scholarly journals Low regularity for the fifth order Kadomtsev–Petviashvili-I type equation

2017 ◽  
Vol 263 (9) ◽  
pp. 5696-5726 ◽  
Author(s):  
Boling Guo ◽  
Zhaohui Huo ◽  
Shaomei Fang
Keyword(s):  
Type Equation ◽  
Low Regularity ◽  
Fifth Order

Low regularity Cauchy problem for the fifth-order modified KdV equations on 𝕋

2018 ◽  
Vol 15 (03) ◽  
pp. 463-557 ◽  
Author(s):  
Chulkwang Kwak
Keyword(s):  
Kdv Equation ◽  
Main Tool ◽  
Well Posedness ◽  
Current Form ◽  
Low Regularity ◽  
Integrable Structure ◽  
Fourier Restriction ◽  
Fourier Restriction Norm Method ◽  
Modified Kdv Equation ◽  
Fifth Order

We consider the fifth-order modified Korteweg–de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in [Formula: see text], [Formula: see text], via the energy method. The main tool is the short-time Fourier restriction norm method, which was first introduced in its current form by Ionescu, Kenig and Tataru [Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math. 173(2) (2008) 265–304]. Besides, we use the frequency localized modified energy to control the high-low interaction component in the energy estimate. We remark that under the periodic setting, the integrable structure is very useful (but not necessary) to remove harmful terms in the nonlinearity and this work is the first low regularity well-posedness result for the fifth-order modified KdV equation.

scholarly journals Local Analyticity in the Time and Space Variables and the Smoothing Effect for the Fifth-Order KdV-Type Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-39
Author(s):  
Kyoko Tomoeda
Keyword(s):  
Kdv Equation ◽  
Type Equation ◽  
Smoothing Effect ◽  
Initial Value ◽  
Time And Space ◽  
Third Order ◽  
Bootstrap Argument ◽  
Real Analytic ◽  
Fifth Order ◽  
Bourgain Space

We consider the initial value problem for the reduced fifth-order KdV-type equation: , , , . This equation is obtained by removing the nonlinear term from the fifth-order KdV equation. We show the existence of the local solution which is real analytic in both time and space variables if the initial data satisfies the condition , for some constant . Moreover, the smoothing effect for this equation is obtained. The proof of our main result is based on the contraction principle and the bootstrap argument used in the third-order KdV equation (K. Kato and Ogawa 2000). The key of the proof is to obtain the estimate of on the Bourgain space, which is accomplished by improving Kenig et al.'s method used in (Kenig et al. 1996).

Quasi-periodic wave solutions and asymptotic properties for a fifth-order Korteweg–de Vries type equation

2016 ◽  
Vol 30 (18) ◽  
pp. 1650223 ◽  
Author(s):  
Chun-Yan Qin ◽  
Shou-Fu Tian ◽  
Xiu-Bin Wang ◽  
Tian-Tian Zhang
Keyword(s):  
Asymptotic Properties ◽  
Soliton Solutions ◽  
Type Equation ◽  
Periodic Wave ◽  
Ocean Dynamics ◽  
Nonlinear Phenomena ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Fifth Order

Under investigation in this paper is a fifth-order Korteweg–de Vries type (fKdV-type) equation with time-dependent coefficients, which can be used to describe many nonlinear phenomena in fluid mechanics, ocean dynamics and plasma physics. The binary Bell polynomials are employed to find its Hirota’s bilinear formalism with an extra auxiliary variable, based on which its [Formula: see text]-soliton solutions can be also directly derived. Furthermore, by considering multi-dimensional Riemann theta function, a lucid and straightforward generalization of the Hirota–Riemann method is presented to explicitly construct the multiperiodic wave solutions of the equation. Finally, the asymptotic properties of these periodic wave solutions are strictly analyzed to reveal the relationships between periodic wave solutions and soliton solutions.

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