Doktorov E.V., Leble S.B. A Dressing Method in Mathematical Physics
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10.5 Davey–Stewartson II equation 349
where
b(k,
¯
k)=
i
4π
dz
R
dz
I
¯u(z, ¯z)M
1
(z, ¯z,k,
¯
k)e
i(kz+
¯
k¯z)
.
Comparing (10.115) with (10.112), we conclude that the
¯
∂ equation for M
takes the form
¯
∂M = b(k,
¯
k)N. (10.116)
To obtain the closure relation which connects N and M, we write the second
equation in (10.112) as
N
2
=e
−i(kz+
¯
k¯z)
1+
1
2πi
dz
′
∧ d¯z
′
¯z
′
− ¯z
1
2
(QN)
2
(z
′
, ¯z
′
,k,
¯
k)e
i(kz
′
+
¯
k¯z
′
)
and compare it with the equation (10.111) for
M
1
.Asaresult,weget
N =
0 −1
10
Me
−i(kz+
¯
k¯z)
. (10.117)
10.5.2 Discrete spectrum and inverse problem solution
The discrete spectrum for the spectral problem (10.108) is given by a set of iso-
lated eigenvalues k
j
,j =1,...,N, for which homogeneous integral equations
allow nontrivial solutions. As distinct from the KP I equation, the eigenvalue
k
j
for the DS II equation has double degeneracy with eigenstates Φ
j
(z, ¯z)and
χ
j
(z, ¯z) [31]. The bound state vector Φ
(j)
=(Φ
(j)
1
,Φ
(j)
2
)
T
is a solution of the
equations
Φ
(j)
1
(z, ¯z)=
1
2πi
dz
′
∧ d¯z
′
z
′
− z
1
2
u(z
′
, ¯z
′
)Φ
(j)
2
(z
′
, ¯z
′
), (10.118)
Φ
(j)
2
(z, ¯z)=−
1
2πi
dz
′
∧ d¯z
′
¯z
′
− ¯z
1
2
¯u(z
′
, ¯z
′
)Φ
(j)
1
(z
′
, ¯z
′
)e
−ik
j
(z−z
′
)−i
¯
k
j
(¯z−¯z
′
)
and χ
(j)
is expressed through Φ
(j)
as
χ
(j)
=
0 −1
10
Φ
(j)
e
−i(k
j
z+
¯
k
j
¯z )
. (10.119)
It is seen from (10.118) that it is natural to normalize bound eigenfunctions
by the condition
Φ
(j)
→
1
z
Q
j
0
, |z|→∞
under the constraint
dz ∧ d¯z¯u(z, ¯z)Φ
(j)
1
(z, ¯z)e
−i(k
j
z+
¯
k
j
¯z )
=0.
350 10 Generating solutions via
¯
∂ problem
Here the index Q
j
is defined as
Q
j
=
1
2π
dxdyu(x, y)Φ
(j)
2
(x, y). (10.120)
We will see that the index Q
j
plays the same role as that for the KP I equation.
The existence of the discrete spectrum indicates that the eigenvector M is
singular and has (in general, multiple) poles in the points k
j
. First we consider
thecaseofsimplepoles:
M =
N
j=1
(k − k
j
)
−1
Φ
(j)
(z, ¯z)+M
reg
,
where M
reg
is regular in k
j
. In order to determine residues Φ
(j)
, we need to
find the limit of M when k → k
j
. We have around k = k
j
M =
Φ
(j)
k − k
j
+ µ, (10.121)
where µ is regular in k
j
and tends to (1, 0)
T
at |k|→∞.TakingfirstQ
j
=1,
we substitute (10.121) into (10.113). Following the procedure described in
Sect. 9.2.2 and accounting for the double degeneracy of k
j
yields [31]
lim
k→k
j
M(z, ¯z,k,
¯
k) −
1
k − k
j
Φ
(j)
(z, ¯z)
= −i(z + z
j
)Φ
(j)
(z, ¯z) −ic
j
χ
(j)
(z, ¯z),
(10.122)
where z
j
and c
j
are constants and Φ
(j)
and χ
(j)
are related by (10.119).
Combining now the contributions of both the continuous and the discrete
spectra, we can write the eigenvector M as
M(z, ¯z,k,
¯
k)=
1
0
+
N
j=1
Φ
(j)
(z, ¯z)
k − k
j
+
1
2πi
dℓ ∧ d
¯
ℓ
ℓ − k
b(ℓ,
¯
ℓ)N(z, ¯z, ℓ,
¯
ℓ).
(10.123)
The limit k → k
j
gives in a ccordance with (10.122) and for Q
j
=1
−i(z + z
j
)Φ
(j)
(z, ¯z) − ic
j
χ
(j)
(10.124)
=
1
0
+
N
i=j
Φ
(i)
(z, ¯z)
k
j
− k
i
+
1
2πi
dℓ ∧ d
¯
ℓ
ℓ − k
j
b(ℓ,
¯
ℓ)N(z, ¯z, ℓ,
¯
ℓ).
Equations (10.123) and (10.124) comprise the closed system of equations
which enable us to solve the inverse problem. Indeed, owing to (10.106), the
potential u(z, ¯z) is retrieved as
u(z, ¯z)=2i
N
j=1
Φ
(j)
2
(z, ¯z) −
1
π
dk ∧ d
¯
k
¯
b(k,
¯
k)N
2
(z, ¯z,k,
¯
k). (10.125)
10.5 Davey–Stewartson II equation 351
10.5.3 Lump solutions
As an example, we derive the one-lump solution to the DS II equation. In this
case b(k,
¯
k)=0,N = 1 and the eigenvector takes the form
M =
1
0
+
Φ
(1)
k − k
1
. (10.126)
In v irtue of (10.119), equation (10.124) reduces to
(z + z
1
)Φ
(1)
+ c
1
0 −1
10
e
−i(k
1
z+
¯
k
1
¯z )
Φ
(1)
=i
1
0
.
Solving this algebraic equation yields
Φ
(1)
=
i
|z + z
1
|
2
+ |c
1
|
2
¯z +¯z
1
c
1
exp
−i(k
1
z +
¯
k
1
¯z)
. (10.127)
Hence, from (10.125) we obtain the lump potential
u
1
(x, y)=
2¯c
1
|z + z
1
|
2
+ |c
1
|
2
e
i(k
1
z+
¯
k
1
¯z)
. (10.128)
As usual, in order to have a lump solution of the DS II equation, we need
to know the temporal evolution of the parameters k
1
, z
1
,andc
1
. Substituting
the vector function M (10.126) with the residue (10.127) into (10.105) and
taking into account (10.100), we obtain
k
1
≡ ξ+iη =const,z
1
(t)=z
1
(0)+2k
1
t, c
1
(t)=c
1
(0) exp
−2i(ξ
2
− η
2
)t
.
(10.129)
Therefore, the lump solu tion o f the DS II equation is written as [31]
u
1
(x, y, t)=2¯c
1
(0)
exp
2i(ξx − ηy +(ξ
2
− η
2
)t)
[x +2ξt + z
1R
(0)]
2
+[y +2ηt + z
1I
(0)]
2
+ |c
1
(0)|
2
.
(10.130)
It describes a localized nonsingular object which moves on the (x, y)-plane
with constant velocity (−2ξ, −2η) and decays as (x
2
+ y
2
)
−1
.
The solution (10.130) corresponds to Q = 1. A novel situation arises for
the DS II equation, as distinct from the KP I equation. Namely, there exist
meromorphic functions of the type (10.126) related to higher indices. For
example, for Q = 2 and the simple pole k
1
the analog of the limit relation
(10.122) takes a more complicated form [440]:
1
2
(z + z
1
)
2
− δ
Φ
(1)
+¯c
1
(¯z +¯z
1
+¯ǫ)e
−i(k
1
z+
¯
k
1
¯z )
0 −1
10
Φ
(1)
=i
z + z
1
c
1
e
−i(k
1
z+
¯
k
1
¯z )
.
352 10 Generating solutions via
¯
∂ problem
Solving this equation with respect to
Φ
(1)
2
, we obtain from (10.125) the lump
solution with Q =2:
u
2
(x, y, t)=∆
−1
¯c
1
(z + z
1
)
2
− 2c
1
(z + z
1
)(z + z
1
+ ǫ) − ¯c
1
δ
e
i(k
1
z+
¯
k
1
¯z)
,
(10.131)
with additional complex parameters ǫ and δ.Here
∆ =
1
4
|(z + z
1
)
2
− δ|
2
+ |c
1
|
2
|z + z
1
+ ǫ|
2
(10.132)
and the complex parameters k
1
, z
1
, c
1
, δ,andǫ evolve in accordance with
(10.129) and δ
t
=2i,ǫ
t
= 0. The solution (10.131) is nonsingular and decays as
(x
2
+y
2
)
−1
. Analysis of the denominator in (10.131) shows that asymptotically
this solution decomposes into two separate lumps [440]. These lumps attract
each other but the attractive force is not strong enough to form a bound state.
After collision the lumps scatter at the angle π/2 (Figs. 10.1–10.3).
The simplest example of a mer omorphic column function with a double
pole is gi ven by
M(k)=
1
0
+
Φ
k − k
1
+
Ψ
(k − k
1
)
2
. (10.133)
In the same way as for the KP I equation, we substitute (10.133) into (10.113)
and expand the Green function (10.114) up to the second order in (k − k
1
).
As a result, the limit relation takes the form for Q
1
=2
Φ = −i(z + z
1
)Ψ +i¯c
1
e
−i(k
1
z+
¯
k
1
¯z )
0 −1
10
Ψ,
1
0
=
1
2
−(z + z
1
)
2
+ δ
Ψ + c
1
(z + z
1
+ ǫ)e
−i(k
1
z+
¯
k
1
¯z )
0 −1
10
Ψ.
Solving this algebraic system, we obtain a solution to the DS II equation for
thedoublepolewithQ = 2. It turns out that this solution coincides with
the solution (10.130) after letting t →−t and taking complex conjugation
Fig. 10.1. Two lumps described by the solution (10.131) before interaction [440]
10.5 Davey–Stewartson II equation 353
0
2
4
x
0
2
4
y
0
2
4
6
8
10
t=0
-4
-2
-4
-2
Fig. 10.2. Interaction of lumps described by the solution (10.131). Two lumps are
merged into a single rotationally symmetric object with an amplitude equal to the
sum of those of the separate lumps [440]
0
2
4
x
0
2
4
y
0
1
2
3
4
-4
-2
-2
-4
t=4
Fig. 10.3. Two lumps after interaction. The positions of lumps after scattering
differ from those before interaction at the angle π/2 [440]
[440]. This result implies that the physically related solutions u(x, y, t)and
¯u(x, y, −t)forQ = 2 have different p ole structures. Examples of meromorphic
vector functions with a more complicated combination of poles can be found
in the paper by Villarroel and Ablowitz [440].
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