Nonlinear Schrodinger-Airy system: Difference between revisions

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The ''nonlinear Schrodinger-Airy system''
The '''nonlinear Schrodinger-Airy system'''


<center><span class="SpellE">u_t</span> + <span class="SpellE">i</span> c <span class="SpellE">u_xx</span> + <span class="SpellE">u_xxx</span> = <span class="SpellE">i</span> gamma |u|^2 u + delta |u|^2 <span class="SpellE">u_x</span> + epsilon u^2 <span class="SpellE"><u>u</u>_x</span></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>


<span class="GramE">on</span> 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 <span class="SpellE">propogation</span> of pulses in an optical fiber [[references.html#Kod1985 Kod1985]], [[references.html#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]].


<span style="mso-fareast-font-family: Symbol; mso-bidi-font-family: Symbol"><font face="Symbol"><span style="mso-list: Ignore">·</span></font></span>When c=delta=epsilon = 0, scaling is s=-1.When c=gamma=0, scaling is –1/2.
When <math>c=\delta=\epsilon = 0\,</math>, scaling is <math>s=-1\,</math>.When <math>c=\gamma=0\,</math>, scaling is -1/2.


<span style="mso-fareast-font-family: Symbol; mso-bidi-font-family: Symbol"><font face="Symbol"><span style="mso-list: Ignore">·</span></font></span>LWP is known when s >= ¼ [[references.html#St1997d St1997d]]
LWP is known when <math>s \geq 1/4\,</math>. [[St1997d]]


<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><span style="mso-list: Ignore">o</span></font></span>For s > ¾ this is in [[references.html#Lau1997 Lau1997]], [[references.html#Lau2001 Lau2001]]
For <math>s > 3/4\,</math> this is in [[Lau1997]], [[Lau2001]]


<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><span style="mso-list: Ignore">o</span></font></span>The s>=1/4 result is also known when c 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]]


<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><span style="mso-list: Ignore">o</span></font></span>For s < -1/4 and delta or epsilon non-zero, the solution map is not C^3 [<span class="SpellE">CvLi</span>-p]
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>.


<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><span style="mso-list: Ignore">o</span></font></span>When delta = epsilon = 0 LWP is known for s > -1/4 [[references.html#Cv2004 Cv2004]]
When <math>\delta = \epsilon = 0\,</math> LWP is known for <math>s > -1/4\,</math> [[Cv2004]]


<span style="mso-fareast-font-family: Wingdings; mso-bidi-font-family: Wingdings"><font face="Wingdings"><span style="mso-list: Ignore">§</span></font></span>For s < -1/4 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: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