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.

Global Well-Posedness for the Fifth-Order KdV Equation in $$H^{-1}(\pmb {\mathbb {R}})$$

Annals of PDE ◽  
2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Bjoern Bringmann ◽  
Rowan Killip ◽  
Monica Visan
Keyword(s):  
Kdv Equation ◽  
Well Posedness ◽  
Fifth Order

GLOBAL WELL-POSEDNESS FOR THE KDV EQUATIONS ON THE REAL LINE WITH LOW REGULARITY FORCING TERMS

2006 ◽  
Vol 08 (05) ◽  
pp. 681-713 ◽  
Author(s):  
KOTARO TSUGAWA
Keyword(s):  
Water Waves ◽  
Low Frequency ◽  
A Priori Estimate ◽  
Delta Function ◽  
Well Posedness ◽  
Nonlinear Water Waves ◽  
Low Regularity ◽  
Dirac Delta ◽  
Fourier Restriction ◽  
Kdv Equations

We consider the initial value problem for the KdV equations with low regularity forcing terms. The case that the forcing term f(x) equals pδ′(x) appears in the study of the excitation of long nonlinear water waves by a moving pressure distribution, where δ′(x) is the first derivative of the Dirac delta function and p is a constant. We have the time global well-posedness with f(x) ∈ L2 by the L2a priori estimate. However, we cannot apply it to the case f(x) ∈ Hσ, σ < 0. To overcome this difficulty, we divide f into the high frequency part and the low frequency part and use the scaling argument. Our results include the time local well-posedness with f(x) ∈ Hσ, σ ≥ -3 and the time global well-posedness with f = pδ′(x) or f(x) ∈ Hσ, σ ≥ -3/2. Our main tools are the Fourier restriction norm method and the I-method.

scholarly journals Well-posedness of stochastic modified Kawahara equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
P. Agarwal ◽  
Abd-Allah Hyder ◽  
M. Zakarya
Keyword(s):  
Cauchy Problem ◽  
Fixed Point ◽  
Water Wave ◽  
Well Posedness ◽  
Kawahara Equation ◽  
Fourier Restriction ◽  
The Cauchy Problem ◽  
Restriction Method ◽  
Fifth Order ◽  
Point Argument

AbstractIn this paper we consider the Cauchy problem for the stochastic modified Kawahara equation, which is a fifth-order shallow water wave equation. We prove local well-posedness for data in $H^{s}(\mathbb{R})$Hs(R), $s\geq -1/4$s≥−1/4. Moreover, we get the global existence for $L^{2}( \mathbb{R})$L2(R) solutions. Due to the non-zero singularity of the phase function, a fixed point argument and the Fourier restriction method are proposed.

Higher-order rational soliton solutions for the fifth-order modified KdV and KdV equations

2021 ◽  
pp. 2150036
Author(s):  
Zhi-Jie Pei ◽  
Hai-Qiang Zhang
Keyword(s):  
Darboux Transformation ◽  
Fourth Order ◽  
Taylor Expansion ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Higher Order ◽  
Kdv Equations ◽  
Perturbation Formula ◽  
Modified Kdv Equation ◽  
Fifth Order

In this paper, we construct the generalized perturbation ([Formula: see text], [Formula: see text])-fold Darboux transformation of the fifth-order modified Korteweg-de Vries (KdV) equation by the Taylor expansion. We use this transformation to derive the higher-order rational soliton solutions of the fifth-order modified KdV equation. We find that these higher-order rational solitons admit abundant interaction structures. We graphically present the dynamics behaviors from the first- to fourth-order rational solitons. Furthermore, by the Miura transformation, we obtain the complex rational soliton solutions of the fifth-order KdV equation.

scholarly journals GLOBAL WELL-POSEDNESS OF THE PERIODIC CUBIC FOURTH ORDER NLS IN NEGATIVE SOBOLEV SPACES

2018 ◽  
Vol 6 ◽  
Author(s):  
TADAHIRO OH ◽  
YUZHAO WANG
Keyword(s):  
Fourth Order ◽  
Sobolev Spaces ◽  
Energy Functional ◽  
Well Posedness ◽  
Initial Condition ◽  
Energy Functionals ◽  
Fourier Restriction ◽  
Fourier Restriction Norm Method ◽  
The Difference ◽  
Negative Sobolev Spaces

We consider the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation (4NLS) on the circle. In particular, we prove global well-posedness of the renormalized 4NLS in negative Sobolev spaces $H^{s}(\mathbb{T})$, $s>-\frac{1}{3}$, with enhanced uniqueness. The proof consists of two separate arguments. (i) We first prove global existence in $H^{s}(\mathbb{T})$, $s>-\frac{9}{20}$, via the short-time Fourier restriction norm method. By following the argument in Guo–Oh for the cubic NLS, this also leads to nonexistence of solutions for the (nonrenormalized) 4NLS in negative Sobolev spaces. (ii) We then prove enhanced uniqueness in $H^{s}(\mathbb{T})$, $s>-\frac{1}{3}$, by establishing an energy estimate for the difference of two solutions with the same initial condition. For this purpose, we perform an infinite iteration of normal form reductions on the $H^{s}$-energy functional, allowing us to introduce an infinite sequence of correction terms to the $H^{s}$-energy functional in the spirit of the $I$-method. In fact, the main novelty of this paper is this reduction of the $H^{s}$-energy functionals (for a single solution and for the difference of two solutions with the same initial condition) to sums of infinite series of multilinear terms of increasing degrees.

scholarly journals A Refined Well-Posedness Result for the Modified KdV Equation in the Fourier–Lebesgue Spaces

2021 ◽  
Author(s):  
Andreia Chapouto
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Mkdv Equation ◽  
Well Posedness ◽  
Lebesgue Spaces ◽  
Correct Model ◽  
Modified Kdv Equation ◽  
Conservation Of Momentum ◽  
Complex Valued ◽  
Math Phys

AbstractWe study the well-posedness of the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle at low regularity. In our previous work (2021), we introduced the second renormalized mKdV equation, based on the conservation of momentum, which we proposed as the correct model to study the complex-valued mKdV outside $$H^\frac{1}{2}({\mathbb {T}})$$ H 1 2 ( T ) . Here, we employ the method introduced by Deng et al. (Commun Math Phys 384(1):1061–1107, 2021) to prove local well-posedness of the second renormalized mKdV equation in the Fourier–Lebesgue spaces $${\mathcal {F}}L^{s,p}({\mathbb {T}})$$ F L s , p ( T ) for $$s\ge \frac{1}{2}$$ s ≥ 1 2 and $$1\le p <\infty $$ 1 ≤ p < ∞ . As a byproduct of this well-posedness result, we show ill-posedness of the complex-valued mKdV without the second renormalization for initial data in these Fourier–Lebesgue spaces with infinite momentum.

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