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 [[ | '''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]] | ||
When <math>c=\delta=\epsilon = 0</math>, scaling is <math>s=-1</math>.When <math>c=\gamma=0</math>, scaling is | When <math>c=\delta=\epsilon = 0</math>, scaling is <math>s=-1</math>.When <math>c=\gamma=0</math>, scaling is \u20131/2. | ||
LWP is known when <math>s \geq 1/4</math>. [[ | LWP is known when <math>s \geq 1/4</math>. [[Bibliography#St1997d|St1997d]] | ||
For <math>s > 3/4</math> this is in [[ | For <math>s > 3/4</math> this is in [[Bibliography#Lau1997|Lau1997]], [[Bibliography#Lau2001|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] | ||
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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 [[ | When delta = epsilon = 0 LWP is known for s > -1/4 [[Bibliography#Cv2004|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 C^3 [<span class="SpellE">CvLi</span>-p] | ||
[[Category:Equations]] | [[Category:Equations]] | ||
Revision as of 19:46, 28 July 2006
The nonlinear Schrodinger-Airy system
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 propogation of pulses in an optical fiber Kod1985, HasKod1987
When , scaling is .When , scaling is \u20131/2.
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 delta = epsilon = 0 LWP is known for s > -1/4 Cv2004
For the solution map is not C^3 [CvLi-p]
