Nonlinear schrödinger equations with the third order dispersion on modulation spaces

X. Carvajal () and M. Panthee ()
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X. Carvajal: UFRJ
M. Panthee: IMECC-UNICAMP

Partial Differential Equations and Applications, 2022, vol. 3, issue 5, 1-21

Abstract: Abstract We consider the initial value problems (IVPs) associated to the extended nonlinear Schrödinger (e-NLS) equation $$\begin{aligned} \partial _{t}v+i\alpha \partial ^{2}_{x}v- \partial ^{3}_{x}v+i\beta |v|^{2}v = 0, \quad x,t \in \mathbb R, \end{aligned}$$ ∂ t v + i α ∂ x 2 v - ∂ x 3 v + i β | v | 2 v = 0 , x , t ∈ R , and the higher order nonlinear Schrödinger (h-NLS) equation $$\begin{aligned} \partial _{t}u-i\alpha \partial ^{2}_{x}u+ \partial ^{3}_{x}u-i\beta |u|^{2}u+\gamma |u|^{2}\partial _{x}u+\delta \partial _{x}(|u|^2)u = 0, \quad x,t \in \mathbb R, \end{aligned}$$ ∂ t u - i α ∂ x 2 u + ∂ x 3 u - i β | u | 2 u + γ | u | 2 ∂ x u + δ ∂ x ( | u | 2 ) u = 0 , x , t ∈ R , for given data in the modulation space $$M_s^{2,p}(\mathbb R)$$ M s 2 , p ( R ) . We derive a trilinear estimate for functions with negative Sobolev regularity and use it in the contraction mapping principle to prove that the IVPs associated to the e-NLS equation and the h-NLS equation are locally well-posed for $$s>-\frac{1}{4}$$ s > - 1 4 and $$s\ge \frac{1}{4}$$ s ≥ 1 4 respectively.

Keywords: Schrödinger equation; Korteweg-de vries equation; Initial value problem; Well-posedness; Sobolev spaces; Fourier–lebesgue spaces; Modulation spaces; 35G20; 35Q53; 35Q55 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s42985-022-00200-4

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