The nonlinear Schr.C3.B6dinger equation in water waves Nonlinear Schrödinger equation

a hyperbolic secant (sech) envelope soliton surface waves on deep water.

blue line: water waves.

red line: envelope soliton.

for water waves, nonlinear schrödinger equation describes evolution of envelope of modulated wave groups. in paper in 1968, vladimir e. zakharov describes hamiltonian structure of water waves. in same paper zakharov shows, modulated wave groups, wave amplitude satisfies nonlinear schrödinger equation, approximately. value of nonlinearity parameter к depends on relative water depth. deep water, water depth large compared wave length of water waves, к negative , envelope solitons may occur.

for shallow water, wavelengths longer 4.6 times water depth, nonlinearity parameter к positive , wave groups envelope solitons not exist. in shallow water surface-elevation solitons or waves of translation exist, not governed nonlinear schrödinger equation.

the nonlinear schrödinger equation thought important explaining formation of rogue waves.

the complex field ψ, appearing in nonlinear schrödinger equation, related amplitude , phase of water waves. consider modulated carrier wave water surface elevation η of form:

η
=
a
(

x

0


,

t

0


)

cos
⁡

[

k

0




x

0


−

ω

0




t

0


−
θ
(

x

0


,

t

0


)
]

,


{\displaystyle \eta =a(x_{0},t_{0})\;\cos \left[k_{0}\,x_{0}-\omega _{0}\,t_{0}-\theta (x_{0},t_{0})\right],}

where a(x0, t0) , θ(x0, t0) modulated amplitude , phase. further ω0 , k0 (constant) angular frequency , wavenumber of carrier waves, have satisfy dispersion relation ω0 = Ω(k0). then

ψ
=
a

exp
⁡

(
i
θ
)

.


{\displaystyle \psi =a\;\exp \left(i\theta \right).}

so modulus |ψ| wave amplitude a, , argument arg(ψ) phase θ.

the relation between physical coordinates (x0, t0) , (x, t) coordinates, used in nonlinear schrödinger equation given above, given by:

x
=

k

0



[

x

0


−

Ω
′

(

k

0


)


t

0


]

,

t
=

k

0


2



[
−

Ω
″

(

k

0


)
]



t

0




{\displaystyle x=k_{0}\left[x_{0}-\omega (k_{0})\;t_{0}\right],\quad t=k_{0}^{2}\left[-\omega (k_{0})\right]\;t_{0}}

thus (x, t) transformed coordinate system moving group velocity Ω (k0) of carrier waves, dispersion-relation curvature Ω (k0) – representing group velocity dispersion – negative water waves under action of gravity, water depth.

for waves on water surface of deep water, coefficients of importance nonlinear schrödinger equation are:

κ
=
−
2

k

0


2


,

Ω
(

k

0


)
=


g

k

0




=

ω

0






{\displaystyle \kappa =-2k_{0}^{2},\quad \omega (k_{0})={\sqrt {gk_{0}}}=\omega _{0}\,\!}

   




Ω
′

(

k

0


)
=


1
2





ω

0



k

0




,


Ω
″

(

k

0


)
=
−


1
4





ω

0



k

0


2




,




{\displaystyle \omega (k_{0})={\frac {1}{2}}{\frac {\omega _{0}}{k_{0}}},\quad \omega (k_{0})=-{\frac {1}{4}}{\frac {\omega _{0}}{k_{0}^{2}}},\,\!}

where g acceleration due gravity @ earth s surface.

in original (x0,t0) coordinates nonlinear schrödinger equation water waves reads:

i


∂


t

0




a
+
i


Ω
′

(

k

0


)


∂


x

0




a
+



1
2




Ω
″

(

k

0


)


∂


x

0



x

0




a
−
ν


|

a


|


2



a
=
0
,


{\displaystyle i\,\partial _{t_{0}}a+i\,\omega (k_{0})\,\partial _{x_{0}}a+{\tfrac {1}{2}}\omega (k_{0})\,\partial _{x_{0}x_{0}}a-\nu \,|a|^{2}\,a=0,}

with



a
=

ψ

∗




{\displaystyle a=\psi ^{*}}

(i.e. complex conjugate of



ψ


{\displaystyle \psi }

) ,



ν
=
κ


k

0


2




Ω
″

(

k

0


)
.


{\displaystyle \nu =\kappa \,k_{0}^{2}\,\omega (k_{0}).}





ν
=



1
2




ω

0



k

0


2




{\displaystyle \nu ={\tfrac {1}{2}}\omega _{0}k_{0}^{2}}

deep water waves.