Well-Posedness for the Two-Dimensional Modified Zakharov–Kuznetsov Equation

Linares, Felipe; Pastor, Ademir

doi:10.1137/080739173

Cited by 119 publications

(140 citation statements)

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“…Moreover, results for the Zakharov-Kuznetsov equation (m = 1) and the modified Zakharov-Kuznetsov equation (m = 2) are also obtained. See, for example, [1,3,9,20] for the Zakharov-Kuznetsov equation and [1,4,16,17,22] for the modified Zakharov-Kuznetsov equation.…”

Section: The Generalized Zakharov-kuznetsov Equation On Two Dimensionsmentioning

confidence: 99%

“…where 2 r = 3 4 1 2 − 1 p , were given by Faminskii [3] and Linares and Pastor [16], respectively. It is known that these two estimates are optimal.…”

Section: The Generalized Zakharov-kuznetsov Equation On Two Dimensionsmentioning

confidence: 99%

“…It is known that these two estimates are optimal. On the other hand, Linares and Pastor [16] proved the following maximal function estimate…”

Section: The Generalized Zakharov-kuznetsov Equation On Two Dimensionsmentioning

confidence: 99%

See 2 more Smart Citations

Well-Posedness for the Generalized Zakharov–Kuznetsov Equation on Modulation Spaces

Kato

2016

J Fourier Anal Appl

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We consider the Cauchy problem for the generalized Zakharov-Kuznetzov equation ∂ t u + ∂ x u = ∂ x (u m+1 ) on two or three space dimensions. We mainly study the two dimensional case and give the local well-posedness and the small data global well-posedness in the modulation space M 2,1 (R 2 ) for m ≥ 4. Moreover, for the quartic case (namely, m = 3), the local well-posedness in M 1/4 2,1 (R 2 ) is given. The well-posedness on three dimensions is also considered.

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Section: The Generalized Zakharov-kuznetsov Equation On Two Dimensionsmentioning

confidence: 99%

“…where 2 r = 3 4 1 2 − 1 p , were given by Faminskii [3] and Linares and Pastor [16], respectively. It is known that these two estimates are optimal.…”

Section: The Generalized Zakharov-kuznetsov Equation On Two Dimensionsmentioning

confidence: 99%

See 1 more Smart Citation

Well-Posedness for the Generalized Zakharov–Kuznetsov Equation on Modulation Spaces

Kato

2016

J Fourier Anal Appl

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“…The mZK equation has attracted the attention of many researchers in the past few years. For instance, from the mathematical point of view, local and global existence for the Cauchy problem was studied in [16][17][18] .…”

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

The Improved Riccati Equation Method and Exact Solutions to mZK Equation

2012

International Journal of Differential Equations

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We utilize the improved Riccati equation method to construct more general exact solutions to nonlinear equations. And we obtain the travelling wave solutions involving parameters, which are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. When the parameters are taken as special values, the method provides not only solitary wave solutions but also periodic waves solutions. The method appears to be easier and more convenient by means of a symbolic computation system. Of course, it is also effective to solve other nonlinear evolution equations in mathematical physics.

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“…For = = = 1 and = 0 in (1), one obtains the classical Zakharov-Kuznetsov (ZK) equation [5], which is a mathematical model to describe the propagation of nonlinear ion-acoustic waves in magnetized plasma. Solitary wave solutions and the Cauchy problem to ZK equation have extensively been studied in the literature ( [6][7][8][9][10][11]). Panthee [12] proved that if a sufficiently smooth solution to the initial value problem associated with the ZK equation is supported compactly in a nontrivial time interval, then it vanishes identically.…”

Section: Introductionmentioning

confidence: 99%