text{0, }} $$ \mu = - 20, \delta = 20 $

    Fig. 7.  Two-order line rogue wave$ \left| u \right| $respectively at the propagation distance $ z = - {\text{1}}{\text{.5, }} - 0.{\text{295, 1}} $: (a), (b), (c) Three-dimensional plots; (d), (e), (f) the corresponding contour plots in the x-y plane; (a), (d) A = 1.72; (b), (e) A = 4.39; (c), (f) A = 2.39; the parameters are$ {w}_{0}=\gamma ={\beta }_{0}=1, \text{ }{\varsigma }_{0}={\zeta }_{0}={\phi }_{0}=\text{0, } $$ \mu = - 20, \delta = 20 $.

    图 8  $ z = 0 $时, 一阶线怪波$ \left| u \right| $x-y平面上不同参数$ \gamma $下的形态 (a) $ \gamma = 1 $; (b) $ \gamma = {\text{3}} $; (c) $ \gamma = {\text{6}} $; (d) $ \gamma = {\text{10}} $; (e) $ \gamma = {\text{15}} $; (f) $ \gamma = {\text{2}}0 $

    Fig. 8.  One-order rogue wave$ \left| u \right| $in the x-y plane under the different$ \gamma $at the propagation distance$ z = 0 $: (a) $ \gamma = 1 $; (b) $ \gamma = {\text{3}} $; (c) $ \gamma = {\text{6}} $; (d) $ \gamma = {\text{10}} $; (e) $ \gamma = {\text{15}} $; (f) $ \gamma = {\text{2}}0 $.

  • [1]

    Divinsky B V, Levin B V, Lopatukhin L I, Pelinovsky E N, Slyunyaev A V 2004 Dokl. Earth Sci. 395 438

    [2]

    Onorato M, Osborne A R, Serio M, Bertone S 2001 Phys. Rev. Lett. 86 5831Google Scholar

    [3]

    Ginzburg N S, Rozental R M, Sergeev A S, Fedotov A E, Zotova I V, Tarakanov V P 2017 Phys. Rev. Lett. 119 034801Google Scholar

    [4]

    Akhmediev N, Dudley J M, Solli D R, Turitsyn S K 2013 J. Opt. 15 060201Google Scholar

    [5]

    Bludov Y V, Konotop V V, Akhmediev N 2009 Phys. Rev. A 80 033610Google Scholar

    [6]

    Moslem W M 2011 Phys. Plasm. 18 032301Google Scholar

    [7]

    Stenfl L, Marklund M 2010 J. Plasm. Phys. 76 293Google Scholar

    [8]

    Onorato M, Residori S, Bortolozzo U, Montina A, Arecchi F 2013 Phys. Rep. 528 47Google Scholar

    [9]

    Müller P, Garrett C, Osborne A 2005 Oceanography 18 66Google Scholar

    [10]

    Hohmann R, Kuhl U, Stockmann H J, Kaplan L, Heller E J 2010 Phys. Rev. Lett. 104 093901Google Scholar

    [11]

    Ganshin A N, Efimov V B, Kolmakov G V, Mezhov Deglin P V, McClintock E 2008 Phys. Rev. Lett. 101 065303Google Scholar

    [12]

    Yan Z Y 2010 Commun. Theor. Phys. 54 947Google Scholar

    [13]

    Shats M, Punzmann H, Xia H 2010 Phys. Rev. Lett. 104 104503Google Scholar

    [14]

    Xia H, Maimbourg T, Punzmann H, Shats M 2012 Phys. Rev. Lett. 109 114502Google Scholar

    [15]

    Pu J C, Li J, Chen Y 2021 Chin. Phys. B 30 060202Google Scholar

    [16]

    Zhou H J, Chen Y 2021 Nonlinear. Dynam. 106 3437Google Scholar

    [17]

    Solli R, Ropers C, Koonath P, Jalali B 2007 Nature 450 1054Google Scholar

    [18]

    Chabchoub A, Hoffmann N P, Akhmediev N 2011 Phys. Rev. Lett. 106 204502Google Scholar

    [19]

    Peregrine D H 1983 J. Aust. Math. Soc. Ser. B:Appl. Math. 25 16Google Scholar

    [20]

    Akhmediev N, Ankiewicz A, Soto Crespo J M 2009 Phys. Rev. E 80 026601Google Scholar

    [21]

    Kedziora D J, Ankiewicz A, Akhmediev N 2012 Phys. Rev. E 86 056602Google Scholar

    [22]

    张解放, 金美贞, 胡文成 2020 物理学报 69 244205Google Scholar

    Zhang J F, Jin M Z, Hu W C 2020 Acta Phys. Sin. 69 244205Google Scholar

    [23]

    张解放, 金美贞 2020 物理学报 69 214203Google Scholar

    Zhang J F, Jin M Z 2020 Acta Phys. Sin. 69 214203Google Scholar

    [24]

    Zakharov V E (edited by Bullough R, Caudrey P) 1980 The Inverse Scattering Method (Vol. 17) (Berlin: Springer) pp243–285 DOI: 10.1007/978-3-642-81448-8_7

    [25]

    Radha R, Lakshmanan M 1994 Inverse Problems 10 L29Google Scholar

    [26]

    Strachan I A B 1992 Inverse Problems 8 L21Google Scholar

    [27]

    Radha R, Lakshmanan M 1997 J. Phys. A:Math. Gen. 30 3229Google Scholar

    [28]

    沈守枫, 张隽 2008 应用数学和力学 29 1254Google Scholar

    Shen S F, Zhang J 2008 Appl. Math. Mech. 29 1254Google Scholar

    [29]

    Wang J, Chen L W, Liu C F 2014 Appl. Math. Comput. 249 76

    [30]

    程丽, 张翼 2016 长江大学学报(自科版) 13 35

    Chen L, Zhang Y 2016 J. Yangtze Univ. (Nat. Sci. Ed.) 13 35 (in Chinese)

    [31]

    Wang X B, Tian S F, Zhang T T 2018 Proceedings of Roceedings of the American Society 146 3353Google Scholar

    [32]

    Chen M D, Li B 2017 Modern Phys. Lett. B 31 1750298

    [33]

    Fokas A S 1994 Inverse Problems 10 19Google Scholar

    [34]

    Chen J C, Chen Y 2014 J. Nonlinear Math. Phys. 21 454

    [35]

    Radha R, Lakshmanan M 1997 Chaos, Solitons and Fractals 8 17Google Scholar

    [36]

    Villarroel J, Prada J, Estévez P G 2009 Stud. Appl. Math. 122 395Google Scholar

    [37]

    Estévez P G 1999 J. Math. Phys. 40 1406Google Scholar

    [38]

    Rao J G, Wang L H, Zhang Y, He J S 2015 Commun. Theor. Phys. 64 605Google Scholar

    [39]

    Chen T T, Hu P Y, He J S 2019 Commun. Theor. Phys. 71 496Google Scholar

    [40]

    Rao J G, Mihalacheb D, Cheng Y, He J S 2019 Phys. Lett. A 383 1138Google Scholar

  • [1] 杨利霞, 刘超, 李清亮, 闫玉波. 斜入射非线性电离层Langmuir扰动的电磁波传播特性. 物理学报, 2022, 71(6): 064101. doi: 10.7498/aps.71.20211204
    [2] 张解放, 俞定国, 金美贞. 二维自相似变换理论和线怪波激发. 物理学报, 2022, 71(1): 014205. doi: 10.7498/aps.71.20211417
    [3] 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
    [4] 张文玲, 马松华, 陈晶晶. (2+1)维Korteweg-de Vries方程的复合波解及局域激发. 物理学报, 2014, 63(8): 080506. doi: 10.7498/aps.63.080506
    [5] 林福忠, 马松华. (2+1)维色散长波方程的新精确解及其复合波激发. 物理学报, 2014, 63(4): 040508. doi: 10.7498/aps.63.040508
    [6] 尹君毅. (2+1)维改进的Zakharov-Kuznetsov方程的无穷序列复合型类孤子新解. 物理学报, 2014, 63(23): 230202. doi: 10.7498/aps.63.230202
    [7] 欧阳成, 石兰芳, 林万涛, 莫嘉琪. (2+1)维扰动时滞破裂孤波方程行波解的摄动方法. 物理学报, 2013, 62(17): 170201. doi: 10.7498/aps.62.170201
    [8] 杨征, 马松华, 方建平. (2+1)维 Zakharov-Kuznetsov 方程的精确解和孤子结构. 物理学报, 2011, 60(4): 040508. doi: 10.7498/aps.60.040508
    [9] 廖龙光, 付虹, 傅秀军. 十二次对称准周期结构的自相似变换及准晶胞构造. 物理学报, 2009, 58(10): 7088-7093. doi: 10.7498/aps.58.7088
    [10] 吴勇旗. (2+1)维Boussinesq方程的新的周期解. 物理学报, 2008, 57(9): 5390-5394. doi: 10.7498/aps.57.5390
    [11] 王悦悦, 杨 琴, 戴朝卿, 张解放. 考虑量子效应的Zakharov方程组的孤波解. 物理学报, 2006, 55(3): 1029-1034. doi: 10.7498/aps.55.1029
    [12] 曾 昕, 张鸿庆. (2+1)维色散长波方程的新的类孤子解. 物理学报, 2005, 54(2): 504-510. doi: 10.7498/aps.54.504
    [13] 何宝钢, 徐昌智, 张解放. (2+1)维非线性KdV方程的新孤子结构. 物理学报, 2005, 54(12): 5525-5529. doi: 10.7498/aps.54.5525
    [14] 智红燕, 王 琪, 张鸿庆. (2+1) 维Broer-Kau-Kupershmidt方程一系列新的精确解. 物理学报, 2005, 54(3): 1002-1008. doi: 10.7498/aps.54.1002
    [15] 曾 昕, 张鸿庆. (2+1)维Boussinesq方程的Backlund变换与精确解. 物理学报, 2005, 54(4): 1476-1480. doi: 10.7498/aps.54.1476
    [16] 徐昌智, 张解放. (2+1)维非线性Burgers方程变量分离解和新型孤波结构. 物理学报, 2004, 53(8): 2407-2412. doi: 10.7498/aps.53.2407
    [17] 赵长海, 盛正卯. Zakharov方程的显式行波解. 物理学报, 2004, 53(6): 1629-1634. doi: 10.7498/aps.53.1629
    [18] 那仁满都拉, 王克协. (2+1)维耗散长波方程与(2+1)维Broer-Kaup方程新的类多孤子解. 物理学报, 2003, 52(7): 1565-1568. doi: 10.7498/aps.52.1565
    [19] 赵熙强, 唐登斌, 王利民, 张耀明. 高阶(2+1)维Broer-Kaup方程的孤波解. 物理学报, 2003, 52(8): 1827-1831. doi: 10.7498/aps.52.1827
    [20] 郑春龙, 张解放. (2+1)维Camassa-Holm方程的相似约化与解析解. 物理学报, 2002, 51(11): 2426-2430. doi: 10.7498/aps.51.2426
计量
  • 文章访问数:  2428
  • PDF下载量:  66
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-06-23
  • 修回日期:  2021-12-09
  • 上网日期:  2022-01-26
  • 刊出日期:  2022-04-20

/

返回文章
返回