Nonlinear Schrodinger-Airy system: Difference between revisions

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<center><math>\partial_t u  + i c \partial_x^2 u + \partial_x^3 u = i \gamma |u|^2 u + \delta |u|^2 \partial_x u + \epsilon u^2 \partial_x u </math></center>
<center><math>\partial_t u  + i c \partial_x^2 u + \partial_x^3 u = i \gamma |u|^2 u + \delta |u|^2 \partial_x u + \epsilon u^2 \partial_x u </math></center>


'''R''' is a combination of the [[nls-3 on R|cubic NLS equation]], the [[dnls-3 on R|derivative cubic NLS equation]], [[modified Korteweg-de Vries on R|complex mKdV]], and a cubic nonlinear [[Airy equation]].  This equation is a general model for propogation of pulses in an optical fiber [[Bibliography#Kod1985|Kod1985]], [[Bibliography#HasKod1987|HasKod1987]].
on '''R''' is a combination of the [[cubic nls|cubic NLS equation]], the [[cubic DNLS on R|derivative cubic NLS equation]], [[modified Korteweg-de Vries on R|complex mKdV]], and a cubic nonlinear [[Airy equation]].  This equation is a general model for propagation of pulses in an optical fiber [[Kod1985]], [[HasKod1987]].


When <math>c=\delta=\epsilon = 0</math>, scaling is <math>s=-1</math>.When <math>c=\gamma=0</math>, scaling is \u20131/2.
When <math>c=\delta=\epsilon = 0\,</math>, scaling is <math>s=-1\,</math>.When <math>c=\gamma=0\,</math>, scaling is -1/2.


LWP is known when <math>s \geq 1/4</math>. [[Bibliography#St1997d|St1997d]]
LWP is known when <math>s \geq 1/4\,</math>. [[St1997d]]


For <math>s > 3/4</math> this is in [[Bibliography#Lau1997|Lau1997]], [[Bibliography#Lau2001|Lau2001]]
For <math>s > 3/4\,</math> this is in [[Lau1997]], [[Lau2001]]


The <math>s\geq1/4 </math> result is also known when <math>c</math> is a time-dependent function [Cv2002], [CvLi2003]
The <math>s\geq1/4 \,</math> result is also known when <math>c</math> is a time-dependent function [[Cv2002]], [[CvLi2003]]


For <math>s < -1/4</math> and <math>\delta</math> or <math>\epsilon</math> non-zero, the solution map is not <math>C^3</math>.  
For <math>s < -1/4\,</math> and <math>\delta\,</math> or <math>\epsilon\,</math> non-zero, the solution map is not <math>C^3</math>.  


When delta = epsilon = 0 LWP is known for s > -1/4 [[Bibliography#Cv2004|Cv2004]]
When <math>\delta = \epsilon = 0\,</math> LWP is known for <math>s > -1/4\,</math> [[Cv2004]]


For <math>s < -1/4</math> the solution map is not C^3 [<span class="SpellE">CvLi</span>-p]
For <math>s < -1/4\,</math> the solution map is not <math>C^3\,</math> [[CvLi-p]]


[[Category:Equations]]
[[Category:Equations]]
[[Category:Schrodinger]]
[[Category:Schrodinger]]
[[Category:Airy]]
[[Category:Airy]]

Latest revision as of 12:45, 11 July 2007

The nonlinear Schrodinger-Airy system

on R is a combination of the cubic NLS equation, the derivative cubic NLS equation, complex mKdV, and a cubic nonlinear Airy equation. This equation is a general model for propagation of pulses in an optical fiber Kod1985, HasKod1987.

When , scaling is .When , scaling is -1/2.

LWP is known when . St1997d

For this is in Lau1997, Lau2001

The result is also known when is a time-dependent function Cv2002, CvLi2003

For and or non-zero, the solution map is not .

When LWP is known for Cv2004

For the solution map is not CvLi-p