On versions of the Kadomtsev-Petviashvili equation

On versions of the Kadomtsev-Petviashvili
equation
Karima Khusnutdinova
Department of Mathematical Sciences, Loughborough University, UK
K.Khusnutdinova@lboro.ac.uk
in collaboration with
C. Klein*, V.B. Matveev* and A.O. Smirnov**
*Insitute de Mathématique de Bourgogne, Dijon, France
**St. Petersburg Aerospace University, Russia
Nonlinear Evolution Equations and Wave Phenomena
March 25 - 28, 2013, Athens, USA
Overview
I
Introduction: Kadomtsev-Petviashvili (KP) and cylindrical KP (cKP)
equations.
I
Derivation of the elliptic cylindrical KP (ecKP) equation for long
surface gravity waves.
I
Transformations between KP, cKP and ecKP equations.
I
New soliton solutions for surface waves.
I
Lax pair.
I
Concluding remarks.
1. Introduction: Euler equations for incompressible fluid
Let (u, v , w ) be the three components of the velocity vector in the
Cartesian coordinates (x, y , z), t - time, p - pressure (pa is the constant
atmospheric pressure at the surface, and Γ is the coefficient of the
surface tension), ρ - constant density, g - gravitational acceleration,
z = 0 - the bottom, and z = h(x, y , t) - the free surface elevation.
ρ(ut + uux + vuy + wuz ) = −px ,
ρ(vt + uvx + vvy + wvz ) = −py ,
ρ(wt + uwx + vwy + wwz ) = −pz − ρg ,
ux + vy + wz = 0,
p|z=h(x,y ,t) = pa − Γ
(1 + hy2 )hxx + (1 + hx2 )hyy − 2hx hy hxy
,
(1 + hx2 + hy2 )3/2
w |z=h(x,y ,t) = ht + uhx + vhy ,
w |z=0 = 0.
(1)
1. Introduction: Kadomtsev-Petviashvili (KP) and
cylindrical KP (cKP) equations
The original KP equation (KP, 1970) (Ablowitz and Segur, 1979 for
surface waves; Grimshaw, 1985 for internal waves with shear flow)
(Uτ + 6UUξ + Uξξξ )ξ + 3α2 UYY = 0
(2)
and cylindrical KP (cKP) equation (Johnson, 1980 for surface waves;
Lipovskii, 1985 for internal waves)
W
3α2
Wτ + 6WWχ + Wχχχ +
(3)
+ 2 WVV = 0
2τ χ
τ
describe the evolution of nearly-plane and nearly-concentric waves,
respectively.
1. Introduction: Kadomtsev-Petviashvili (KP) and
cylindrical KP (cKP) equations
Transformations between the KP and cKP equations were found by
Johnson (1980) and rediscovered by Lipovskii, Matveev, Smirnov (1989).
The map
Y2 Y
W (τ, χ, V ) → U(τ, ξ, Y ) := W τ, ξ +
,
12α2 τ τ
transforms any solution of the cKP equation (3) into a solution of the
KP equation (2). Conversely, the map
τV 2
U(τ, ξ, Y ) → W (τ, χ, V ) := U τ, χ −
,
τ
V
12α2
transforms any solution of the KP equation (2) into a solution of the
cKP equation (3).
The transformation has been used to construct some special solutions of
the cKP equation by Klein, Matveev, Smirnov (2007).
2. Derivation of the elliptic cylindrical KP equation
We aim to consider waves with the nearly-elliptic front, and we write this
set of equations in the elliptic cylindrical coordinate system:
x = d cosh α cos β,
y = d sinh α sin β,
z = z,
where d has the meaning of half a distance between the foci of the
coordinate lines, and change the two horizontal components of the
velocity vector: u → u cos β − v sin β, v → u sin β + v cos β.
We nondimensionalise the variables
λ
t,
x → λx, y → λy , z → h0 z, t → √
gh0
√
p
p
h0 gh0
u → gh0 u, v → gh0 v , w →
w,
λ
h → h0 + hs η, p → pa + ρg (h0 − z) + ρgh0 p,
which leads to the appearance of the two usual nondimensional
parameters in the problem: the long wavelength parameter δ = hλ0 and
the small amplitude parameter = hh0s , as well as a new nondimensional
parameter γ = λd (ellipticity parameter).
Scaling: u → u,
v → v ,
w → w ,
p → p.
2. Derivation of the elliptic cylindrical KP equation
Equation for the linear waves (in the long-wave approximation) is
obtained for = δ = 0 as
ηαα + ηββ
.
ηtt =
2
γ (sinh2 α + sin2 β)
(4)
Equation (4) reduces to the equation
1
1
ηtt − (ηrr + ηr + 2 ηββ ) = 0
(5)
r
r
for the long linear waves in the polar cylindrical coordinates in the limit
1 α
γe → r being finite.
(6)
α → ∞, γ → 0 with
2
The derivation of the cylindrical KP (cKP) equation (also known as the
nearly-concentric KP equation or Johnson’s equation) is based on the
existence of the exact reduction of the equation (5) to the equation
1
ηtt − (ηrr + ηr ) = 0.
r
Unlike (5), equation (4) does not have an exact reduction to the
equation with no dependence on β. However, there is an asymptotic
reduction.
2. Derivation of the elliptic cylindrical KP equation
We introduce the variables:
2
6
δ
(γ
cosh
α
−
t)
,
R
=
γ cosh α, ν = 2 sin β,
δ2
δ4
3
3
3
5
5
η = 2 H, p = 2 P, u = 2 U, w = 4 W , v = 3 V ,
δ
δ
δ
δ
δ
ζ=
which generalise a change of variables for the cylindrical coordinates.
Here, 2γ cosh α is the sum of the distances from a point on an ellipse to
its foci. Thus, ζ is an asymptotic characteristic coordinate for the waves
with the nearly-elliptic front, and it becomes a characteristic coordinate
for the concentric waves in the limit (6).
Notations:
∆=
4
1,
δ2
A=γ
6
.
δ4
2. Derivation of the elliptic cylindrical KP equation
We now seek a solution of the system of equations and boundary
conditions in the form of asymptotic multiple-scales expansions:
H = H0 + ∆H1 + O(∆2 ),
with similar expansions for U, V , W and P.
We obtain the elliptic cylindrical KP equation (ecKP)
«
«
„
„
1
R
ν2
2
2H0R + 3H0 H0ζ +
− We H0ζζζ + 2
H
−
A
H
0
0ζ
3
R − A2
R 2 − A2
ζ
+
1
H0νν = 0,
R 2 − A2
Γ
where We = ρgh
2 (Weber number).
0
A scaling transformation brings the derived equation to the form
τ
a2 ν 2
3σ 2
H
−
H
+ 2
H = 0,
Hτ + 6HHζ + Hζζζ +
ζ
2
2
2
2
2
2 νν
2(τ − a )
12σ (τ − a )
ζ τ −a
where σ 2 = sign
1
3
− We .
3. Transformations between KP, cKP and ecKP equations
The geometry of a wave with the nearly-elliptic front, considered
simultaniously in the Cartesian and elliptic cylindrical coordinates,
suggests the introduction of the sum and the difference of the distances
from a point on the wave front to the two foci of the coordinate system
d1 + d2 = 2γ cosh α,
d1 − d2 = 2γ cos β,
where the foci have the following Cartesian coordinates: F1 (−γ, 0) and
F2 (γ, 0). We recall that the variables have been nondimensionalised, as
discussed in section 2, and
d
γ= .
λ
Note that 21 (d1 + d2 ) − t corresponds, up to the scaling, to the
asymptotic characteristic variable ζ.
3. Transformations between KP, cKP and ecKP equations
Then, for the area satisfying
asymptotic behaviour
y
y
x−γ , x+γ
→ 0, we obtain the following
1
1
(d1 + d2 ) − t ∼ x − t + y 2
2
4
„
1
1
+
x +γ
x −γ
«
.
Next, for sufficiently large α and small β, our nondimensional variable
4
x = γ cosh α cos β ∼ δ6 R, and the previous asymptotics can be rewritten
as
1
1
R
(d1 + d2 ) − t ∼ ξ + Y 2 2
,
2
2
R − A2
where ξ = x − t, Y =
3
δ2 y
6
and A = γ δ4 . Similarly,
A Y2
1
(d1 − d2 ) ∼ γ −
.
2
2 R 2 − A2
This asymptotic behaviour of the geometrically meaningful objects
motivates the change of variables:
Rν 2 p 2
H0 (R, ζ, ν) = η(R, ζ −
, R − A2 ν).
2
3. Transformations between KP, cKP and ecKP equations
We write the KP equation in the form
(Uτ + 6UUξ + Uξξξ )ξ + 3α2 UYY = 0,
the cKP equation in the form
3α2
1
W
+ 2 WVV = 0,
Wτ + 6WWχ + Wχχχ +
2τ
τ
χ
and the ecKP equation as
„
Hτ + 6HHζ + Hζζζ +
τ
a2 ν 2
H−
Hζ
2
2
2
2(τ − a )
12σ (τ 2 − a2 )
«
+
ζ
3σ 2
Hνν = 0.
− a2
τ2
The map
τV 2
, τV
U(τ, ξ, Y ) → W (τ, χ, V ) := U τ, χ −
12α2
transforms any solution of the KP equation into a solution of the cKP
equation, and the map
τ ν2 p 2
2ν
U(τ, ξ, Y ) → H(τ, ζ, ν) := U τ, ζ −
,
τ
−
a
12α2
transforms any solution of the KP eq. into a solution of the ecKP eq.
4. New soliton solutions for surface waves
We return to the original nondimensional variables x, y , t:
x = γ cosh α cos β, y = γ sinh α sin β,
r
4
a
η = 1/3 (1 − 3We )1/3
H(τ, ζ, ν),
γ
6
where
„
Hτ + 6HHζ + Hζζζ +
τ
a2 ν 2
H−
Hζ
2
2
2
2(τ − a )
12σ (τ 2 − a2 )
σ 2 = sign (1 − 3We )
«
+
ζ
and τ = R = a cosh α,
1/3
ζ=
6 a
(γ cosh α − t),
γ∆(1 − 3We )1/3
ν=
62/3
sin β.
∆1/2 |1 − 3We |1/6
Here, t is the physical time (nondimensional), We =
We also have
r
γ
γ
=
∆, δ = ∆3/2 .
a
a
Γ
.
ρgh02
3σ 2
Hνν = 0,
− a2
τ2
4. New soliton solutions for surface waves
The 1-soliton solution of the ecKP-II equation is explicitly written in the
form
H(τ, ζ, ν) =
» „
«–
p
τ ν2
K2
K
ζ−
sech2
+ L τ 2 − a2 ν − (K 2 + 3L2 )τ + δ0
,
2
2
12
where K , L, δ0 are arbitrary constants. The corresponding surface wave
elevation η is plotted below for γ = 1, a = 2, ∆ = 1/2 and
We = 0, δ0 = 0.
Figure: Surface wave corresponding to the one-soliton solution of the ecKP-II
equation with K = 1, L = 0 for t = 0 (top left), t = 0.25 (top right), t = 0.5
(bottom left), t = 1 (bottom right).
4. New soliton solutions for surface waves
Figure: Surface wave corresponding to the one-soliton solution of the ecKP-II
equation with K = 1, L = 0.1 for t = 0 (top left), t = 0.25 (top right), t = 0.5
(bottom left), t = 1 (bottom right).
4. New soliton solutions for surface waves
Figure: Surface wave corresponding to the one-soliton solution of the ecKP-II
equation with K = 1, L = −0.5 for t = 0 (top left), t = 0.25 (top right),
t = 0.5 (bottom left), t = 2 (bottom right).
4. New soliton solutions for surface waves
Figure: Surface wave corresponding to the one-soliton solution of the ecKP-II
eq. with K = 1.5, L = 0 for t = 0 (left), t = 2 (right).
Figure: Surface wave corresponding to the one-soliton solution of the ecKP-II
eq. with K = 1.6, L = 0.1 for t = 0 (left), t = 2 (right).
Figure: Surface wave corresponding to the exceptional one-soliton solution of
the ecKP-II eq. with K = 1.5 and L ≈ 0.1 for t = 0 (left) and t = 1 (right).
4. New soliton solutions for surface waves
Figure: Surface wave corresponding to a two-soliton solution of the ecKP-II
equation with for t = 1 (top left), t = 2 (top right), t = 3 (bottom left), t = 4
(bottom right).
5. Lax pair
The ecKP equation can be obtained as a compatibility condition of the
following linear problem (Lax pair):
p
τ 2 − a2 ψζζ +
“p
τ 2 − a2 H(τ, ζ, ν)
«
τζ
a2 ν 2
√
− √
+
ψ,
12 τ 2 − a2
144σ 2 τ 2 − a2
„
«
a2 ν 2
ψτ = −4ψζζζ − 6H(τ, ζ, ν) −
ψζ
12σ 2 (τ 2 − a2 )
!
e ζ, ν)
3σ H(τ,
a2 ζν
− 3Hζ (τ, ζ, ν) + √
−
ψ.
12σ(τ 2 − a2 )3/2
τ 2 − a2
σψν =
When a = 0 we recover the Lax pair of the cKP equation (Dryuma,
1983).
6. Concluding remarks
I
Applications of cKP and ecKP?
I
Wave instabilities? (Previous results for KP and cKP by Zakharov
(1975), and Ostrovsky and Shrira (1976).)
I
Other solutions of ecKP and Euler equations? (using the map from
KP and recent results by Biondini, Chakravarty, and Kodama for the
KP).
I
Other versions of the KP and other admissible maps, associated with
the physical problem formulation?
References
1. B.P. Kadomtsev, V.I. Petviashvili, Sov. Phys. Dokl. 15 (1970)
539-541.
2. M.J. Ablowitz and H. Segur, J. Fluid Mech. 92 (1979) 691-715.
3. R.S. Johnson, J. Fluid Mech. 97 (1980) 701-719.
4. V.S. Dryuma, Dokl. Akad. Nauk SSSR, 268 (1982) 15-17.
5. R. Grimshaw, Stud. Appl. Math. 73 (1985) 1- 33.
6. V.D. Lipovskii, Izv. Acad. Nauk SSSR, Ser. Fiz. Atm. Okeana 21
(1985) 864-871.
7. V.D. Lipovskii, V.B. Matveev, and A.O. Smirnov, J. Soviet Math. 46
(1989) 1609-1612.
8. C. Klein, V.B. Matveev, and A.O. Smirnov, Theor. Math. Phys. 152
(2007) 1132-1145.
9. K.R. Khusnutdinova, C. Klein, V.B. Matveev, A.O. Smirnov, Chaos
23, 013126 (2013) 13 pages.