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近年来, 尘埃等离子体的研究在太空、工业和实验室等领域中有着重要的作用. 该文从双温尘埃等离子体的控制方程组出发, 通过运用多尺度分析与约化摄动方法, 推导了(2+1)维的Kadomtsev-Petviashvili (KP)方程来描述双温尘埃等离子体声波的传播. 接下来, 利用半逆方法和分数变分原理, 将(2+1) 维KP方程推广到时空分数阶KP方程; 分数阶KP方程对于描述实际问题中的物理现象具有潜在的应用价值. 进一步, 基于李对称分析方法, 讨论了时间分数阶KP方程的守恒律, 得到了双温尘埃等离子体声波的守恒量. 最后, 基于双线性方法, 获得了分数阶KP方程的Lump解. 该解的存在说明双温尘埃等离子体中存在怪波, 特别地, 分析了分数阶阶数对怪波的影响.
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关键词:
- 尘埃等离子体 /
- 分数阶Kadomtsev-Petviashvili方程 /
- 双线性方法 /
- Lump解
In recent years, the dust plasma research plays an important role in the field of space, industry, and laboratory. In this paper, starting from the control equations of the double temperature dust plasma, we derive the (2+1)-dimensional Kadomtsev-Petviashvili (KP) equation to describe the double temperature dust plasma sound waves by using the multi-scale analysis, and reduce it by using the perturbation method. Then by using the semi inverse method and fractional variational principle, the (2+1)-dimensional KP equation is introduced into the time-space fractional KP equation (TFS-KP). The fractional KP equation has potential applications in describing physical phenomena in practical problems. Furthermore, based on the symmetrical analysis method, by which lie discussed the time fractional KP (TF-KP) equation of the conservation law, the dual temperature dust plasma acoustic conserves quantity. Finally, based on the bilinear method, the lump solution of fractional KP equation is obtained. The existence of this solution indicates the rogue waves existing in double temperature dusty plasma. The influence of fractional order on rogue wave is also analyzed.-
Keywords:
- dust plasma /
- fractional Kadomtsev-Petviashvili equation /
- bilinear method /
- Lump solutions
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图 1 当a1 = 1.8, a2 = 2.1, a4 = 0, a5 = –0.3 , a6 = 0.9, a8 = 0时A关于
$\tau$ 和$\xi$ 的图像Fig. 1. When a1 = 1.8, a2 = 2.1, a4 = 0, a5 = –0.3 , a6 = 0.9, a8 = 0, the graph of A with respect to
$\xi$ and$\eta$ .
图 2 当a1 = 1.3, a2 = 1.1, a4 = 0, a5 = –0.3 , a6 = 0.9, a8 = 0时A关于
$\xi$ 和$\eta$ 的图像Fig. 2. When a1 = 1.3, a2 = 1.1, a4 = 0, a5 = –0.3 , a6 = 0.9, a8 = 0, the graph of A with respect to
$\xi$ and$\eta$ .
图 3 当分数阶阶数为
${1}/{2}$ 时A关于$\xi$ 的图像Fig. 3. Graph of A with respect to
$\xi$ when the fractional order is${1}/{2}$ .
图 4 当分数阶阶数为1时A关于
$\xi$ 的图像Fig. 4. Graph of A with respect to
$\xi$ when the fractional order is 1. -
[1] Tian R H, Fu L, Yang H W 2019 Math. Meth. Appl. Sci. doi: 10.1002/mma.5823
[2] Seadawy A R 2017 Pramana-J. Phys. 89 49
Google Scholar
[3] Guo M, Fu C, Zhang Y, Liu J X, Yang H W 2018 Complexity 2018 6852548
Google Scholar
[4] Selwyn G S, Singh J, Bennett R S 1989 Jour. Vacu. Sci. Tech. A 7 4
Google Scholar
[5] Barkan A, Merlino R L, D'Angelo N 1995 Phys. Plas. 2 10
Google Scholar
[6] Lazar M, Kourakis I, Poedts S, Fichtne H 2018 Plan. Space Sci. 156 130
Google Scholar
[7] Duan W S 2002 Chaos Solitons Frac. 14 503
Google Scholar
[8] 高梦涵, 张艳锋, 王钧峰, 杨红卫 2015 数学建模及其应用 4 4
Gao M H, Zhang Y F, Wang J F, Yang H W 2015 Math. Model. Its Appl. 4 4
[9] Liu Q S, Zhang R G, Yang L G, Song J 2019 Phys. Lett. A 383 514
Google Scholar
[10] Zhang R G, Yang L G, Liu Q S, Yin X J 2019 Appl. Math. Comp. 346 666
Google Scholar
[11] Guo M, Dong H Y, Liu J X, Yang H W 2019 Nonl. Anal. Model. Cont. 24 1
Google Scholar
[12] Fu L, Chen Y D, Yang H W 2019 Mathematics 7 41
Google Scholar
[13] Ozkan G, Hasan A 2016 Optik 127 10076
Google Scholar
[14] 白占兵 2017 数学建模及其应用 6 2
Bai Z B 2017 Math. Model. its Appl. 6 2
[15] Yang X J, Gao F, Srivastava H M 2018 J. Comp. Appl. Math. 339 285
Google Scholar
[16] Serife M E, Emine M 2017 New. Trends. Math. Sci. 5 225
Google Scholar
[17] Song F X, Yang H W 2019 Math. Model. Nat. Phenom. 14 301
Google Scholar
[18] Yang H W, Sun J C, Fu C, 2019 Comm. Nonl. Sci. Nume. Simu. 71 187
Google Scholar
[19] Meng S M, Cui Y J 2019 Mathematics 7 186
Google Scholar
[20] Song Q L, Bai Z B 2018 Adv. Differ. Equ. 2018 183
Google Scholar
[21] Zhong Y D, Zhao Q L, Li X Y 2019 Appl. Math. Lett. 98 359
Google Scholar
[22] Zhang L J, Wang Y, Khalique C M, Bai Y Z 2018 J. Appl. Anal. Comp. 8 1938
Google Scholar
[23] Meng X Z, Zhang L 2018 Math. Meth. Appl. Sci. 39 177
Google Scholar
[24] Wang D S, Zhang H Q 2005 Chaos. Solitons Frac. 25 601
Google Scholar
[25] Shang N, Zheng B 2013 Int. J. Appl. Math. 43 1
[26] Kaplan M, Bekir A 2016 Optik 127 8209
Google Scholar
[27] Kilic B, Inc M 2015 Appl. Math. Comp. 254 70
Google Scholar
[28] 马文秀, 董焕河 2017 数学建模及其应用 6 3
Mang W X, Dong H H 2017 Math. Model. Its Appl. 6 3
[29] Tao M S, Zhang N, Gao D Z, Yang H W 2018 Adv. Differ. Equ. 2018 300
Google Scholar
[30] Ren Y W, Tao M S, Dong H H, Yang H W 2019 Adv. Differ. Equ. 2019 13
Google Scholar
[31] Lie S 1890 Teub. Leip. 2 645
[32] Noether E 1971 Tran. Theo. Stat. Phys. 1 3
Google Scholar
[33] Lu C N, Xie L X, Yang H W 2019 Compu. Math. Appl. 77 3154
Google Scholar
[34] Yang H W, Guo M, He H L 2019 Int. J. Nonl. Sci. Nume. Simu. 20 17
Google Scholar
[35] El-Tantawy S A 2018 Chaos Solitons Frac. 113 356
Google Scholar
[36] El-Tantawy S A, Elgendy A T, Ismail S 2017 Phys. Lett. A 381 3465
Google Scholar
[37] Zhang X E, Chen Y 2017 Comm. Nonl. Sci. Numer. Simu. 52 24
Google Scholar
[38] Zaki M, Hosseini M 2019 Optik 186 259
Google Scholar
[39] Zhang X E, Chen Y 2019 Appl. Math. Lett. 98 306
Google Scholar
[40] Li M M, Duan W S 2005 Chaos Solitons Frac. 23 929
Google Scholar
[41] Bailung H, Sharma S K, Nakamura Y 2011 Phys. Rev. Lett. 107 255005
Google Scholar
[42] Deng S F 2012 Appl. Math. Comp. 218 5974
Google Scholar
[43] Zhang X E, Chen Y, Tang X Y 2018 Comp. Math. Appl. 76 1938
Google Scholar
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