Nonlinear Schrodinger-Airy system: Difference between revisions
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'''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 [[references.html#Kod1985 Kod1985]], [[references.html#HasKod1987 HasKod1987]] | '''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 [[references.html#Kod1985 Kod1985]], [[references.html#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 –1/2. | ||
LWP is known when <math>s \geq 1/4</math>. [[references.html#St1997d St1997d]] | LWP is known when <math>s \geq 1/4</math>. [[references.html#St1997d St1997d]] | ||
Revision as of 15:24, 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 references.html#Kod1985 Kod1985, references.html#HasKod1987 HasKod1987
When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c=\delta=\epsilon = 0} , scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s=-1} .When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c=\gamma=0} , scaling is –1/2.
LWP is known when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq 1/4} . references.html#St1997d St1997d
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 3/4} this is in references.html#Lau1997 Lau1997, references.html#Lau2001 Lau2001
The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s\geq1/4 } result is also known when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c} is a time-dependent function [Cv2002], [CvLi2003]
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s < -1/4} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon} non-zero, the solution map is not Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C^3} .
When delta = epsilon = 0 LWP is known for s > -1/4 references.html#Cv2004 Cv2004
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s < -1/4} the solution map is not C^3 [CvLi-p]
