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Integrable effective dynamics for a non linear wave equation
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Universite´dOrle´ans-Fe´d´erationDenisPoisson WORSHOPHANDDYSEPT. 2011
jointwork with rardP. Ge´is´tvire(nUd)issuePar
Sandrine Grellier
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Consider the half wave equation
itu− |D|u=|u|2u,tR,xT
where
=X |D|kXZckeikx!:kZck|k|eikx.
Toy model for NLS on degenerate geometries leading to lack of dispersion. Admits the same conservation laws as NLS : H(u) =12(|D|u,u)L2+14kukL44, Q(u) =kukL22, M(u) = (Du,u)L2.
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itu− |D|u=|u|2u,tR,xT
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Toy model for NLS on degenerate geometries leading to lack of dispersion. Admits the same conservation laws as NLS : H(u) =12(|D|u,u)L2+41kukL44, 2 Q(u) =kukL2, M(u) = (Du,u)L2.
where
|D|kXZckeikx!:=kXZck|k|eikx.
Consider the half wave equation
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itu− |D|u=|u|2u,tR,xT
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Toy model for NLS on degenerate geometries leading to lack of dispersion. Admits the same conservation laws as NLS : 4 H(u) =12(|D|u,u)L2+14kukL4, Q(u) =kuk2L2, M(u) = (Du,u)L2.
a
where
u
D|Xc |kZkeikx!:=kXZck|k|eikx.
i
Consider the half wave equation
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whereu±:= Π±uand Π+(Xckeikx) :=Xckeikx, kZk0 Π(Xceikx) :=Xckeikx k. nZk<0
Compared to the ”usual” 1D-NLS
itu+x2u=|u|2u,
this is anon dispersiveequation : (i(t+x)u+= Π+(|u|2u), i(tx)u= Π(|u|2u),
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itu+2u|u|2u, x=
this is anon dispersiveequation : (i(t+x))uu+=ΠΠ=+((||uu||22uu)),, i(tx
whereu±:= Π±uand Π+(Xckeikx) :=Xcke, ikx kZk0 Π(Xckeikx) :=Xckeikx. nZk<0
Compared to the ”usual” 1D-NLS
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