scholarly journals Well-posedness for the fourth-order Schrödinger equation with third order derivative nonlinearities

2021 ◽  
Vol 28 (5) ◽  
Author(s):  
Hiroyuki Hirayama ◽  
Masahiro Ikeda ◽  
Tomoyuki Tanaka
Keyword(s):  
Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Order Derivative ◽  
Well Posedness ◽  
Third Order

On well-posedness of the third-order nonlinear Schrödinger equation with time-dependent coefficients

2015 ◽  
Vol 17 (04) ◽  
pp. 1450031 ◽  
Author(s):  
Xavier Carvajal ◽  
Mahendra Panthee ◽  
Marcia Scialom
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Scaling Limit ◽  
Nonlinear Schrodinger Equation ◽  
Time Dependent ◽  
Well Posedness ◽  
Third Order ◽  
Nonlinear Schrödinger ◽  
The Third

We consider the Cauchy problem associated to the third-order nonlinear Schrödinger equation with time-dependent coefficients. Depending on the nature of the coefficients, we prove local as well as global well-posedness results for given data in L2-based Sobolev spaces. We also address the scaling limit to fast dispersion management and prove that it converges in H1to the solution of the averaged equation.

scholarly journals GLOBAL WELL-POSEDNESS FOR THE GENERALISED FOURTH-ORDER SCHRÖDINGER EQUATION

2012 ◽  
Vol 85 (3) ◽  
pp. 371-379 ◽  
Author(s):  
YUZHAO WANG
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Vries Equation ◽  
Well Posedness ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Image Position ◽  
Appl Math

AbstractWe study the Cauchy problem for the generalised fourth-order Schrödinger equation for data u0 in critical Sobolev spaces $\dot {H}^{1/2-3/2k}$. With small initial data we obtain global well-posedness results. Our proof relies heavily on the method developed by Kenig et al. [‘Well-posedness and scattering results for the generalised Korteweg–de Vries equation via the contraction principle’, Commun. Pure Appl. Math.46 (1993), 527–620].

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