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April  2021, 20(4): 1699-1719. doi: 10.3934/cpaa.2021037

KdV-type equation limit for ion dynamics system

1. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

2. 

School of Mathematics and Statistics, Chongqing University, Chongqing, 400044, China

* Corresponding author

Received  July 2020 Revised  January 2021 Published  April 2021 Early access  March 2021

Fund Project: The first author (Rong Rong) was supported by the Innovation Research for the Postgraduates of Guangzhou University under Grant (2020GDJC-D09)

In this paper, we consider the KdV-type limit for ion dynamics system. Under the Gardner-Morikawa type transforms, we derive the KdV-type equation by the scaling $ \varepsilon^{\frac{1}{4}}(x-t) \rightarrow X $, $ \varepsilon^{\frac{3}{4}}t\rightarrow T $ for ion dynamics system in one dimension. By proving the uniform estimates for the remainders system, we show that when $ \varepsilon \rightarrow 0 $, the solutions to the ion dynamics system converge globally in time to the solutions of the KdV-type equation.

Citation: Rong Rong, Yi Peng. KdV-type equation limit for ion dynamics system. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1699-1719. doi: 10.3934/cpaa.2021037
References:
[1]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modifified KdV on R and T, J. Amer. Math. Soci., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[2]

H. K. Daniel, From Vlasov-Poisson to Korteweg-de Vries and Zakharov-Kuznetsov, Commun. Math. Phys., 324 (2012), 961-993.  doi: 10.1007/s00220-013-1825-8.  Google Scholar

[3]

R. Fedele, Envelope solitons versus solitons, Phys. Scripta, 65 (2002), 502-508.   Google Scholar

[4]

Z. T. FuZ. Chen and S. K. Liu, New solutions to generalized mKdV equation, Commun. Theor. Phys., 41 (2004), 25-28.  doi: 10.1088/0253-6102/41/1/25.  Google Scholar

[5]

Z. T. FuS. K. Liu and S. D. Liu, A new approach to solve nonlinear wave equations, Commun. Theor. Phys., 39 (2003), 27-30.  doi: 10.1088/0253-6102/39/1/27.  Google Scholar

[6]

C. S. Gardner and G. M. Morikawa, Similarity in the asymptotic behaviour of collision-free hydromagnetic wave and water waves, New York Univ., Courant Inst. Math. Soci., Res. Rept., NYO-9082, (1960). Google Scholar

[7]

Z. Guo, Global well-posedness of Korteweg-de Vries equation in $H^{-3/4}(R)$, J. Math. Pures Appl., 91 (2009), 583-597.  doi: 10.1016/j.matpur.2009.01.012.  Google Scholar

[8]

Y. Guo and B. Pausader, Global smooth ion dynamics in the Euler-Poisson system, Commun. Math. Phys., 303 (2011), 89-125.  doi: 10.1007/s00220-011-1193-1.  Google Scholar

[9]

Y. Guo and X. Pu, KdV limit of the Euler-Poisson system, Arch. Ration. Mech. Anal., 211 (2014), 673-710.  doi: 10.1007/s00205-013-0683-z.  Google Scholar

[10] E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge Univ. Press, Cambridge, 1990.   Google Scholar
[11]

C. E. KenigG. Ponce and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Math. J., 59 (1989), 585-610.  doi: 10.1215/S0012-7094-89-05927-9.  Google Scholar

[12]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Commun. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[13]

D. LannesF. Linares and J. C. saut, The cauchy problem for the Euler poisson system and derivation of the Zakharov-Kuznetsov equation, Nonlinear Differ. Equ. Appl., 84 (2003), 181-213.  doi: 10.1007/978-1-4614-6348-1_10.  Google Scholar

[14]

F. Linares and G. Ponce, Introduction to nonlinear dispersive equations, Universitext. Springer, New York, 2009.  Google Scholar

[15]

S. Y. Lou and H. Y. Ruan, Infinite conservation laws for KdV equation and mKdV equation with variable coefficients, J. Phy. Sci., 02 (1992), 8-13.   Google Scholar

[16]

X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions, SIAM J. Math. Anal., 45 (2013), 834-878.  doi: 10.1137/120875648.  Google Scholar

[17]

P. Rosenau and J. M. Hyman, Compactons: solitons with finite wavelength, Phys. Rev. Lett. vol., 70 (1993), 564-567.  doi: 10.1016/0375-9601(95)00933-7.  Google Scholar

[18]

H. Schamel, A modified Korteweg-de Vries equation for ion acoustic waves due to resonant electrons, J. Plasma Phys., 9 (1973), 377-387.   Google Scholar

[19]

Ta ogetusang and Si rendaoreji, New exact solitary wave solutions to generalized mKdV equation and generalized Zakharov-Kuzentsov equation, Chin. phy. B, 015 (2006), 1143-1148.   Google Scholar

[20]

V. E. Zakharov and E. A. Kuznetzov, On three dimensional solitons, Sov. Phys. JETP, 39 (1974), 285-286.   Google Scholar

show all references

References:
[1]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modifified KdV on R and T, J. Amer. Math. Soci., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[2]

H. K. Daniel, From Vlasov-Poisson to Korteweg-de Vries and Zakharov-Kuznetsov, Commun. Math. Phys., 324 (2012), 961-993.  doi: 10.1007/s00220-013-1825-8.  Google Scholar

[3]

R. Fedele, Envelope solitons versus solitons, Phys. Scripta, 65 (2002), 502-508.   Google Scholar

[4]

Z. T. FuZ. Chen and S. K. Liu, New solutions to generalized mKdV equation, Commun. Theor. Phys., 41 (2004), 25-28.  doi: 10.1088/0253-6102/41/1/25.  Google Scholar

[5]

Z. T. FuS. K. Liu and S. D. Liu, A new approach to solve nonlinear wave equations, Commun. Theor. Phys., 39 (2003), 27-30.  doi: 10.1088/0253-6102/39/1/27.  Google Scholar

[6]

C. S. Gardner and G. M. Morikawa, Similarity in the asymptotic behaviour of collision-free hydromagnetic wave and water waves, New York Univ., Courant Inst. Math. Soci., Res. Rept., NYO-9082, (1960). Google Scholar

[7]

Z. Guo, Global well-posedness of Korteweg-de Vries equation in $H^{-3/4}(R)$, J. Math. Pures Appl., 91 (2009), 583-597.  doi: 10.1016/j.matpur.2009.01.012.  Google Scholar

[8]

Y. Guo and B. Pausader, Global smooth ion dynamics in the Euler-Poisson system, Commun. Math. Phys., 303 (2011), 89-125.  doi: 10.1007/s00220-011-1193-1.  Google Scholar

[9]

Y. Guo and X. Pu, KdV limit of the Euler-Poisson system, Arch. Ration. Mech. Anal., 211 (2014), 673-710.  doi: 10.1007/s00205-013-0683-z.  Google Scholar

[10] E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge Univ. Press, Cambridge, 1990.   Google Scholar
[11]

C. E. KenigG. Ponce and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Math. J., 59 (1989), 585-610.  doi: 10.1215/S0012-7094-89-05927-9.  Google Scholar

[12]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Commun. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[13]

D. LannesF. Linares and J. C. saut, The cauchy problem for the Euler poisson system and derivation of the Zakharov-Kuznetsov equation, Nonlinear Differ. Equ. Appl., 84 (2003), 181-213.  doi: 10.1007/978-1-4614-6348-1_10.  Google Scholar

[14]

F. Linares and G. Ponce, Introduction to nonlinear dispersive equations, Universitext. Springer, New York, 2009.  Google Scholar

[15]

S. Y. Lou and H. Y. Ruan, Infinite conservation laws for KdV equation and mKdV equation with variable coefficients, J. Phy. Sci., 02 (1992), 8-13.   Google Scholar

[16]

X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions, SIAM J. Math. Anal., 45 (2013), 834-878.  doi: 10.1137/120875648.  Google Scholar

[17]

P. Rosenau and J. M. Hyman, Compactons: solitons with finite wavelength, Phys. Rev. Lett. vol., 70 (1993), 564-567.  doi: 10.1016/0375-9601(95)00933-7.  Google Scholar

[18]

H. Schamel, A modified Korteweg-de Vries equation for ion acoustic waves due to resonant electrons, J. Plasma Phys., 9 (1973), 377-387.   Google Scholar

[19]

Ta ogetusang and Si rendaoreji, New exact solitary wave solutions to generalized mKdV equation and generalized Zakharov-Kuzentsov equation, Chin. phy. B, 015 (2006), 1143-1148.   Google Scholar

[20]

V. E. Zakharov and E. A. Kuznetzov, On three dimensional solitons, Sov. Phys. JETP, 39 (1974), 285-286.   Google Scholar

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