Acta Polytechnica


doi:10.14311/AP.2016.56.0180
Acta Polytechnica 56(3):180–192, 2016 © Czech Technical University in Prague, 2016

available online at http://ojs.cvut.cz/ojs/index.php/ap

ON IMMERSION FORMULAS FOR SOLITON SURFACES

Alfred Michel Grundlanda, b, ∗, Decio Levic, Luigi Martinad

a Centre de Recherches Mathématiques, Université de Montréal, Montréal CP 6128 (QC) H3C 3J7, Canada
b Department of Mathematics and Computer Science, Université du Québec, Trois-Rivières, CP 500 (QC) G9A
5H7, Canada

c Dipartimento di Mathematica e Fisica dell’Università Roma Tre, Sezione INFN di Roma Tre, Via della Vasca
Navale 84, Roma, 00146 Italy

d Dipartimento di Mathematica e Fisica dell’Università del Salento, Sezione INFN di Lecce, Via Arnesano, C.P.
193 Lecce, 73100 Italy

∗ corresponding author: grundlan@crm.umontreal.ca

Abstract. This paper is devoted to a study of the connections between three different analytic
descriptions for the immersion functions of 2D-surfaces corresponding to the following three types of
symmetries: gauge symmetries of the linear spectral problem, conformal transformations in the spectral
parameter and generalized symmetries of the associated integrable system. After a brief exposition
of the theory of soliton surfaces and of the main tool used to study classical and generalized Lie
symmetries, we derive the necessary and sufficient conditions under which the immersion formulas
associated with these symmetries are linked by gauge transformations. We illustrate the theoretical
results by examples involving the sigma model.

Keywords: integrable systems; soliton surfaces; immersion formulas; generalized symmetries.

AMS Mathematics Subject Classification: 35Q35, 22E60, 53A05.

1. Introduction
Soliton surfaces associated with integrable systems
have been shown to play an essential role in many
problems with physical applications (see e.g. [2, 5–
9, 11–13, 15, 19, 29, 30, 33–35]). We say that a surface
is integrable if the Gauss-Mainardi-Codazzi equations
corresponding to it are integrable, i.e. if they can be
represented as the compatibility conditions for some
“non-fake” Linear Spectral Problem (LSP) [2, 3, 5, 17,
18, 21, 22, 24–26, 31–35]. The possibility of using an
LSP to represent a moving frame on the integrable
surface has yielded many new results concerning the
intrinsic geometric properties of such surfaces (see e.g.
[4, 29]). In the present state of development, it has
proved most fruitful to extend such characterizations
of soliton surfaces via their immersion functions (see
e.g. [1, 2, 19, 23, 27, 30] and references therein).
The construction of surfaces related to completely

integrable models was initiated by A. Sym [33–35].
This construction makes use of the conformal invari-
ance of the zero-curvature representation of the system
with respect to the spectral parameter. Another ap-
proach for finding such surfaces has been formulated
by J. Cieslinski and A. Doliwa [5–7] using gauge sym-
metries of the LSP. A third approach, using the LSP
for integrable systems and their symmetries has been
introduced by Fokas and Gel’fand [8, 9], to construct

families of soliton surfaces. Most recently, in a series
of papers [11–13, 15], a reformulation and extension of
the Fokas-Gel’fand immersion formula has been per-
formed through the formalism of generalized vector
fields and their actions on jet spaces. This extension
has provided the necessary and sufficient conditions
for the existence of soliton surfaces in terms of the
symmetries of the LSP and integrable models. The
objective of this paper is to investigate and construct
the relation between the three approaches concerning
2D-soliton surfaces associated with integrable systems.

This paper is organized as follows. Section 2 con-
tains a brief summary of the results concerning the
construction of soliton surfaces and symmetries. Us-
ing the Sym-Tafel (ST) formula, we get the surface by
differentiating the solution of the LSP with respect
to the spectral parameter. Through the Cieslinski-
Doliwa (CD) formula we apply a gauge transformation
and through the Fokas-Gel’fand (FG) formula we con-
sider the generalized symmetries of the associated
integrable equation. In Section 3 we demonstrate that
the immersion problem can be mapped through a
gauge to any of the three immersion formulas listed
above and we show that these formulas correspond to
possibly different parametrizations of the same sur-
face. Then, in Section 4, we apply the results on the
example of the sigma model. Section 5 contains the
concluding remarks.

180

http://dx.doi.org/10.14311/AP.2016.56.0180
http://ojs.cvut.cz/ojs/index.php/ap


vol. 56 no. 3/2016 On Immersion Formulas for Soliton Surfaces

2. Summary of results
on the construction
of soliton surfaces

In this section we recall the main tools used to study
symmetries suitable for the use of Fokas-Gel’fand
formulas for the construction of 2D surfaces. We
make use of the formalism of vector fields and their
prolongations as presented in [29]. More specifically,
we rewrite the formula for the immersion functions of
2D surfaces in terms of the prolongation formalism of
the vector fields instead of the Fréchet derivatives.

2.1. Classical and generalized
Lie symmetries

Let X (with coordinates xα, α = 1, . . . ,p) and U (with
coordinates uk, k = 1, . . . ,q) be differential manifolds
representing spaces of independent and dependent
variables, respectively. Let Jn = Jn(X ×U) denote
the n-jet space over X ×U. The coordinates of Jn
are given by xα, uk and ukJ =

∂nuk

∂xj1...∂xjn
where J =

(j1, . . . ,jn) is a symmetric multi-index. We denote
these coordinates by x and u(n). On Jn we can define
a system of partial differential equations (PDEs) in
p independent and q dependent variables given by m
equations of the form

Ωµ(x,u(n)) = 0, µ = 1, . . . ,m. (2.1)

We consider a vector field v tangent to J0 = X ×U

v = ξα(x,u)∂α + ϕk(x,u)∂k, (2.2)

where ∂α = ∂/∂xα, ∂k = ∂/∂uk and we adopt the
summation convention over repeated indices. Such a
field defines vector fields, pr(n) v on Jn [29]

pr(n) v = ξα∂α + ϕkJ
∂

∂ukJ
. (2.3)

The functions ϕkJ are given by

ϕkJ = DJR
k + ξαukJ,α, R

k = ϕk − ξαukα, (2.4)

where the operators DJ correspond to multiple total
derivatives, each of which is a combination of total
derivatives of the form

Dα = ∂α + ukJ,α
∂

∂ukJ
, α = 1, . . . ,p (2.5)

and Rk are the so-called characteristics of the vector
field v. In the following, the representation of v can
be written equivalently as

v = ξαDα + ωR, ωR = Rk
∂

∂uk
. (2.6)

One says that the vector field v is a classical Lie point
symmetry of a nondegenerate system of PDEs (2.1)
if and only if its n-th prolongation of v is such that

pr(n) vΩµ(x,u(n)) = 0, µ = 1, . . . ,m (2.7)

whenever Ωµ(x,u(n)) = 0, µ = 1, . . . ,m are satisfied.
By a totally non-degenerate system we mean a system
for which all their prolongations have maximal rank
and are locally solvable [29]. It follows from the well-
known properties of the symmetries of a differential
system that the commutator of two symmetries is
again a symmetry. Thus, such symmetries form a
Lie algebra g, which locally defines an action of a Lie
group G on J0.
Every solution of (2.1) can be represented by its

graph, uk = θk(x), which is a section of J0. The
symmetry group G transforms solutions into solutions.
This means that a graph corresponding to one solution
is transformed into a graph associated with another
solution. If the graph is preserved by the group G or
equivalently, if the vector fields v from the algebra g
are tangent to the graph, then the related solution
is said to be G-invariant. Invariant solutions satisfy,
in addition to the equations (2.1), the characteristic
equations equated to zero

ϕka(x,θ) − ξ
α
a (x,θ)θ

k
,α = 0, a = 1, . . . ,r, (2.8)

where the index a runs over the generators of g.
It may happen that invariant solutions are restricted

in number or trivial if the full symmetry group is small.
To extend the number of symmetries, and thus of
solutions, one looks for generalized symmetries. They
exist only if the nonlinear equation (2.1) is integrable
[28], i.e. it has been obtained as the compatibility
of a Lax pair (see (2.13) and in the following). A
generalized vector field is expressed in terms of the
characteristics

ωR = Rk[u]
∂

∂uk
, (2.9)

where [u] = (x,u(n)) ∈ Jn. The prolongation of an
evolutionary vector field ωR is given by

pr ωR = ωR + DJRk
∂

∂ukJ
. (2.10)

A vector field ωR is a generalized symmetry of a non-
degenerated system of PDEs (2.1) if and only if [29]

pr ωRΩµ(x,u(n)) = 0, (2.11)

whenever Ω(x,u(n)) = 0 and its differential conse-
quences are satisfied.

2.2. The immersion formulas
for soliton surfaces

In order to analyse the Fokas-Gel’fand immersion
formula for a surface in 2D, we briefly summarize the
results obtained in [2, 5–9, 11–13, 15, 33–35].

Let us consider an integrable system of partial differ-
ential equations (PDEs) in two independent variables
x1,x2 and m dependent variables uk(x1,x2) written
as

Ω[u] = 0. (2.12)

181



A. M. Grundland, D. Levi, L. Martina Acta Polytechnica

Suppose that the system (2.12) is obtained as the
compatibility of a matrix LSP written in the form [33]

∂αΦ(x1,x2,λ) −Uα([u],λ)Φ(x1,x2,λ) = 0,
α = 1, 2. (2.13)

In what follows, the potential matrices Uα can be
defined on the extended jet space N = (Jn,λ), where
λ is the spectral parameter. The compatibility condi-
tion of the LSP (2.13), often called the Zero-Curvature
Condition (ZCC)

D2U1 −D1U2 + [U1,U2] = 0, (2.14)

which is assumed to be valid for all values of λ, implies
(2.12). The bracket in (2.14) denotes the Lie algebra
commutator. Equation (2.14) provides a representa-
tion for the initial system (2.12) under consideration.
The m-dimensional matrix functions Uα take values
in some semisimple Lie algebra g and the wavefunc-
tion Φ takes values in the corresponding Lie group
G. We can say that, as long as the potential matri-
ces Uα([u],λ) satisfy the ZCC (2.14), there exists a
group-valued function Φ which satisfies (2.13). There
exists a subclass of Φ which can be defined formally
on the extended jet space N . For Φ belonging to this
subclass we can write formally Φ = Φ([u],λ) ∈ G,
meaning that Φ depends functionally on [u] and mero-
morphically in λ. When Φ = Φ([u],λ) the LSP (2.13)
can be written as

Λα([u],λ) ≡ DαΦ([u],λ) −Uα([u],λ)Φ([u],λ) = 0,
α = 1, 2, (2.15)

which is a convenient form for the analysis we carry
out in the following.

In reference [9], the authors looked for a simultane-
ous infinitesimal deformation of the associated LSP
(2.15) which preserves the ZCC (2.14)

Ũ1Ũ2
Φ̃


 =


U1U2

Φ


 + �


A1A2

Ψ


 + O(�2), (2.16)

where the matrices Ũ1, Ũ2, A1 and A2 take values
in the Lie algebra g, while Φ̃ and Ψ belong to the
corresponding Lie group G. The parameter λ is left
invariant under the transformation (2.16) and 0 <
� � 1. The corresponding infinitesimal generator
formally takes the evolutionary form

X̂e = A1∂U1 + A
2∂U2 + Ψ∂Φ. (2.17)

Equation (2.17) can be written as

X̂� = A1j∂Uj1 + A
2j∂

U
j
2

+ Ψj∂Φj , (2.18)

where we decompose the matrix functions A1 and A2
in the basis ej, j = 1, . . . ,s for the Lie algebra g

Uα = Ujαej ∈ g, [ei,ej] = c
k
ijek, (2.19)

where [ , ] is the Lie algebra commutator and ckij are
the structural constants of g. Since this generator
X̂e does not transform λ, these symmetries preserve
the singularity structure of the potential matrices
Uα in the spectral parameter λ. The infinitesimal
deformation of the LSP (2.13) and the ZCC (2.14)
under the infinitesimal transformation (2.16) requires
that the matrix functions Uα and Ψ satisfy, at first
order in �, the equations

DαΨ = UαΨ + AαΦ, α = 1, 2 (2.20)

and

D2A1 −D1A2 + [A1,U2] + [U1,A2] = 0, (2.21)

The equation (2.21) coincides with the compatibility
condition for (2.20). For the given matrix functions
Uα, Aα ∈ g and Φ ∈ G satisfying equations (2.14),
(2.15) and (2.21), an infinitesimal symmetry of the
matrix system of the integrable PDEs (2.12) allows us
to generate a 2D-surface immersed in the Lie algebra
g. According to [8] this result is formulated as follows.

Theorem 1. If the matrix functions Uα ∈ g, α = 1, 2
and Φ ∈ G of the LSP (2.15) satisfy the ZCC (2.14)
and Aα ∈ g are linearly independent matrix func-
tions which satisfy (2.21) and Φ ∈ G satisfies the
LSP (2.15), then there exists (up to affine transforma-
tions) a 2D-surface with a g-valued immersion function
F ([u],λ) such that the tangent vectors to this surface
are given by

DαF([u],λ) = Φ−1Aα([u],λ)Φ, α = 1, 2. (2.22)

Proof. The compatibility condition of (2.22) coincides
with (2.21). So an immersion function F ([u],λ) exists
and can be assumed to take its values in the Lie
algebra g. If we define the matrix function

Ψ = ΦF, (2.23)

then, using (2.22), the function Ψ satisfies (2.20).
Hence, since F = Φ−1Ψ, the formula Φ̃ = Φ + εΨ =
Φ(I + εF) implies that Φ̃ is in the Lie group G. The
immersion function

F =
(

Φ−1([u],λ)
dΦ̃([u],λ,�)

d�

)∣∣∣∣
�=0

(2.24)

is an element of the Lie algebra g. This shows that
we have constructed an appropriate infinitesimal de-
formation of the wavefunction Φ.

In [5, 33] it was shown that the admissible symme-
tries of the ZCC (2.14) include a conformal transfor-
mation of the spectral parameter λ, a gauge trans-
formation of the wavefunction Φ in the LSP (2.13)
and generalized symmetries of the integrable system
(2.12). All these symmetries can be used to determine
explicitly a g-valued immersion function F of a 2D-
surface. Thus, a generalization of the FG formula for
immersion can be formulated as follows [9, 11].

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vol. 56 no. 3/2016 On Immersion Formulas for Soliton Surfaces

Theorem 2. Let the set of scalar functions {uk}
satisfy a system of integrable PDEs Ω[u] = 0. Let the
G-valued function Φ([u],λ) satisfy the LSP (2.15) of g-
valued potentials Uα([u],λ). Let us define the linearly
independent g-valued matrix functions Aα([u],λ) (α =
1, 2) by the equations

Aα([u],λ) = β(λ)DλUα + (DαS + [S,Uα])
+ pr ωRUα + (pr ωR(DαΦ −UαΦ)) Φ−1. (2.25)

Here β(λ) is an arbitrary scalar function of λ, S =
S([u],λ) is an arbitrary g-valued matrix function de-
fined on the jet space N , ωR = Rk[u]∂uk is the vector
field, written in evolutionary form, of the generalized
symmetries of the integrable PDEs Ω[u] = 0 given
by the ZCC (2.14). Then there exists a 2D-surface
with immersion function F([u],λ) in the Lie algebra
g given by the formula (up to an additive g-valued
constant)

F([u],λ) = Φ−1 (β(λ)DλΦ + SΦ + pr ωRΦ) . (2.26)

The integrated form of the surface (2.26) defines a
mapping F : N → g and we will refer to it as the ST
immersion formula (when S = 0,ωR = 0) [33–35]

FST ([u],λ) = β(λ)Φ−1(DλΦ) ∈ g, (2.27)

the CD immersion formula (when β = ωR = 0) [5–7]

FCD([u],λ) = Φ−1S([u],λ)Φ ∈ g, (2.28)

or the FG immersion formula (when β = 0,S = 0)
[8, 9]

FFG([u],λ) = Φ−1(pr ωRΦ) ∈ g. (2.29)

2.3. Application of the method
The construction of soliton surfaces requires three ele-
ments for an explicit representation of the immersion
function F ∈ g:
(1.) An LSP (2.13) for the integrable PDE.
(2.) A generalized symmetry ωR of the integrable
PDE.

(3.) A solution Φ of the LSP associated with the
soliton solution of the integrable PDE.
Note that item (1.) is always required. In its pres-

ence, even without one of the remaining two objects,
we can obtain an immersion function F.

When a solution Φ of the LSP is unknown, the
geometry of the surface F can be obtained using the
non-degenerate Killing form on the Lie algebra g.
The 2D-surface with the immersion function F can
be interpreted as a pseudo-Riemannian manifold.
When the generalized symmetries ωR of the inte-

grable PDE are unknown but we know a solution Φ
of the LSP then we can define the 2D-soliton surface
using the gauge transformation and the λ-invariance
of the ZCC

F = Φ−1(β(λ)DλΦ + SΦ), (2.30)

where β(λ) is an arbitrary scalar function of λ and S
is an arbitrary g-valued matrix function defined on the
extended jet space N . Equation (2.30) is consistent
with the tangent vectors

DαF = β(λ)Φ−1(DλUα) + Φ−1(DαS + [S,Uα])Φ.
(2.31)

In all cases, the tangent vectors, given by (2.22),
and the unit normal vector to a 2D-surface expressed
in terms of matrices are

DαF = Φ−1AαΦ ∈ g, N =
Φ−1[A1,A2]Φ

( 12 tr[A1,A2]
2)1/2

∈ g,

(2.32)
where the g-valued matrices Aα are given by (2.25).
The first and second fundamental forms are given by

I = gijdxidxj, II = bijdxidxj, i = 1, 2, (2.33)

where

gij =
�

2
tr(AiAj),

bij =
�

2
tr
(
(DjAi + [Ai,Uj])N

)
, � = ±1. (2.34)

This gives the following expressions for the mean and
Gaussian curvatures

H =
1
∆

(
tr(A22) tr

(
(D1A1 + [A1,U1])N

)
− 8 tr(A1A2) tr

(
(D2A1 + [A1,U2])N

)
+ tr(A21) tr

(
(D2A2 + [A2,U2])N

))
,

K =
1
∆

(
tr
(
(D1A1 + [A1,U1])N

)
· tr
(
(D2A2 + [A2,U2])N

)
− 2 tr2

(
(D2A1 + [A1,U2])N

))
,

∆ = tr(A21) tr(A
2
2) − 4 tr(A1A2), (2.35)

which are expressible in terms of Uα and Aα only.
The study of soliton surfaces defined via the FG

formula for immersion provides a unique mechanism
for studying the relationship between these various
characteristics of integrable systems. For example,
do the infinite families of conservation laws have a
geometric characterization? What is the geometry
behind the Hamiltonian structure? Is there a geo-
metric interpretation of the family of surfaces and
frames associated with the spectral parameter? These
answers will not only serve to construct surfaces with
interesting geometric quantities but can also help to
clarify some problems in the theory of integrable sys-
tem.
Following the three terms in (2.26) for the immer-

sion of 2D-soliton surfaces in Lie algebras, we now
show that there exists a relation between the Sym-
Tafel, the Cieslinski-Doliwa and the Fokas-Gel’fand
formulas.

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A. M. Grundland, D. Levi, L. Martina Acta Polytechnica

3. Mapping between
the Sym-Tafel,
the Cieslinski-Doliwa
and the Fokas-Gel’fand
immersion formulas

3.1. λ-conformal symmetries
and gauge transformations

In this subsection we show that the ST immersion
formula can always be represented by a gauge transfor-
mation through the CD formula for immersion. The
converse statement is also true: from a specific gauge
it is always possible to determine the ST immersion
formula for soliton surfaces.

Proposition 1. A symmetry of the ZCC (2.14) of
the LSP associated with an integrable system Ω[u] = 0
is a λ-conformal symmetry if and only if there exists
a g-valued matrix function S1 = S1([u],λ) which is a
solution of the system of differential equations

DαS1 + [S1,Uα] = β(λ)DλUα, α = 1, 2. (3.1)

Proof. First we show that for any λ-conformal sym-
metry of the ZCC of the LSP associated with the
integrable system Ω[u] = 0, there exists a g-valued
matrix function S1 = S1([u],λ) which is a solution of
the system of differential equations (3.1). Indeed, the
linearly independent g-valued matrix functions

Aα([u],λ) = β(λ)DλUα([u],λ), (3.2)

associated with the λ-conformal symmetry of the ZCC
(2.14) satisfy the infinitesimal deformation of the ZCC
(2.21) and the corresponding ST immersion function
is

FST ([u],λ) = β(λ)Φ−1DλΦ ∈ g, (3.3)
with linearly independent tangent vectors

DαF
ST = β(λ)Φ−1(DλUα)Φ, α = 1, 2. (3.4)

On the other hand any g-valued matrix function
can be written as the adjoint group action on its
Lie algebra. This implies the existence of a g-valued
matrix function S1([u],λ) for which the ST immersion
formula (3.3) is the CD formula, i.e.

FCD([u],λ) = Φ−1S1([u],λ)Φ ∈ g, (3.5)

whose tangent vectors are found to be

DαF
CD = Φ−1

(
DαS1 +[S1,Uα]

)
Φ, α = 1, 2. (3.6)

By comparing the tangent vectors (3.4) and (3.6) we
obtain (3.1). It remains to show that the system
(3.1) is a solvable one. Indeed, from the compatibility
condition of (3.1) we get

β(λ)D2(DλU1) −β(λ)D1(DλU2)
−
[
β(λ)DλU2 − [S1,U2],U1

]
− [S1,D2U1]

+
[
β(λ)DλU1 − [S1,U1],U2

]
+ [S1,D1U2] = 0,

(3.7)

which has to be satisfied whenever (3.1) holds. Using
the ZCC (2.14) and the Jacobi identity it is easy to
show that (3.7) is identically satisfied. So if we can
find a gauge S1([u],λ) which satisfies (3.1), then the
ST immersion formula (3.3) can always be represented
by a gauge.
Conversely, we show that for any g-valued matrix

function S1 defined as a solution of the system of
PDEs (3.1), there exists a λ-conformal symmetry of
the ZCC of the LSP associated with the integrable
system Ω[u] = 0. Indeed, comparing the immersion
formulas (3.3) with (3.5) we find a linear matrix equa-
tion for the wavefunction Φ

β(λ)DλΦ = S1([u],λ)Φ. (3.8)

If the gauge function S1([u],λ) is known, by solv-
ing (3.8) we can determine the wavefunction Φ and
consequently obtain the ST immersion formula for
2D-soliton surfaces. Therefore, the ST formula for
immersion (2.27) is equivalent to the CD immersion
formula (2.28) for the gauge S1, which satisfies differ-
ential equation (3.1).

3.2. Generalized symmetries
and gauge transformations

In this subsection we discuss the links between gauge
transformations and generalized symmetries of the
ZCC associated with the integrable partial differential
system Ω[u] = 0. We show that the immersion formula
associated with the generalized symmetries (2.29) can
always be obtained by a gauge transformation and
the converse statement is also true.

Proposition 2. A vector field ωR is a generalized
symmetry of the ZCC (2.14) of the LSP associated
with an integrable system Ω[u] = 0 if and only
if there exists a g-valued matrix function (gauge)
S2 = S2([u],λ) which is a solution of the system
of differential equations

DαS2 + [S2,Uα] = pr ωRUα
+
(
pr ωR(DαΦ −UαΦ)

)
Φ−1. (3.9)

Proof. First we demonstrate that for every infinitesi-
mal generator ωR which is a generalized symmetry of
the ZCC (2.14), there exists a g-valued matrix func-
tion (gauge) S2 = S2([u],λ) which is a solution of
the system of differential equations (3.9). Indeed an
evolutionary vector field ωR is a generalized symmetry
of the ZCC (2.14) if and only if

pr ωR
(
D2U1 −D1U2 + [U1,U2]

)
= 0, (3.10)

whenever the ZCC (2.14) holds. Equation (3.10) is
equivalent to the infinitesimal deformation of the ZCC
(2.14) given by (2.21), with linearly independent g-
valued matrix functions

Aα([u],λ) = pr ωRUα
+
(
pr ωR(DαΦ −UαΦ)

)
Φ−1, α = 1, 2. (3.11)

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vol. 56 no. 3/2016 On Immersion Formulas for Soliton Surfaces

In the derivation of (3.11) we use the fact that the
total derivatives Dα commute with the prolongation
of a vector field ωR written in the evolutionary form
[29]

[Dα, pr ωR] = 0, α = 1, 2. (3.12)

Using the LSP (2.15), equation (3.11) can be written
in the equivalent form

Aα([u],λ) =
[
−Uα(pr ωRΦ) + pr ωR(DαΦ)

]
Φ−1.

α = 1, 2 (3.13)

Substituting (3.13) into (2.21) we obtain(
−D2U1 + D1U2 − [U1,U2]

)
(pr ωRΦ)Φ−1 = 0,

which is satisfied identically whenever the ZCC (2.14)
holds.
An integrated form of the immersion function

FFG([u],λ) of a 2D-surface associated with a general-
ized symmetry ωR of the ZCC (2.14) and the tangent
vectors (2.22) is given by the FG formula

FFG([u],λ) = Φ−1(pr ωRΦ) ∈ g. (3.14)

The fact that any g-valued matrix function can be
written under the adjoint group action implies that
there exists a g-valued gauge S2, such that (3.5) holds
for the CD immersion function and its tangent vectors
DαF

CD given by (3.6). Comparing equations (2.22)
and (3.11) with (3.6) we get (3.9).

Let us show that the system (3.9) always possesses
a solution. The compatibility condition of (3.9), when-
ever (3.9) and (2.21) hold, implies the relation

[S2,D2U1 −D1U2] +
[
[S2,U1],U2

]
−
[
[S2,U2],U1

]
= 0, (3.15)

which is identically satisfied in view of the ZCC (2.14)
and the Jacobi identity. So, if we can find a gauge
function S2([u],λ) which satisfies (3.9), then the FG
formula (3.14) can always be represented by a gauge
transformation.

The converse statement is also true. We show that
for any g-valued matrix function S2 defined as a solu-
tion of the system of PDEs (3.9), there exists a gener-
alized symmetry ωR of the ZCC of the LSP associated
with Ω[u] = 0. Indeed, let the LSP of Ω[u] = 0 admit
a gauge symmetry. If the gauge S2([u],λ) is given,
then the immersion function FCD of a 2D-surface can
be integrated explicitly [5–7]

FCD([u],λ) = Φ−1S2([u],λ)Φ ∈ g, (3.16)

whenever the tangent vectors

DαF
CD = Φ−1

(
DαS2 + [S2,Uα]

)
Φ. (3.17)

are linearly independent. It is straightforward to verify
that the characteristics of a generalized vector field

ωSR, written in evalutionary form, associated with a
gauge symmetry S2, can be expressed as

Aα = DαS2 + [S2,Uα] ∈ g. (3.18)

The matrices Aα identically satisfy the determining
equations (2.21) which are required for ωSR to be a
generalized symmetry of the ZCC Ω[u] = 0

D2A1 −D1A2 + [A1,U2] + [U1,A2]
= pr ωSR

(
D2U1 −D1U2 + [U1,U2]

)
= 0, (3.19)

whenever Ω[u] = 0 holds. Hence, the vector field ωSR
associated with a gauge symmetry S2 is given by

ωSR =
(
DαS2 + [S2,Uα]

)j ∂
∂U

j
α

, (3.20)

where we have decomposed the matrix functions Aα
and Uα in the basis {ej}n1 for the Lie algebra

Uα = Ujαej ∈ g,

DαS2 + [S2,Uα] =
(
DαS2 + [S2,Uα]

)j
ej. (3.21)

Hence, for any smooth g-valued gauge S2([u],λ) there
exists a generalized symmetry ωSR of the ZCC (2.14)
and the converse statement holds as well.
Comparing the FG formula for immersion (3.14)

with the CD immersion formula (3.5) we find the
gauge

S2 = (pr ωRΦ)Φ−1. (3.22)
Hence the FG formula for immersion (2.29) is equiva-
lent to the CD immersion formula (2.28) for the gauge
S2 satisfying (3.9).

3.3. The Sym-Tafel immersion formula
versus the Fokas-Gel’fand
immersion formula

Under the assumptions of Propositions 1 and 2, we
have the following result.

Proposition 3. Let S1 and S2 be the two g-valued
matrix functions determined in Propositions 1 and 2,
respectively in terms of a λ-conformal symmetry and
a generalized symmetry of the ZCC (2.14) of the LSP
associated with an integrable system Ω[u] = 0.

If the gauge S2 is a non-singular matrix then there
exists a matrix M = S1S−12 such that

β(λ)(DλΦ) = M(pr ωRΦ). (3.23)

The matrix M defines a mapping from the FG im-
mersion formula (3.14) to the ST immersion formula
(3.3).

Alternatively, if the gauge S1 is a non-singular ma-
trix then there exists a matrix M−1 such that

(pr ωRΦ) = M−1β(λ)(DλΦ). (3.24)

The matrix M−1 defines a mapping from the ST im-
mersion formula (3.3) to the FG immersion formula
(3.14).

185



A. M. Grundland, D. Levi, L. Martina Acta Polytechnica

FST = β(λ)Φ−1(DλΦ) ∈ g

Φ ∈ G

FFG = Φ−1(pr ωRΦ) ∈ g

S2◦S−11

S1∈g

S2∈g
S1◦S−12

Figure 1. Representation of the relations between
the wavefunction Φ ∈ G and the g-valued ST and FG
formulas for immersions of 2D-soliton surfaces.

Proof. Equation (3.23) or (3.24) is obtained by elimi-
nating the wavefunction Φ from the right-hand side
of equations (3.8) and

pr ωRΦ = S2Φ, (3.25)

respectively. So the link between the immersion func-
tions FST and FFG exists, up to a g-valued gauge
function.

It should be noted that in order to recover soliton
surfaces, we have to perform an integration with re-
spect to the curvilinear coordinates in the case of the
FG formula. Alternatively, by using the ST immer-
sion formula, we obtain the same soliton surface by
differentiating the wavefunction Φ with respect to the
spectral parameter λ. The connection between the
FG and ST approaches for determining the immersion
functions FST and FFG of 2D-surfaces is obtained
through the gauge matrix functions M or M−1 from
the equation (3.23) or (3.24), respectively (see Fig. 1).
We can also write direct equations relating the

generalized symmetries ωR with the Sym-Tafel λ-
conformal symmetry for the ZCC (2.14), eliminating
the gauge S2([u],λ) in (3.9) by using (3.8). So we get

β(λ)(DλΦ)Uα −β(λ)ΦUαΦ−1(DλΦ)
+ β(λ)(DλUα)Φ + Φ

[
−pr ωR(DαΦ)

+ Uα(pr ωRΦ)
]
Φ−1 = 0. (3.26)

However, equations (3.26) are nonlinear differential
equations for the wavefunction Φ, which in general
are not easy to solve.
To conclude, in all three cases we give explicit ex-

pressions for 2D-soliton surfaces immersed in the Lie
algebra g and demonstrate that one such surface can
be transformed to another one through a gauge.

4. The sigma model
and soliton surfaces

For the sake of generality we start by considering the
general CPN−1 model. The problem of constructing
integrable surfaces associated with the CPN−1 models
and their deformations under various types of dynam-
ics have generated a great deal of interest over the
past decades [1, 23, 37]. The most fruitful approach
to the study of general properties of this model has

been formulated through descriptions of the model
in terms of rank-one Hermitian projectors. A matrix
P (z, z̄) is said to be a rank-one Hermitian projector if

P 2 = P, P = P†, tr P = 1. (4.1)

The target space of the projector P is determined by
a complex line in CN, i.e. by a one-dimensional vector
function f(z, z̄) given by

P =
f ⊗f†

f†f
, (4.2)

where f is the mapping C ⊇ Ω 3 z = x + iy 7→
f = (f0,f1, . . . ,fN−1)CN\{0}. Equation (4.2) gives
an isomorphism between the equivalence classes of
the CPN−1 model and the set of rank-one Hermitian
projectors P . The equations of motion

Ω(P) = [∂+∂−P,P ] = 0,

∂± =
1
2

(∂1 ± i∂2), ∂1 = ∂x, ∂2 = ∂y (4.3)

and other properties of the model take a compact form
when the model is written in terms of the projector.

Now we present some examples which illustrate the
theoretical considerations presented in the previous
section. Our first example shows that the integrated
form of the surface associated with the CPN−1 model
admits conformal symmetries which depend on two
arbitrary functions of one complex variable. This
model is defined on the Riemann sphere S2 = C∪{∞}
and its action functional is finite [37]. An entire class
of solutions of (4.3) is obtained by acting on the
holomorphic (or anti-holomorphic) solution P [10]
with raising and lowering operators. These operators
are given by

Π±(P) =




(∂±P)P(∂∓P)
tr(∂±PP∂∓P)

for (∂±P)P(∂∓P) 6= 0,

0 for (∂±P)P(∂∓P) = 0,

Π−(Pk) = Pk−1, Π+(Pk) = Pk+1. (4.4)

The set of N rank-1 projectors {P0, . . . ,PN−1} acts on
orthogonal complements of one-dimensional subspaces
in CN and satisfy the orthogonality and completeness
relations

PjPk = δjkPj, (no summation) and
N−1∑
j=0

Pj = IN,

(4.5)
where IN is the N ×N identity matrix on CN. These
projectors provide a basis of commuting elements in
the space of the Hermitian matrices on CN and satisfy
the Euler-Lagrange equation (written in the form of
a conservation law)

∂[∂̄Pk,Pk] + ∂̄[∂Pk,Pk] = 0, k = 0, 1, . . . ,N − 1,
(4.6)

where ∂ = 12 (∂x − i∂y) and ∂̄ =
1
2 (∂x + i∂y). For

a given set of rank-1 projector solutions Pk of (4.6)

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vol. 56 no. 3/2016 On Immersion Formulas for Soliton Surfaces

the su(N)-valued generalized Weierstrass formula for
immersion (GWFI) [20]

Fk(z, z̄) = i
∫
γ

(−[∂Pk,Pk]dz + [∂̄Pk,Pk]dz̄),

k = 0, 1, . . . ,N − 1 (4.7)

(where γ is a curve locally independent of the trajec-
tory in C) can be explicitly integrated [10]

Fk(z, z̄) = −i
(
Pk + 2

k−1∑
j=0

Pj

)
+

1 + 2k
N

IN. (4.8)

The immersion functions Fk satisfy the algebraic con-
ditions

[Fk − ickIN ][Fk − i(ck−1)IN ][Fk − i(ck−2)IN ] = 0,
0 < k < N − 1,

[F0 − ic0IN ][F0 − i(c0 − 1)IN ] = 0,
[FN−1 + ic0IN ][FN−1 + i(c0 − 1)IN ] = 0,

N−1∑
j=0

(−1)jFj = 0, ck =
1
N

(1 + 2k). (4.9)

The LSP associated with (4.6) is given by [27, 36]

∂αΦk = UαkΦk, Uαk =
2

1 ±λ
[∂αPk,Pk],

(U1k)† = −U2k, (4.10)

(where α = 1, 2 stands for ±) with soliton solution
Φk = Φk([P ],λ) ∈ SU(N) which goes to IN as λ →∞
[36, 37]

Φk = IN +
4λ

(1 −λ)2
k−1∑
j=0

Pj −
2

1 −λ
Pk,

Φ−1k = IN −
4λ

(1 + λ)2
k−1∑
j=0

Pj −
2

1 + λ
Pk,

λ = it, t ∈ R. (4.11)

The recurrence relation (4.4) is expressed in terms
of rank-1 projectors Pk, without any reference to the
sequence of functions fk as in (4.2). For the sake
of simplicity, in this section, we drop the index k
attributed to the N projectors Pk.
It is convenient for computational purposes to ex-

press the CPN−1 model in terms of the matrix

θ ≡ i
(
P −

1
N

IN
)

= θses ∈ su(N),

[ej,el] = Csjles, j, l,s = 1, . . . ,N
2 − 1, (4.12)

where Csjl are the structural constants of g and es is
the basis element for the su(N) algebra. Due to the
indempotency of the projector P we get the following
algebraic restriction on θ:

θ ·θ = −i
2 −N
N

θ +
1 −N
N2

IN ⇐⇒ P 2 = P. (4.13)

The equations of motion in terms of the matrix θ are

Ωj[θ] =
[
(∂21 + ∂

2
2 )θ,θ

]j = 0, j = 1, . . . ,N2 − 1
(4.14)

where [·, ·]j denotes the coefficients of the commutator
with respect to the jth basis element ej for the su(N)
algebra. The potential matrices Uα in terms of θ are

U1 =
−2

1 −λ2
(
[∂1θ,θ] − iλ[∂2θ,θ]

)
,

U2 =
−2

1 −λ2
(
iλ[∂1θ,θ] + [∂2θ,θ]

)
,

λ = it, t ∈ R. (4.15)

The wavefunction Φ in terms of θ is

Φ([θ],λ) = IN +
4λ

(1 −λ)2
k−1∑
j=0

Πj−(θ)

−
2

1 −λ

( 1
N

IN − iθ
)
∈ SU (N), (4.16)

where Π± are the raising and lowering operators acting
on the elements θ of the algebra su(N)

Π−(θk) = θk−1, Π+(θk) = θk+1. (4.17)

In what follows, we use the simplified notation of
Π±(θk) by Π±(θ), where the index k is suppressed.
The operators (4.17) are written explicitly as

Π−(θ) =
∂̄θ(E − iθ)∂θ

tr(∂̄θ(E − iθ)∂θ)
,

Π+(θ) =
∂θ(E − iθ)∂̄θ

tr(∂θ(E − iθ)∂̄θ)
,

E =
1
N

IN, (4.18)

where the traces in the denominators are different
from zero unless the whole matrix is zero. For any
functions f and g of one variable, the equations of
motion (4.14) and their LSP (2.15) (with the potential
matrices (4.15)) admit the conformal symmetries

ωCi =
[
f(x)∂1θj + g(y)∂2θj

] ∂
∂θj

, i = 1, 2. (4.19)

The vector fields ωCi are related to the fields ηci
defined on the jet space M = [(Φ,Uα)]

ηCi = (∂iΦ
j)

∂

∂Φj
+ (∂iUjα)

∂

∂U
j
α

, i = 1, 2, (4.20)

which are conformal symmetries of the LSP (2.15).
The integrated form of the surface is given by the FG
formula [15]

FFG = Φ−1
(
f(x)U1 + g(y)U2

)
Φ ∈ su(N). (4.21)

4.1. Soliton surfaces associated
with the CP 1 sigma model

We give a simple example to illustrate the construc-
tion of 2D-soliton surfaces associated with the CP 1

187



A. M. Grundland, D. Levi, L. Martina Acta Polytechnica

model (N = 2) introduced in previous section. The
only solutions with finite action of the CP 1 model
are holomorphic and antiholomorphic projectors [37].
The rank-one Hermitian projectors (i.e. holomorphic
P0 and antiholomorphic P1) based on the Veronese
sequence f0 = (1,z), take the form

P0 =
f0 ⊗f

†
0

f
†
0f0

=
1

1 + |z|2

(
1 z̄
z |z|2

)
,

P1 =
f1 ⊗f

†
1

f
†
1f1

=
1

1 + |z|2

(
|z|2 −z̄
−z 1

)
, (4.22)

where f1 = (I2 − P0)∂f0. The corresponding inte-
grated forms of the surfaces are given by the GWFI
(4.8)

F0 = i(
1
2
I2 −P0)

=
i

1 + |z|2

(1
2 (|z|

2 − 1) −z̄
−z 12 (1 −|z|

2)

)
∈ su(2),

F1 = −i(P1 + 2P0) +
3i
2
I2 = F0. (4.23)

From equation (4.10) the potential matrices Uαk be-
come

U10 = U11 =
2

(λ + 1)(1 + |z|2)2

(
−z̄ −z̄2
1 z̄

)
,

U20 = U21 =
2

(λ− 1)(1 + |z|2)2

(
−z 1
−z2 z

)
,

λ = it, t ∈ R. (4.24)

The SU(2)-valued soliton wavefunctions Φk in the
LSP (4.10) for the CP 1 model have the form

Φ0 =
1

1 + |z|2

(
−i+t+(i+t)|z|2

t−i
−2iz̄
t−i

−2iz
t+i

i+t+(t−i)|z|2
t+i

)
,

Φ1 =
1

1 + |z|2

(
1+t2+(t+i)2|z|2

(t−i)2
2(1−it)z̄
(t−i)2

−2i(t−i)z
(t+i)2

1+t2+(t−i)2|z|2
(t+i)2

)
.

(4.25)

Let us now consider separately four different ana-
lytic descriptions for the immersion functions of 2D-
soliton surfaces in Lie algebras which are related to
four different types of symmetries.

I. The ZCC (2.14) of the CP 1 model admits a con-
formal symmetry in the spectral parameter λ. The
tangent vectors DαFSTk associated with this symme-
try are given by

DαF
ST
k = Φ

−1
k (DλUαk)Φk, α,k = 1, 2

and are linearly independent. The integrated forms of
the 2D-surfaces in su(2) are given by the ST formulas

(2.27)

FST0 = Φ
−1
0 (DλΦ0) =

2i
(1 + t2)2(1 + |z|2)2

·
(
−|z|2[t2 − 3 + |z|2(1 + t2)]
z[(t− i)2 + |z|2(3 − 2it + t2)]

z̄[(t + i)2 + |z|2(3 + 2it + t2)]
|z|2[t2 − 3 + |z|2(t2 + 1)]

)
,

FST1 = Φ
−1
1 (DλΦ1) =

2i
(1 + t2)2(1 + |z|2)2

·
(
−[(t2 + 1)(1 + 2|z|4) + 3|z|2(t2 − 3)]
z[t2 − 6it− 5 + |z|2(7 + t2 − 6it)]
z̄[6it− 5 + t2 + |z|2(7 + 6it + t2)]
(t2 + 1)(1 + 2|z|4) + 3|z|2(t2 − 3)

)
, (4.26)

where, without loss of generality, we can put β(λ) = 1
in the expression (2.27). The surfaces FSTk satisfy

(FSTk )
2 +

1
4
I2 = 0,

for k = 0, 1 and have positive constant Gaussian and
mean curvatures. Hence, they are spheres (see Fig. 2a)

K0 = K1 = 4, H0 = H1 = 4. (4.27)

The surfaces FSTk in Cartesian coordinates (x,y) take
the form for z = x + iy

FSTk =
{ x

1 + x2 + y2
,

y

1 + x2 + y2
,

1 −x2 −y2

2(1 + x2 + y2)

}
(4.28)

The su(2)-valued gauges SSTk associated with the ST
immersion functions FSTk (4.26) take the form

SST0 = (DλΦ0)Φ
−1
0 =

2i
1 + |z|2

(
−|z|2
t2+1

z̄
(t−i)2

z
(t+i)2

|z|2
t2+1

)
,

SST1 = (DλΦ1)Φ
−1
1 =

2i
1 + |z|2

(
−(1+2|z|2)

t2+1
z̄(t+i)2
(t−i)4

z(t−i)2
(t+i)4

1+2|z|2
t2+1

)
,

det SSTk 6= 0, tr S
ST
k = 0. (4.29)

II. The surfaces Fgk ∈ su(2) associated with the
scaling symmetries of the ZCC (2.14) associated with
equations of the CP 1 model (4.6)

ω
g
k =

(
D1(zU1k) + z̄(D2U1k)

) ∂
∂θ1

+
(
z(D1U2k) + D2(z̄U2k)

) ∂
∂θ2

, (4.30)

have the integrated form [14]

F
g
k = Φ

−1
k (zU1k + z̄U2k)Φk, k = 0, 1 (4.31)

where θ1 and θ2 are complex-valued functions deter-
mined from the su(2) Lie algebra (4.12). The surfaces
F
g
k also have constant positive curvatures

K0 = K1 = −4λ2, H0 = H1 = −4iλ, iλ ∈ R.
(4.32)

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vol. 56 no. 3/2016 On Immersion Formulas for Soliton Surfaces

The surfaces Fgk are not spheres (as in the previous
cases (4.27)) since they have boundaries (see Fig. 2b).
The surfaces Fgk can be given in the parametric form

F
g
k =

(x3 − 2x2y + x(y2 − 1) − 2y(1 + y2)
(1 + x2 + y2)2

,

−
2x3 + x2y + y(y2 − 1) + 2x(1 + y2)

(1 + x2 + y2)2
,

2(x2 + y2)
(1 + x2 + y2)2

)
. (4.33)

The su(2)-valued gauges Sgk = (pr ω
gΦk)Φ−1k associ-

ated with the scaling symmetries ωgk are given by

S
g
0 = S

g
1 =

2
(t2 + 1)(1 + |z|2)2

·
(

2it|z|2 iz̄[i− t + |z|2(t + i)]
z[1 − it + |z|2(1 + it)] −2it|z|2

)
,

(4.34)

where det Sgk 6= 0.

III. In the case of surfaces associated with the con-
formal symmetries

ωck = −gk(z)∂ − ḡk(z̄)∂̄, (4.35)

where for simplicity we have assumed that gk(z) =
1 + i, the su(2)-valued immersion functions Fck are
given by [12]

Fck = Φ
−1
k (U1k + U2k)Φk, (4.36)

where

U10 + U20 = U11 + U21 =
2

(t2 + 1)(1 + |z|2)2

·
(

2z + i(t + i)(z + z̄) −1 − it + iz̄2(t + i)
1 + z2 + it(z2 − 1) −[2z + i(t + i)(z + z̄)]

)
.

(4.37)

By further computation it can be verified that the
Gaussian curvature and the mean curvature corre-
sponding to the surfaces Fck are not constant for any
value of λ. The fact that the surfaces Fck have the
Euler-Poincaré characters [10]

χk =
−1
π

∫∫
S2
∂∂̄ ln [tr(∂Pk · ∂̄Pk)]dx1dx2 (4.38)

equal to 2 and positive Gaussian curvature K > 0
means that the surfaces Fck are homeomorphic to
ovaloids (see Fig. 2c). The surfaces Fck associated with
the conformal symmetries ωck are cardioid surfaces
which can be parametrized as follows

Fck =
(x2 − 1 − 4xy −y2

(1 + x2 + y2)2
,−

2(1 + x2 + xy −y2)
(1 + x2 + y2)2

,

2(2x−y)
(1 + x2 + y2)2

)
(4.39)

The su(2)-valued gauges Sck associated with the
conformal symmetries ωcktake the form

Sc0 = (pr ω
cΦ0)Φ−10 =

2
(1 + |z|2)2

·

(
−i(1−i)(t−i)z+(1+i)(1−it)z̄

t2+1
i(1+i)(t+i)−(1−i)z2(t−i)

(t+i)2

(1−i)(1+it)+(1+i)(1−it)z̄2
(t−i)2

(1−i)(1+it)z+i(1+i)(t+i)z̄
t2+1

)
,

Sc1 = (pr ω
cΦ1)Φ−11 =

2
(1 + |z|2)2

·

(
−i(1−i)(t−i)z+(1+i)(1−it)z̄

t2+1
i(t−i)2[(1+i)(t+i)−(1−i)(t−i)z2]

(t+i)4

(t+i)2[(1−i)(1+it)+(1+i)(1−it)z̄2]
(t−i)4

(1−i)(1+it)z+i(1+i)(t+i)z̄
t2+1

)
, (4.40)

where det Sck 6= 0.

IV. If the generalized symmetries of the CP 1 model
(4.6) are written in the evolutionary form

ωRk =
(
D21U1k + D

2
2U1k + [D1U1k,U1k]

+[D2U1k,U1k]
) ∂
∂θ1

+
(
D21U2k +D

2
2U2k +[D2U2k,U2k]

+ [D1U2k,U2k]
) ∂
∂θ2

, k = 1, 2 (4.41)

(where θ1 and θ2 are complex-valued functions ob-
tained from (4.12)) then the su(2)-valued integrated
form of the immersion becomes [14]

FFGk = Φ
−1(pr ωRk Φk) = Φ

−1
k (D1U1k + D2U2k)Φk.

(4.42)
The tangent vectors to this surface are given by

D1F
FG
k = Φ

−1
k (pr ω

R
k U1k)Φk = Φ

−1
k (D

2
1U1k + D

2
2U1k

+[D1U1k,U1k] + [D2U1k,U1k])Φk,

D2F
FG
k = Φ

−1
k (pr ω

R
k U2k)Φk = Φ

−1
k (D

2
1U2k + D

2
2U2k

+[D2U2k,U2k] + [D1U2k,U2k])Φk. (4.43)

The surfaces FFGk also have positive Gaussian curva-
tures K > 0 and the Euler-Poincaré characters are
equal to 2. In the parametrization x,y, the surfaces
FFGk take the form

FFGk =
(
−
x3 − 6x2y −x(1 + 3y2) + 2y(1 + y2)

(1 + x2 + y2)3
,

2x3 + y + 3x2y −y3 + x(2 − 6y2)
(1 + x2 + y2)3

,

−
2(x2 − 4xy −y2)

(1 + x2 + y2)3
)

(4.44)

and they are homeomorphic to ovaloids.
The su(2)-valued gauges SFGk associated with the

generalized symmetry ωRk take the form

SFGk = (pr ω
R
k Φk)Φ

−1
k = D1U1k + D2U2k. (4.45)

189



A. M. Grundland, D. Levi, L. Martina Acta Polytechnica

Under the assumption that Pk are holomorphic or
antiholomorphic projectors (4.22) the gauges SFGk
take the form

SFG0 = S
FG
1 =

4
(t2 + 1)(1 + |z|2)3

·
(
−z2(1 + it) + z̄2(1 − it) z̄3(1 − it) + z(it + 1)
−iz3(t− i) + iz̄(t + i) z2(1 + it) − z̄2(1 − it)

)
,

(4.46)

where det SFGk 6= 0. Hence the mappings Mk =
SSTk (S

FG
k )

−1 from the FG immersion formulas to the
ST immersion formulas are given by

M0 = SST0 (S
FG
0 )

−1 =
1

2(t2 + 1)

·

(
−2iz3(t−i)+z̄[(t+i)2+|z|2(t2+1)]

z(t−i)

− [z
3(1+t2)+2z̄(1−it)+z(t−i)2]

t+i
z[(t+i)2+(t2+1)|z|2+2(1+it)]

t−i
z(t−i)2+|z|2z(t2+1)+2iz̄3(t+i)

z̄(t+i)

)
,

M1 = SST1 (S
FG
1 )

−1 =
i

2|z|2

(
−(1+2|z|2)

t2+1
(i+t)2z̄
(t−i)4

z(t−i)2
(t+i)4

1+2|z|2
t2+1

)

·
(
z2(it + 1) + z̄2(it− 1) −z(it + 1) + z̄3(it− 1)
z3(it + 1) + (1 − it)z̄ −z2(1 + it) + (1 − it)z̄2

)
,

(4.47)

where det Mk 6= 0. Conversely, the gauges M−1k =
SFGk (S

ST
k )

−1 do exist. Hence there exist mappings
from the ST immersion formulas to the FG immersion
formulas

M−10 = S
FG
0 (S

ST
0 )

−1 =
2

(1 + |z|2)3

(
m11 m12
m21 m22

)
,

(4.48)
where

m11 =
(t− i)2z + (1 + t2)|z|2z + 2(it− 1)z̄3

(i + t)z̄
,

m12 =
2(1 + it)z2 + (i + t)2z̄2 + (1 + t2)|z|2z̄2

(i− t)z
,

m21 =
(t− i)2z2 + (1 + t2)|z|2z2 + 2(1 − it)z̄2

(i + t)z̄
,

m22 =
−2(1 + it)z3 + (i + t)2z̄ + (1 + t2)|z|2z̄

(t− i)z
,

and

M−11 =S
FG
1 (S

ST
1 )

−1 =
2

(t2 + 1)(1 + |z|2)3(1 + 4|z|2)

·
(
−z2(1 + it) + z̄2(1 − it) z̄3(1 − it) + z(1 + it)
−z3(1 + it) + z̄(it− 1) z2(1 + it) − z̄2(1 − it)

)
·

(
(1 + 2|z|2)(1 + it)(i + t) −iz̄(i+t)

4

(t−i)2
−iz(t−i)4

(i+t)2 (1 + 2|z|
2)(1 − it)(−i + t)

)
.

(4.49)

(a) (b)

(c) (d)

Figure 2. Surfaces F ST0 in (a), F
g
0 in (b), F

c
0 in

(c) and F F G0 in (d) for k = 0, 1 and λ = i/2, and
ξ± = x ± iy with x,y ∈ [−5, 5]. The axes indi-
cate the components of the immersion function in

the basis for su(2), e1 =
(

0 i
i 0

)
, e2 =

(
0 −1
1 0

)
,

e3 =
(
i 0
0 −i

)
.

5. Concluding remarks
In this paper we have shown how three different
analytic descriptions for the immersion function of
2D-soliton surfaces can be related through different
g-valued gauge transformations. The existence of
such gauges is demonstrated by reducing the problem
to that of mappings between different forms of the
immersion formulas for three types of symmetries :
conformal transformations in the spectral parameter,
gauge symmetries of the LSP and generalized symme-
tries of the integrable systems. We have investigated
the geometric consequences of these mappings and
rephrased them as requirements for the existence of
the corresponding vector fields and their prolonga-
tions acting on a solution Φ of the associated LSP for
an integrable PDE. The explicit expressions for these
relations, which we have established, have provided
us with a tool for distinguishing between the cases
in which soliton surfaces can be or cannot be related
among them, see Proposition 3.

The task of finding an increasing number of soliton
surfaces associated with integrable systems is related
to the symmetry properties of these systems. The
construction of soliton surfaces started with the con-
tribution of Sym [33, 34] and Tafel [35] providing
a formula for the immersion of integrable surfaces
which are extensively used in the literature (see e.g.

190



vol. 56 no. 3/2016 On Immersion Formulas for Soliton Surfaces

[2, 5–16, 30, 33–35]). In this paper we have addressed
the question and formulated easily verifiable condi-
tions which ensure that the ST formula produces a
desired result. This advance can assist future studies
of 2D-soliton surfaces with integrable models, which
can describe more diverse types of surfaces than the
ones discussed in three-dimensional Euclidean space
for the CP 1 sigma model. It may be worthwhile to
extend the investigation of soliton surfaces to the case
of the sigma models defined on other homogeneous
spaces via Grassmann models and possibly to models
associated with octonion geometry. This case could
lead to different classes and types of surfaces than
those studied in this paper. This task will be explored
in a future work.

Acknowledgements
AMG has been partially supported by a research grant
from NSERC of Canada and would also like to thank the
Dipartimento di Mathematica e Fisica of the Universitá
Roma Tre and the Dipartimento di Mathematica e Fisica
of Universitá del Salento for its warm hospitality.

DL has been partly supported by the Italian Ministry
of Education and Research, 2010 PRIN Continuous and
discrete nonlinear integrable evolutions: from water waves
to symplectic maps.

LM has been partly supported by the Italian Ministry
of Education and Research, 2011 PRIN Teorie geometriche
e analitiche dei sistemi Hamiltoniani in dimensioni finite
e infinite.

DL and LM are also supported by INFN IS-CSN4 Math-
ematical Methods of Nonlinear Physics.

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	Acta Polytechnica 56(3):180–192, 2016
	1 Introduction
	2 Summary of results on the construction of soliton surfaces
	2.1 Classical and generalized Lie symmetries
	2.2 The immersion formulas for soliton surfaces
	2.3 Application of the method

	3 Mapping between the Sym-Tafel, the Cieslinski-Doliwa and the Fokas-Gel'fand immersion formulas
	3.1 lambda-conformal symmetries and gauge transformations
	3.2 Generalized symmetries and gauge transformations
	3.3 The Sym-Tafel immersion formula versus the Fokas-Gel'fand immersion formula

	4 The sigma model and soliton surfaces
	4.1 Soliton surfaces associated with the CP1 sigma model

	5 Concluding remarks
	Acknowledgements
	References