Nonlinear Schrodinger-Airy system: Difference between revisions
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The ''nonlinear Schrodinger-Airy system'' | The ''nonlinear Schrodinger-Airy system'' | ||
<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]] | <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]] | ||
Revision as of 15:17, 28 July 2006
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 propogation of pulses in an optical fiber references.html#Kod1985 Kod1985, references.html#HasKod1987 HasKod1987
·When c=delta=epsilon = 0, scaling is s=-1.When c=gamma=0, scaling is –1/2.
·LWP is known when s >= ¼ references.html#St1997d St1997d
oFor s > ¾ this is in references.html#Lau1997 Lau1997, references.html#Lau2001 Lau2001
oThe s>=1/4 result is also known when c is a time-dependent function [Cv2002], [CvLi2003]
oFor s < -1/4 and delta or epsilon non-zero, the solution map is not C^3 [CvLi-p]
oWhen delta = epsilon = 0 LWP is known for s > -1/4 references.html#Cv2004 Cv2004
§For s < -1/4 the solution map is not C^3 [CvLi-p]
