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> | ||
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 | 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 | 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>. [[ | LWP is known when <math>s \geq 1/4\,</math>. [[St1997d]] | ||
For <math>s > 3/4\,</math> this is in [[ | 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 | 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 <math>delta = epsilon = 0\,</math> LWP is known for <math>s > -1/4\,</math> [[ | 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 <math>C^3\,</math> [ | 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 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\,} . 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 Lau1997, 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 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 = \epsilon = 0\,} LWP is known 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\,} 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 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\,} CvLi-p
