# Deformation Conjecture: Deforming Lower Dimensional Integrable Systems to
Higher Dimensional Ones by Using Conservation Laws

S. Y. Lou1, Xia-zhi Hao2 and Man Jia1 1School of Physical Science and
Technology, Ningbo University, Ningbo, 315211, China
2Faculty of Science, Zhejiang University of Technology, Hangzhou, 310014,
China

###### Abstract

Utilizing some conservation laws of $(1+1)$-dimensional integrable local
evolution systems, it is conjectured that higher dimensional integrable
equations may be regularly constructed by a deformation algorithm. The
algorithm can be applied to Lax pairs and higher order flows. In other words,
if the original lower dimensional model is Lax integrable (possesses Lax
pairs) and symmetry integrable (possesses infinitely many higher order
symmetries and/or infinitely many conservation laws), then the deformed higher
order systems are also Lax integrable and symmetry integrable. For
concreteness, the deformation algorithm is applied to the usual
$(1+1)$-dimensional Korteweg-de Vries (KdV) equation and the
$(1+1)$-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) system (including
nonlinear Schrödinger (NLS) equation as a special example). It is interesting
that the deformed (3+1)-dimensional KdV equation is also an extension of the
$(1+1)$-dimensional Harry-Dym (HD) type equations which are reciprocal links
of the (1+1)-dimensional KdV equation. The Lax pairs of the
$(3+1)$-dimensional KdV-HD system and the $(2+1)$-dimensional AKNS system are
explicitly given. The higher order symmetries, i.e., the whole
$(3+1)$-dimensional KdV-HD hierarchy, are also explicitly obtained via the
deformation algorithm. The single soliton solution of the $(3+1)$-dimensional
KdV-HD equation is implicitly given. Because of the effects of the
deformation, the symmetric soliton shape of the usual KdV equation is no
longer conserved and deformed to be asymmetric and/or multi-valued. The
deformation conjecture holds for all the known $(1+1)$-dimensional integrable
local evolution systems that have been checked, and we have not yet found any
counter-example so far. The introduction of a large number of
($D+1$)-dimensional integrable systems of this paper explores a serious
challenge to all mathematicians and theoretical physicists because the
traditional methods are no longer directly valid to solve these integrable
equations.

Key words: Deformation conjecture, Lax integrability, higher order symmetries,
high dimensional integrable systems, $(3+1)$-dimensional KdV-HD hierarchy,
$(2+1)$-dimensional AKNS system

###### pacs:

05.45.Yv,02.30.Ik,47.20.Ky,52.35.Mw,52.35.Sb

Introduction. Integrable systems and/or soliton equations are widely applied
in almost all of the modern physical fields including the condensed matter
physics CMP ; Sci , particle and nuclear physics PNP , field theory FT ,
universe Univ , fluid mechanics FM , geophysics GeoP , planetary and space
science PSS , optics Opt , plasma physics Plsm , and in other scientific and
technological fields such as biology Bio , communications Commun , chemistry
Chem , etc. Usually, the typical soliton equations are in (1+1)-dimensional
cases. To extend (1+1)-dimensional soliton equations to higher dimensions is
one of the important and significant topics both in mathematics and physics
Fofas ; hevenly ; dual ; Ptest . In this manuscript, we try to find higher
dimensional integrable systems from lower dimensional ones by proposing a
deformation algorithm.

Contracting a complex theory to a simpler one is not a difficult work. For
instance, the limiting procedures $\hbar\rightarrow 0$ and $c\rightarrow 0$
(where $\hbar$ is the Planck constant and $c$ is the speed of light) will
render a relativistic quantum theory degenerating into a classical theory.
However, the inverse procedure, i.e., the deformation of a simple theory to a
more complex theory is very difficult. Fortunately, there are some unusual
methods to deal with the inverse problem. In some special circumstances, it is
possible to accomplish the inverse procedure. For example, the classical Yang-
Baxter equation may evolve to the quantum Yang-Baxter equation. Some critical
phenomenon theory treated by conformal field theory, where the mass of the
system is zero, may be deformed to a theory for the nonzero mass case. Would
such a problem occur also in nonlinear equations? That is to say could a
solution of a complex equation be obtained from a simple equation? In deed,
some special solutions of the well known sine-Gordon (SG) equation in $D+1$
dimensions can be deformed to those of the doubled SG (DSG) equation DsG ;
DsG1 . Solutions of the $\phi^{4}$ model may also be deformed to those of the
$\phi^{6}+\phi^{4}$ model and the $\phi^{3}+\phi^{4}$ model (or named T. D.
Lee model) phi4 ; phi41 .

To reduce a high dimensional integrable system to lower dimensional ones, one
can use some different approaches like the classical and nonclassical Lie
approaches Symm ; Symm1 ; Symm2 , symmetry constraints SymC ; SymD  and
decompositions Dec ; Cao . From the studies of the (1+1)- and
(2+1)-dimensional integrable models, one knows that there are several ordinary
differential equation (ODE) reductions such as the Riccati equation and the
Painlevé I-VI equations Pain  for all known integrable lower dimensional
partial differential equations (PDEs). This fact shows that all of the known
integrable models may be considered as different deformations of some ODEs.
For instance, some (1+1)- and (2+1)-dimensional integrable sine-Gordon and
Tzitzeca equations can be considered as the deformations of a simple Riccati
equation via Miura type deformation relations Riccati . Here, we propose a
universal algorithm to deform lower dimensional integrable systems to higher
dimensional ones via a deformation conjecture. For the examples given below, a
central role is played by the following conjecture.

Deformation conjecture. For a general (1+1)-dimensional integrable local
evolution system

$u_{t}=F(u,\ u_{x},\ \ldots,\ u_{xn}),\ u_{xn}=\partial_{x}^{n}u,u=(u_{1},\
u_{2},\ \ldots,\ u_{m}),$ (1)

if there exist some conservation laws

$\rho_{it}=J_{ix},\ i=1,\ 2,\ \ldots,D-1,\ \rho_{i}=\rho_{i}(u),\
J_{i}=J_{i}(u,\ u_{x},\ \ldots,\ u_{xN}),$ (2)

where the conserved densities $\rho_{i}$ are dependent only on the field $u$
while the flows $J_{i}$ can be field derivative dependent, then the deformed
$(D+1)$-dimensional system

$\hat{T}u=F(u,\ \hat{L}u,\ \ldots,\ \hat{L}^{n}u)\ $ (3)

may be integrable with the deformation operators

$\hat{L}\equiv\partial_{x}+\sum_{i=1}^{D-1}\rho_{i}\partial_{x_{i}},\
\hat{T}\equiv\partial_{t}+\sum_{i=1}^{D-1}\bar{J}_{i}\partial_{x_{i}}$ (4)

and the deformed flows

$\bar{J_{i}}=\left.J_{i}\right|_{u_{xj}\rightarrow\hat{L}^{j}u,\ j=1,\ 2,\
\ldots,\ N}.$ (5)

In accordance with the deformation conjecture, any $(1+1)$-dimensional
integrable system possessing conservation laws will result in a higher
dimensional integrable one. Though the conjecture has not yet been strictly
proven, we have already checked the correctness, effectiveness and flexibility
of the conjecture for almost all the known $(1+1)$-dimensional local
integrable evolution systems such as the Korteweg-de Vries (KdV) equation KdV
, the modified KdV equation mKdV , the Harry-Dym equation HD , the Burgers
equation BE , the Sharma-Tasso-Olver equation STO , the Ablowitz-Kaup-Newell-
Segur (AKNS) system (including the nonlinear Schrödinger equation (NLS)) AKNS
; AKNS1 , the fifth order KdV equation 5KdV , the Sawada-Kotera equation SK ,
the Kaup-Kupershmidt equation KK , the Boussinesq equation Bq , the dispersive
long wave system DLWE , the Ito system Ito , the Hirota-Satsuma system HS ,
the Kaup-Broer system KB , the Kaup-Newell system (including the first type of
derivative NLS equation) KN , the Chen-Lee-Liu system (including the second
type of derivative NLS equation) CLL ; CLL1 , the Gerdjikov-Ivanov equation
(including the third type of derivative NLS equation) GI ; GI1 , the Landau-
Lifshitz equation (including the Heisenberg equation) LL ; LL1 , the Levi
system Levi , the Manakov system Mana , and so on.

We should mention that discrete integrable systems, such as Toda lattice
systems, Calogero-Moser-Sutherland systems, etc., have not been considered
here. The deformation conjecture for these discrete integrable systems should
be excluded and need further consideration.

Higher dimensional KdV-HD equations. We first examine the algorithm for
constructing higher dimensional integrable systems on a classical example KdV
, the KdV equation

$u_{t}=u_{xxx}+6uu_{x}.$ (6)

The KdV equation (6) was first introduced by Boussinesq Bq  and rediscovered
by Diederik Korteweg and Gustav de Vries KdV . It has various connections to
physical problems, which approximately describes the evolution of long, one-
dimensional waves in many physical settings, including shallow-water waves
with weak nonlinear restoring forces, long internal waves in a density-
stratified ocean, ion acoustic waves in a plasma, acoustic waves on a crystal
lattice KdVappl  and the two-dimensional quantum gravity gravity . The KdV
equation can be solved using the inverse scattering transform and other
methods such as those applied to other integrable systems IST .

There are infinite number of conservation laws satisfied by solutions to the
KdV equation (6). Only two will be of concern to us

$\displaystyle\rho_{1t}=J_{1x},\ \rho_{1}=u,\ J_{1}=u_{xx}+3u^{2},$ (7)
$\displaystyle\rho_{2t}=J_{2x},\ \rho_{2}=u^{2},\
J_{2}=2uu_{xx}-u_{x}^{2}+4u^{3}.$ (8)

A simple comparison with deformation conjecture leads to the discovery of a
(3+1)-dimensional KdV-HD equation

$\displaystyle\hat{T}u=\hat{L}(\hat{L}^{2}u+3u^{2}),$

or equivalently,

$\displaystyle
u_{t}=\hat{L}(\hat{L}^{2}u+3u^{2})-u_{y}\bar{J}_{1}-u_{z}\bar{J}_{2}\equiv\bar{K}_{3},$
(9)

where the deformed operators $\hat{L}$ and $\hat{T}$ possess the forms

$\displaystyle\hat{L}=\partial_{x}+u\partial_{y}+u^{2}\partial_{z},\
\hat{T}=\partial_{t}+\bar{J}_{1}\partial_{y}+\bar{J}_{2}\partial_{z}$ (10)

with the deformed flows $\bar{J}_{1}=\hat{L}^{2}u+3u^{2}$ and
$\bar{J}_{2}=2u\hat{L}^{2}u-(\hat{L}u)^{2}+4u^{3}$.

To emphasize the non-triviality of the (3+1)-dimensional KdV-HD equation (9),
we impose some reductions to this equation and derive from it some of known
and novel integrable equations in (1+1)- and (2+1)-dimensions. Equation (9)
reduces to the usual (1+1)-dimensional KdV equation (6) provided that $u$ is
independent of $y$ and $z$.

If $u$ is $\\{x,\ z\\}$-independent, then (9) becomes

$\displaystyle u_{t}=\left(u^{3}u_{yy}+u^{3}\right)_{y},$ (11)

which is equivalent to another well known (1+1)-dimensional physical model,
the Harry-Dym (HD) equation HD1 . In other words, (11) is a reciprocal link of
the KdV equation (6) related to the conservation law (7). It is not difficult
to find that the Harry-Dym equation possesses two conservation laws
$\rho^{\pm}_{t}=J^{\pm}_{y}$ with $\rho^{\pm}=u^{\pm 1},\
J^{+}=u^{3}(u_{yy}+1)$ and $J^{-}=-(uu_{y})_{y}-3u$. Applying the deformation
conjecture to (11) with these two conservation laws, one can re-obtain the
(3+1)-dimensional KdV-HD equation (9).

If $u$ is $\\{x,\ y\\}$-independent, then (9) becomes a new reciprocal link of
the KdV equation (6)

$\displaystyle
u_{t}=\left(u^{6}u_{zz}+\frac{1}{2}u^{4}\right)_{z}+3u^{4}u_{z}^{3}$ (12)

related to the conservation law (8). The model (12) possesses the conservation
laws $(u^{-1})_{t}=(-u^{4}u_{zz}-u^{3}u_{z}^{2}-u^{2})_{z}$ and
$(u^{-2})_{t}=(-2u^{3}u_{zz}-3u^{2}u_{z}^{2}-4u)_{z}$. The application of the
deformation conjecture and these two conservation laws to the model (12) will
again lead to the (3+1)-dimensional KdV-HD equation (9).

When $u$ is $z$-independent, (9) becomes a (2+1)-dimensional KdV-HD equation

$\displaystyle
u_{t}=\left(u_{xx}+3u^{2}+3uu_{xy}+\frac{3}{2}u^{2}u_{yy}\right)_{x}+\left(u^{3}u_{yy}+u^{3}+\frac{3}{2}u^{2}u_{xy}\right)_{y},$
(13)

which is an integrable (2+1)-dimensional combination of the KdV equation (6)
and its reciprocal link (11).

When $u$ is $y$-independent, (9) becomes another (2+1)-dimensional KdV-HD
equation

$\displaystyle
u_{t}=\left(u_{xx}+3u^{2}+3u^{2}u_{xz}-6u^{3}u_{z}^{2}\right)_{x}+\left(u^{6}u_{zz}+\frac{1}{2}u^{4}+3u^{4}u_{xz}+6u^{3}u_{x}u_{z}\right)_{z}+3u_{z}(u_{x}+u^{2}u_{z})^{2},$
(14)

which is an integrable (2+1)-dimensional combination of the KdV equation (6)
and its reciprocal link (12).

When $u$ is $x$-independent, (9) becomes an integrable (2+1)-dimensional
combination of two different reciprocal links (11) and (12)

$\displaystyle
u_{t}=\left[u^{3}(1+u_{yy}+3u^{2}u_{zz})+9u^{3}u_{z}v_{-}\right]_{y}+\left[\frac{1}{2}u^{4}(2u^{2}u_{zz}+6u_{yy}+1)-9u^{3}u_{y}v_{-}\right]_{z}+3u^{2}u_{z}v_{+}^{2},$
(15)

where $v_{\pm}=uu_{z}\pm u_{y}$.

When one higher dimensional equation is obtained from a lower dimensional one
via deformation conjecture, the corresponding integrable properties may also
be transferred from the original equation by the similar deformation
algorithm. To prove the Lax integrability of the (3+1)-dimensional KdV
equation (9), one can apply the similar deformation algorithm to the Lax pair
of the original (1+1)-dimensional KdV equation (6)

$\displaystyle(\partial_{x}^{2}+u-\lambda)\psi\equiv{M}\psi=0,$
$\displaystyle(\partial_{t}-4\partial_{x}^{3}-6u\partial_{x}-3u_{x})\psi\equiv{N}\psi=0.$
(16)

The compatibility condition $[{M},\ {N}]={M}{N}-{N}{M}=0$ is equivalent to the
KdV equation (6).

It is interesting that applying the deformation relations

$\partial_{x}\rightarrow\hat{L},\ \partial_{t}\rightarrow\hat{T}$ (17)

on (16), the Lax pair of the (3+1)-dimensional KdV equation (9) can then be
easily put in the form

$\displaystyle(\hat{L}^{2}+u-\lambda)\psi\equiv\hat{M}\psi=0,$
$\displaystyle\big{[}\hat{T}-4\hat{L}^{3}-6u\hat{L}-3(\hat{L}u)\big{]}\psi\equiv\hat{N}\psi=0.$
(18)

Now, it is straightforward to verify that the linear problem (18) is just the
Lax pair of the (3+1)-dimensional KdV equation (9). In other words, the
compatibility condition of (18)

$\displaystyle[\hat{M},\ \hat{N}]=\hat{M}\hat{N}-\hat{N}\hat{M}=0$ (19)

gives the (3+1)-dimensional KdV equation (9).

It is also known that the (1+1)-dimensional KdV equation is symmetry
integrable that means (6) possesses infinitely many higher order symmetries
$K_{2n+1}=\Phi^{n}u_{x}$. In other words, there are infinitely many higher
order commuting flows

$u_{t_{2n+1}}=K_{2n+1}=\Phi^{n}u_{x},\
\Phi=\partial_{x}^{2}+4u+2u_{x}\partial_{x}^{-1},\ n=0,\ 1,\ \ldots,\ \infty,$
(20)

which is called as the KdV hierarchy. For $n=1$, (20) is just the KdV equation
($t_{3}=t$) while the fifth order KdV equation

$u_{t_{5}}=(u_{x4}+10uu_{xx}+5u_{x}^{2}+10u^{3})_{x}$ (21)

is related to (20) with $n=2$. It is known that $u$ and $u^{2}$ are also
conserved densities for the whole KdV hierarchy (20) with the differential
polynomial flows in the forms

$\displaystyle u_{t_{2n+1}}$ $\displaystyle=$ $\displaystyle(J_{2n+1})_{x},\
J_{2n+1}=\partial_{x}^{-1}\Phi^{n}u_{x},\ $ $\displaystyle(u^{2})_{t_{2n+1}}$
$\displaystyle=$ $\displaystyle(G_{2n+1})_{x},\
G_{2n+1}=2\partial_{x}^{-1}u\Phi^{n}u_{x},\ $ (22)

where $J_{2n+1}$ and $G_{2n+1}$ are differential polynomials of $u$ with
respect to $x$. For $n=2$,

$\displaystyle J_{5}$ $\displaystyle=$ $\displaystyle
u_{x4}+10uu_{xx}+5u_{x}^{2}+10u^{3},$ $\displaystyle G_{5}$ $\displaystyle=$
$\displaystyle 2uu_{x4}-2u_{x}u_{x3}+u_{xx}^{2}+20u^{2}u_{xx}+15u^{4}.$ (23)

Owing to the conservation laws (22) and the deformation assumption, we can
obtain the whole (3+1)-dimensional KdV hierarchy

$\displaystyle\hat{T}_{2n+1}u=\left.\left(\Phi^{n}u_{x}\right)\right|_{u_{xm}\rightarrow\hat{L}^{m}u,\
m=1,\ 2,\ \ldots,\ 2n+1},\ $ (24)

where
$\hat{T}_{2n+1}=\partial_{t_{2n+1}}+\bar{J}_{2n+1}\partial_{y}+\bar{G}_{2n+1}\partial_{z},\hat{L}=\partial_{x}+u\partial_{y}+u^{2}\partial_{z},\
\bar{J}_{2n+1}=\left.J_{2n+1}\right|_{u_{xm}\rightarrow\hat{L}^{m}u,\ m=1,\
2,\ \ldots,\ 2n+1}$ and
$\bar{G}_{2n+1}=\left.G_{2n+1}\right|_{u_{xm}\rightarrow\hat{L}^{m}u,\ m=1,\
2,\ \ldots,\ 2n+1}$. The (3+1)-dimensional KdV hierarchy (24) is equivalent to

$u_{t_{2n+1}}=\left.\left(1-u_{y}\partial_{x}^{-1}-2u_{z}\partial_{x}^{-1}u\right)\Phi^{n}u_{x}\right|_{u_{xm}\rightarrow\hat{L}^{m}u,\
m=1,\ 2,\ \ldots,\ 2n+1}\equiv\bar{K}_{2n+1}$ (25)

with the first three $\bar{K}_{2n+1}$ being $\bar{K}_{1}=u_{x}$,
$\bar{K}_{3}=(\hat{L}-u_{y})(\hat{L}^{2}u+3u^{2})-u_{z}\left[2u\hat{L}^{2}u-(\hat{L}u)^{2}+4u^{3}\right]$
and

$\bar{K}_{5}=(\hat{L}-u_{y})\left[\hat{L}^{4}u+10u\hat{L}^{2}u+5(\hat{L}u)^{2}+10u^{3}\right]-u_{z}\left[2u\hat{L}^{4}u-2(\hat{L}u)(\hat{L}^{3}u)+(\hat{L}^{2}u)^{2}+20u^{2}\hat{L}^{2}u+15u^{4}\right].$

One can check that the compatibility conditions,
$u_{t_{2n+1}t_{2m+1}}=u_{t_{2m+1}t_{2n+1}}$, of the (3+1)-dimensional KdV
hierarchy (25) are satisfied, i.e.,

$[\bar{K}_{2n+1},\ \bar{K}_{2m+1}]=\lim_{\epsilon\rightarrow
0}\frac{\mbox{d}}{\mbox{d}\epsilon}\left[\bar{K}_{2n+1}(u+\epsilon\bar{K}_{2m+1})-\bar{K}_{2m+1}(u+\epsilon\bar{K}_{2n+1})\right]=0.$
(26)

The (2+1)-dimensional AKNS system. In additional to the KdV equation, the NLS
equation is another important physical model to describe various physical
phenomena including Bose-Einstein condensates BEC , quantum spin Spin ,
quantum optics Opt1 , nonequilibrium physics in biology Bio1 , string theory
and gravity string , particle physics particle , oceanic waves Ocean ,
nonlinear optics NLOpt , plasma physics plasma , atmospheric science atmos ,
superfluids sfluid , acoustic waves acous , economics yan , and so on. The
more general form of the NLS equation is the so-called AKNS system

$u_{t}=u_{xx}+2u^{2}v,\ v_{t}=-v_{xx}-2v^{2}u.$ (27)

The usual NLS equation is related to the AKNS system (27) by $v=u^{*}$ and
$t\rightarrow\sqrt{-1}t$. Generally, the AKNS system is believed to contain
certain important classes of evolution equations AKNS1  which can be solved by
the inverse scattering method.

We illustrate the algorithm for the case of the AKNS system. According to the
deformation conjecture, to deform the AKNS system (27) to a higher dimensional
integrable one, we have to find some conservation laws of (27). Fortunately,
it is known that the AKNS system (27) possesses the conservation law

$(uv)_{t}=(vu_{x}-uv_{x})_{x}.$ (28)

Thus, applying the deformation conjecture, we can obtain a deformed
(2+1)-dimensional AKNS system

$\displaystyle\hat{T}u=\hat{L}^{2}u+2u^{2}v,\ \hat{T}v=-\hat{L}^{2}v-2v^{2}u,$
(29) $\displaystyle\hat{L}=\partial_{x}+uv\partial_{y},\
\hat{T}=\partial_{t}+\left[v(\hat{L}u)-u(\hat{L}v)\right]\partial_{y},$

i.e.,

$\displaystyle
u_{t}=u_{xx}+2u^{2}v+u^{2}v(vu_{yy}+2u_{y}v_{y})+2u(vu_{xy}+v_{x}u_{y}),$
$\displaystyle
v_{t}=-v_{xx}-2v^{2}u-uv^{2}(uv_{yy}+2u_{y}v_{y})-2v(uv_{xy}+u_{x}v_{y}).$
(30)

Similar to the (3+1)-dimensional KdV equation, the (2+1)-dimensional AKNS
system (30) is also Lax integrable and symmetry integrable. We can go directly
to the Lax pair of the (2+1)-dimensional AKNS system (30)

$\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)_{x}=-uv\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)_{y}+\left(\begin{array}[]{cc}\lambda&-v\\\
u&-\lambda\end{array}\right)\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right),$ (31) $\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)_{t}=(uv_{x}-vu_{x}+u^{2}vv_{y}-v^{2}uu_{y})\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)_{y}+\left(\begin{array}[]{cc}-2\lambda^{2}-uv&v_{x}+uvv_{y}+2\lambda
v\\\ u_{x}+vuu_{y}-2\lambda
u&2\lambda^{2}+uv\end{array}\right)\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)$ (32)

by repeating the algorithm that we followed previous for the (3+1)-dimensional
KdV equation.

When the fields $\psi_{1},\ \psi_{2},\ u$ and $v$ are $y$-independent, the Lax
pair (31)-(32) of the (2+1)-dimensional AKNS system (30) is reduced back to
the Lax pair of the usual (1+1)-dimensional AKNS system (27).

When the fields $u$ and $v$ of (30) are $y$ independent, the (2+1)-dimensional
AKNS system is return back to the usual AKNS equations. For $u_{x}=v_{x}=0$,
the (2+1)-dimensional AKNS system (30) becomes a reciprocal link related to
the conservation law (28)

$\displaystyle u_{t}=u^{2}v(vu_{yy}+2u_{y}v_{y})+2u^{2}v,$ $\displaystyle
v_{t}=-uv^{2}(uv_{yy}+2u_{y}v_{y})-2v^{2}u.$ (33)

The reciprocal link (33) of the AKNS system possesses the following
conservation law

$\displaystyle(u^{-1}v^{-1})_{t}=(uv_{y}-vu_{y})_{y}.$ (34)

By using the deformation conjecture and the conservation law (34) to the model
(33), one can re-obtain the $(2+1)$-dimensional AKNS system (30).

Summary and discussions. In summary, applying the deformation conjecture to
the known (1+1)-dimensional integrable evolution models with $D-1$
conservation laws, $[\rho_{j}(u)]_{t}=[J_{j}(u,\ u_{x},\ \ldots)]_{x}$, one
can obtain many $(D+1)$-dimensional integrable systems. The fact that the
deformation operation that distills the higher dimensional equations from the
lower dimensional ones which possesses the Lax and symmetry integrability, in
some sense, preserves integrability, implies these higher dimensional
equations be Lax integrable and symmetry integrable. The deformation algorithm
will destroy the Painlevé integrability because the reciprocal transformations
are included. The Lax pairs of the $(3+1)$-dimensional KdV-HD equation (9) and
the $(2+1)$-dimensional AKNS (30) are explicitly given. The deformation
conjecture has the advantage that it constructs $(D+1)$-dimensional integrable
systems, as well as integrable hierarchies, in a straightforward manner. The
whole $(3+1)$-dimensional KdV hierarchy has also been obtained via the
deformation algorithm from the $(1+1)$-dimensional KdV hierarchy.

Usually, ($n+1$)-dimensional integrable models for $n\geq 2$ such as the
Kadomtsev-Petviashivili equation and the Davey-Stewartson equation are
nonlocal in their evolution forms. However, the ($D+1$)-dimensional integrable
models obtained from the deformation algorithm are all local.

For every conservation law, $\rho_{t}=J_{x}$, of (1+1)-dimensional local
evolution systems, the deformation algorithm included in the deformation
conjecture implies a reciprocal transformation of the original
$(1+1)$-dimensional model by deformation relations
$\partial_{x}\rightarrow\rho\partial_{y}$ and
$\partial_{t}\rightarrow\partial_{t}+\bar{J}\partial_{y}$.

Though the deformation conjecture requires the conserved density is only the
field $u$-dependent but not the $x$-derivatives of the field, one can make
suitable transformations such that the field derivative dependent conserved
density becomes the field derivative independent for the new models. For
instance, the AKNS system (27) possesses a conservation law with the conserved
density $vu_{x}$. By using the dependent variable transformations,
$vu_{x}=u_{1},\ uv=u_{2}$, the AKNS system (27) is transformed to a variant
form of the AKNS system (VAKNS)

$u_{1t}=[u_{2}^{2}+u_{1x}+2u_{1}u_{2}^{-1}(u_{1}-u_{2x})]_{x},\
u_{2t}=(2u_{1}-u_{2x})_{x}.$ (35)

The VAKNS system (35) possesses three field derivative independent conserved
densities, $u_{1},\ u_{2}$ and $u_{1}u_{2}^{-1}$ with
$(u_{1}u_{2}^{-1})_{t}=(u_{2}^{-1}u_{1x}-u_{1}u_{2}^{-2}u_{2x}+u_{1}^{2}u_{2}^{-2}+2u_{2})_{x}$.
Thus, the VAKNS system can be deformed to a (4+1)-dimensional integrable local
evolution system by means of the deformation algorithm,

$\hat{T}u_{1}=\hat{L}[u_{2}^{2}+\hat{L}u_{1}+2u_{1}u_{2}^{-1}(u_{1}-\hat{L}u_{2})],\
\hat{T}u_{2}=\hat{L}(2u_{1}-\hat{L}u_{2})$ (36)

with
$\hat{L}=\partial_{x}+u_{1}\partial_{y}+u_{2}\partial_{z}+u_{1}u_{2}^{-1}\partial_{\xi}$,
$\hat{T}=\partial_{t}+\bar{J}_{1}\partial_{y}+\bar{J}_{2}\partial_{z}+\bar{J}_{3}\partial_{\xi}$,
and
$\\{\bar{J}_{1}=u_{2}^{2}+\hat{L}u_{1}+2u_{1}u_{2}^{-1}(u_{1}-\hat{L}u_{2}),\
\bar{J}_{2}=2u_{1}-\hat{L}u_{2},\
\bar{J}_{3}=u_{2}^{-1}\hat{L}u_{1}-u_{1}u_{2}^{-2}\hat{L}u_{2}+u_{1}^{2}u_{2}^{-2}+2u_{2}\\}$.
The (4+1)-dimensional VAKNS system (36) is a combination of the VAKNS system
(35) and its three reciprocal links related to the conserved densities,
$u_{1}$, $u_{2}$ and $u_{1}u_{2}^{-1}$, respectively.

In fact, for the AKNS system (27), there are some other important variant
forms such as the KN system KN , the Levi system Levi , the Heisenberg system
LL1 , and so on. For these variant forms, there are three or four field
derivative independent conserved densities. Therefore, some (4+1)- and
(5+1)-dimensional integrable local systems can be found by applying the
deformation conjecture on these variant forms.

It should be mentioned that for any ($D+1$)-dimensional deformed integrable
systems, there are some new (1+1)-dimensional reductions. Applying the
deformation algorithm on these reductions, one can re-discover the
($D+1$)-dimensional deformed integrable systems.

Though we can obtain various higher dimensional local integrable evolution
systems in the form (3), it will be very difficult to find analytical explicit
solutions of these integrable models because almost all the traditional
methods such as the Hirota’s bilinear direct method, Darboux transformations
and inverse scattering transformation, etc. can not be directly applied. This
fact provides us a challenge even for the fundamental problem, what is
integrability? The models obtained by the deformation algorithm of this paper
are Lax integrable and symmetry integrable, however, the traditional methods
on integrable systems are not valid to find exact solutions even for non-
degenerated $n$-soliton solutions for $n\geq 2$.

Figure 1: Different single soliton shapes of (38) with the same wave
parameters $\omega=2$ and $k=1$ and different deformation parameters $p$ and
$q$. (a). A quasi-symmetric soliton with small deformation parameter
selections $p=q=1/30$. (b) An asymmetric soliton with parameter selections
$p=q=-1/3$. (c). A foldon under the larger parameter selections $p=q=-1$ .

For the travelling wave solution, $u=U(X),\ X=kx+py+qz+\omega t+X_{0}$, of the
(3+1)-dimensional KdV equation (9) satisfies the following ordinary
differential equation

$\omega
U_{X}=\left[(qU^{2}+pU+k)^{3}U_{XX}+\frac{1}{2}U^{2}(qU^{2}+2pU+6k)\right]_{X}+3q(qU^{2}+pU+k)^{2}U_{X}^{3}.$
(37)

The general solution of (37) can be expressed by an elliptic integration. The
single soliton solution can be implicitly expressed as

$U=\frac{\omega}{2k}\mbox{\rm
sech}^{2}\left[\frac{\omega}{6k^{3}}\sqrt{1-\frac{2k}{\omega}U}(qkU+3kp+q\omega)+\frac{\sqrt{k\omega}}{2k^{2}}X\right].$
(38)

From the soliton expression (38), we know that when $p=q=0$, i.e., the soliton
is $\\{y,\ z\\}$-independent, (38), it returns back to the standard soliton
solution of the usual KdV equation. As $|p|$ and $|q|$ increasing, the soliton
shape will deform to an asymmetric soliton and a foldon or folded solitary
wave (multi-valued soliton or solitary wave foldon ). Fig. 1a displays the
soliton shape expressed by (38) with small modification parameters $p=q=1/30$
while other parameters are fixed as $\omega=2$ and $k=1$. In this case, the
soliton shape is still quite symmetric. Fig. 1b shows us that the soliton has
deformed to an asymmetric one where the parameters are selected as $p=q=-1/3,\
\omega=2$ and $k=1$. For larger selections of the deformed parameters,
$p=q=-1$, the soliton becomes a foldon as shown in Fig. 1c where the values of
$\omega$ and $k$ are same as in other two figures.

In the deformation conjecture, we have put some restrict conditions on it. The
original model should be a (1+1)-dimensional, local and evolution system and
the conserved density should be field derivative independent. From the
discussions related to the VAKNS system (35), we know that the conserved
density may be field derivative dependent. In fact, other restrict conditions
may also be loosed however, this is not the topic of this paper and would be
discussed elsewhere.

## Acknowledgement

The work was sponsored by the National Natural Science Foundation of China
(Nos. 12235007, 11975131 and 12275144), K. C. Wong Magna Fund in Ningbo
University, Natural Science Foundation of Zhejiang Province No. LQ20A010009. I
would also like to thank Profs. R. X. Yao, X. B. Hu, Q. P. Liu, B. F. Feng and
D. J. Zhang for their valuable discussions.

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# Evaluation of Vortex Criteria by Virtue of the Quadruple Decomposition of
Velocity Gradient Tensor ††thanks: This is an English version of the Chinese
paper published in Acta Phys. Sin. Vol. 63, No. 5 (2014), doi:
10.7498/aps.63.054704. The project was partially supported by the National
Basic Research Program of China (973 Program Grant No. 2012CB720101) and the
National Natural Science Foundation of China (NSFC Grant No. 11072130). The
translator is the first author (Zhen Li), who is now affiliated with the
Beijing Institute of Mathematical Sciences and Applications (BIMSA).

Zhen Li<EMAIL_ADDRESS>School of Aerospace, Tsinghua University, Beijing
100084 China  Yau Mathematical Sciences Center, Tsinghua University, Beijing
100084 China  Xiwen Zhang  School of Aerospace, Tsinghua University, Beijing
100084 China  Feng He<EMAIL_ADDRESS>School of Aerospace, Tsinghua
University, Beijing 100084 China

((Received 9 October 2013; Revised 27 November 2013))

###### Abstract

Based on the analysis of the velocity gradient tensor, we investigate in this
paper the physical interpretation and limitations of four vortex criteria:
$\omega$, $Q$, $\varDelta$ and $\lambda_{ci}$, and reveal the actual physical
meaning of vortex patterns which are usually illustrated by level sets of
various vortex criteria. A quadruple decomposition based on the normality of
the velocity gradient tensor is proposed for the first time, which resolves
the motion of a fluid element into dilation, axial stretch along the normal
frame, in-plane distortion, and simple shear, in order to clarify the
kinematical interpretation of various vortex criteria. The mean rotation
characterized by the vorticity $\omega$ always consists of simple shear; the
$Q$-criterion can reflect the strength of net rotation within the invariant
plane relative to the axial stretch of a fluid element, and it is a sufficient
but unnecessary condition for the existence of net rotation; the
$\varDelta$-criterion can exactly identify the existence of net rotation, but
it is not the strength of net rotation; in the case that net rotation exists,
$\lambda_{ci}$ characterize its absolute strength. net rotation is the total
effect of the normal rotation within the invariant plane and the simple shear,
where the former is the most basic rotation. The newly introduced quadruple
decomposition can improve our understanding of vortices in fluids and their
motions.

Keywords: vortex criteria, velocity gradient tensor, quadruple decomposition

##  1 Introduction

Vortex plays a key role in fluid mechanics. The study of the generation,
development, motion, decline of vortices, and the interactions among vortices
and those between vortices and solids, and the relationship between vortex and
turbulence, as well as the definition and identification of vortices raised in
recent years, all have always been important research topics in fluid
mechanics. Shi-Jia Lu and Küchemann [1] call vortex the muscle and essence of
fluid motion. [2] referred to vortex as the sinew of turbulence. The analysis
of vortices is crucial to understand the motion of fluids. It is a fundamental
problem of fluid mechanics.

However, the definition of vortex in fluid mechanics is still vague. The
difficulty of defining vortex reflects the distinction between our intuitive
understanding and the description methods of classical fluid mechanics. The
intuition usually perceives the large-scale change of configuration of fluids,
which bears the dynamical-system perspective. However, the theory of classical
fluid mechanics usually adopts a field-theoretical description of local
properties of the flow fields.

Here we leave aside the justifiability of the definition of vortex and focus
on the analysis of existing and applied vortex criteria. They view vortices
with a field-theoretical perspective and determine the center of local
vortices, which they regard as regions with relatively high vorticity. For any
point satisfying a criterion, the fluid element in the neighborhood of the
point is regarded as a local vortex rotating about the point. Large vortices
are regions consisting of these local vortices.

In a three-dimensional flow field, consider the motion of a point with
displacement $(\delta x_{1},\delta x_{2},\delta x_{3})$ relative to its
neighbor $O$. The first-order Talyor expansion of its velocity relative to $O$
is

$\displaystyle\bm{V}=\bm{V}_{0}+\delta\bm{V},$ (1)

where the relative velocity $\delta\bm{V}$ is related to the velocity gradient
tensor $\nabla\bm{V}$ as follows:

$\displaystyle\delta\bm{V}=(\delta x_{1},\delta x_{2},\delta
x_{3})\cdot\nabla\bm{V}.$ (2)

The definitions of many vortex criteria are based on the eigen-analysis of the
velocity gradient tensor $\nabla\bm{V}$ of fluid elements, which provides some
understanding of the complicated motion of fluid elements. In this paper, a
quadruple decomposition based on the normality of the velocity gradient tensor
is proposed in an attempt to provide a deeper understanding of the physical
meaning and limitations of popular vortex criteria. Then the kinematical
properties of local vortices, which are reflected by the quantities of vortex
criteria, can be better understood in their applications. It also provides a
better understanding of the vortex pattern illustrated by level sets of the
quantities. All these are helpful to the analysis and understanding of the
patterns of complex flows.

##  2 Popular vortex criteria:
$\omega$,$Q$,$\varDelta$,$\lambda_{ci}$,$\lambda_{2}$

In experimental or numerical simulation studies of flow fields, people often
need to visualize vortical structures. Therefore, various methods are
developed for identifying vortices.

For analyzing vortices in flow fields, the simplest way is to use the
vorticity field $\omega$, which is referred to as the $\omega$-criterion.
$\omega$ is twice the mean angular velocity of the fluid element, hence it
reflects the mean rotation. People usually regard the $\omega$-concentrated
regions as vortices. The value of $\omega$ is easy to obtain through
measurement of velocity profile [3]. It is well-known, however, that the
vorticity of parallel shear flow is also nonzero, but there are no spiral
streamlines in the flow. Hence the criterion has limitations in application.

[4, 5] and [6] introduced the $Q$-criterion for incompressible or low-
compressible flows to determine regions where vorticity tensor exceeds
deformation rate tensor. $Q$ is defined as the second of the principal
invariants $I_{1},I_{2},I_{3}$, i.e., $Q=I_{2}$. The criterion is $Q>0$, which
is widely applied in the identification of vortices in three-dimensional flows
[7].

[8] applied the method for analyzing local topological property of equilibrium
points of ordinary differential equations to the analysis of complex flow
patterns and proposed the $\varDelta$-criterion, where $\varDelta$ is the
discriminant of the characteristic polynomial of $\nabla\bm{V}$. The criterion
is $\varDelta>0$. The $\varDelta$ can be expressed in terms of the tree
principal invariants of $\nabla\bm{V}$ as

$\displaystyle\varDelta=\frac{1}{4}I_{3}^{2}+\left(\frac{1}{27}I_{1}^{3}-\frac{1}{6}I_{1}I_{2}\right)I_{3}+\left(\frac{1}{27}I_{2}^{3}-\frac{1}{108}I_{1}^{2}I_{2}^{2}\right).$
(3)

For any point in the flow field that satisfies $\varDelta>0$, the
$\nabla\bm{V}$ has one real eigenvalue and a pair of complex conjugate
eigenvalues. There is a focus of instantaneous streamlines on its invariant
plane, which is regarded as the center of a vortex. The instantaneous
streamlines around the focus are spiral, which are stretched or compressed
along the direction of the eigenvector corresponding to the real eigenvalue,
as illustrated in Fig. 1.

Figure 1: 3D helical streamlines in the reference frame moving with the focus.

The local vortex presented in the invariant plane of $\nabla\bm{V}$ depends
only on the property of eigenvalues of $\nabla\bm{V}$ at that point. It does
not rely on the reference frame of the observer.

[9] proposed to use the absolute value of the imaginary part of the complex
eigenvalues of $\nabla\bm{V}$ as the strength of vortex, provided that
$\nabla\bm{V}$ has complex conjugate eigenvalues. It reflects the angular
velocity of net rotation within the invariant plane of $\nabla\bm{V}$ of a
fluid element. Since $\lambda_{ci}$ exists only when $\varDelta>0$, it is
equivalent to the $\varDelta$-criterion.

Most of the criteria are constructed with the kinematical perspective. But in
the research and engineering applications of fluid mechanics, people also use
low pressure as an indicator of vortex, for example, this method is widely
used for identifying vortices in meteorology. The principle underlying this
method is that the rotation of a vortex produces centrifugal force, which must
be balanced by the pressure gradient in certain situations. Hence a low-
pressure region is created at the center of the vortex. [10] modified this
method. After some analysis, they concluded that the only reason for the
appearance of sectional minimum pressure is the centrifugal force caused by
rotation if the effects of viscosity and unsteadiness are excluded. Through
some manipulation of the Navier-Stokes equation for incompressible flow, they
obtain a tensor $\mathbf{D}^{2}+\bm{\Omega}^{2}$, where $\mathbf{D}$ is the
deformation rate tensor and $\bm{\Omega}$ is the vorticity tensor. The
eigenvalues of this new tensor are arranged in the decreasing order
$\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}$. The inequality $\lambda_{2}<0$ is
considered as the necessary and sufficient condition for the pressure to
attain its sectional minimum, and it is called the $\lambda_{2}$-criterion.
[11] uses the $\lambda_{2}$-criterion to illustrate their new findings on the
vortices in streamwise steaks of a transient boundary layer.

Many other types of vortex criteria are out of the scope of the current paper.
The analysis of the local variables of flow fields belongs to the field theory
of classical mechanics, which is easier for computation. Hence the $\omega$-,
$Q$-, $\varDelta$-, $\lambda_{ci}$\- and $\lambda_{2}$-criteria are widely
applied.

##  3 Kinematical analysis of vortex criteria in three-dimensional flow fields

The second-order tensor $\nabla\bm{V}$ can be decomposed into a symmetric part
and an anti-symmetric part which, expressed in the Cartesian coordinate
system, are the deformation rate tensor

$\displaystyle D_{ij}=\frac{1}{2}\left(\frac{\partial\dot{x}_{j}}{\partial
x_{i}}+\frac{\partial\dot{x}_{i}}{\partial x_{j}}\right)$

and the vorticity tensor

$\displaystyle\Omega_{ij}=\frac{1}{2}\left(\frac{\partial\dot{x}_{j}}{\partial
x_{i}}-\frac{\partial\dot{x}_{i}}{\partial x_{j}}\right).$

That is

$\displaystyle\nabla\bm{V}=\mathbf{D}+\bm{\Omega}.$

$\mathbf{D}$ can be further decomposed as the sum of a dilation tensor
$E_{ii}=\partial\dot{x}_{i}/\partial x_{i}$ and a stretch rate tensor 111This
is a mistake. It should be $E_{ij}=(\nabla\cdot\bm{V})/3\delta_{ij}$ and
$A_{ij}=D_{ij}-E_{ij}$.

$\displaystyle A_{ij}=\frac{1}{2}\left(\frac{\partial\dot{x}_{j}}{\partial
x_{i}}+\frac{\partial\dot{x}_{i}}{\partial x_{j}}\right),\quad(i\neq j).$

Hence $\nabla\bm{V}$ is usually decomposed into three parts (triple
decomposition)

$\displaystyle\nabla\bm{V}=\mathbf{E}+\mathbf{A}+\bm{\Omega}.$ (4)

The motion of a fluid element in a three-dimensional flow is very complex,
whose dilation, stretch, and mean rotation are coupled with each other.
$\bm{V}$ is generally non-normal. It is hoped that a new understanding of
vortex criteria could be revealed by studying the motion of fluid elements
from the perspective of the normality of $\nabla\bm{V}$.

When $\varDelta\leq 0$, the eigenvalues of $\nabla\bm{V}$
$\lambda_{1},\lambda_{2},\lambda_{3}$ are all real numbers. $\nabla\bm{V}$ is
represented in the normal frame as an upper triangular matrix

$\displaystyle\nabla\bm{V}=\begin{pmatrix}\lambda_{1}&\gamma&\beta\\\
0&\lambda_{2}&\alpha\\\ 0&0&\lambda_{3}\end{pmatrix}.$ (5)

If the eigenvalues are all real numbers, then there is no focus or center as
the topological structure of instantaneous streamlines on the plane spanned by
eigenvectors [6]. Decompose $\nabla\bm{V}$ as the sum of a normal tensor
$\mathbf{N}$ and a nilpotent tensor $\mathbf{S}$, where $\mathbf{S}$ is
regarded as the tensor of shear rate in the normal frame.

$\displaystyle\nabla\bm{V}=\mathbf{N}+\mathbf{S},\quad\mathbf{N}=\begin{pmatrix}\lambda_{1}&0&0\\\
0&\lambda_{2}&0\\\
0&0&\lambda_{3}\end{pmatrix},\quad\mathbf{S}=\begin{pmatrix}0&\gamma&\beta\\\
0&0&\alpha\\\ 0&0&0\end{pmatrix}.$ (6)

The $\mathbf{N}$ can be further decomposed into a dilation rate tensor
$\mathbf{E}$ representing isotropic dilation, a stretch rate tensor
$\mathbf{Z}$ representing axial stretch along some axis (e.g. the eigenvector
associated with $\lambda_{3}$) and an in-plane tensor $\bm{\Psi}$ representing
the stretch in the plane orthogonal to the axis. That is

$\displaystyle\mathbf{N}=\mathbf{E}+\mathbf{Z}+\bm{\Psi},$ (7)
$\displaystyle\mathbf{E}=\frac{\vartheta}{3}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},\quad\mathbf{Z}=\frac{\varepsilon}{3}\begin{pmatrix}-1&0&0\\\
0&-1&0\\\ 0&0&2\end{pmatrix},\quad\bm{\Psi}=\psi\begin{pmatrix}1&0&0\\\
0&-1&0\\\ 0&0&0\end{pmatrix},$ (8)

where $\vartheta=\lambda_{1}+\lambda_{2}+\lambda_{3}$,
$\varepsilon=\lambda_{3}-(\lambda_{1}+\lambda_{2})/2$,
$\psi=(\lambda_{1}-\lambda_{2})/2$.

When $\varDelta>0$, $\nabla\bm{V}$ has a real eigenvalue $\lambda_{r}$ and a
pair of complex conjugate eigenvalues $\lambda_{cr}\pm\mathrm{i}\lambda_{ci}$.
In the normal frame, there is

$\displaystyle\nabla\bm{V}=\begin{pmatrix}\lambda_{cr}&\psi+\gamma&\beta\\\
-\psi&\lambda_{cr}&\alpha\\\ 0&0&\lambda_{r}\end{pmatrix}.$ (9)

The characteristic equation of $\nabla\bm{V}$ is

$\displaystyle(\lambda-\lambda_{r})(\lambda-\lambda_{cr}-\mathrm{i}\lambda_{ci})(\lambda-\lambda_{cr}+\mathrm{i}\lambda_{ci})=0,$
(10)

whose discriminant is

$\displaystyle\varDelta=\frac{1}{27}\lambda_{ci}^{2}\left(\lambda_{ci}^{2}+(\lambda_{r}-\lambda_{cr})^{2}\right)^{2}.$
(11)

Decompose $\nabla\bm{V}$ as the sum of a tensor $\mathbf{N}$ and a nilpotent
tensor $\mathbf{S}$, where the expressions of $\mathbf{N}$ and $\mathbf{S}$
are as follows:

$\displaystyle\nabla\bm{V}=\mathbf{N}+\mathbf{S},\quad\mathbf{N}=\begin{pmatrix}\lambda_{cr}&\psi&0\\\
-\psi&\lambda_{cr}&0\\\
0&0&\lambda_{r}\end{pmatrix},\quad\mathbf{S}=\begin{pmatrix}0&\gamma&\beta\\\
0&0&\alpha\\\ 0&0&0\end{pmatrix}.$ (12)

And $\mathbf{N}$ can also be decomposed into three parts

$\displaystyle\mathbf{N}=\mathbf{E}+\mathbf{Z}+\bm{\Psi},$ (13)
$\displaystyle\mathbf{E}=\frac{\vartheta}{3}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},\quad\mathbf{Z}=\frac{\varepsilon}{3}\begin{pmatrix}-1&0&0\\\
0&-1&0\\\ 0&0&2\end{pmatrix},\quad\bm{\Psi}=\psi\begin{pmatrix}0&1&0\\\
-1&0&0\\\ 0&0&0\end{pmatrix},$ (14)

where $\vartheta=\lambda_{r}+2\lambda_{cr}$,
$\varepsilon=\lambda_{r}-\lambda_{cr}$.

And the $\psi$ is the absolute value of the imaginary part of the complex
conjugate eigenvalues of $\mathbf{N}$. The physical interpretation of the
tensor $\bm{\Psi}$ becomes the normal rotation within its invariant plane.

For both $\varDelta\leq 0$ and $\varDelta>0$, $\nabla\bm{V}$ has the following
quadruple decomposition

$\displaystyle\nabla\bm{V}=\mathbf{E}+\mathbf{Z}+\bm{\Psi}+\mathbf{S}.$ (15)

The typical flow patterns corresponding to the tensors $\mathbf{E}$,
$\mathbf{Z}$, $\bm{\Psi}$, $\mathbf{S}$ are illustrated in Fig. 2.

Figure 2: Typical flow patterns corresponding to the tensors in the quadruple
decomposition. (a) The dilation rate tensor $\mathbf{E}$ stands for isotropic
expansion and compression; (b) The axial stretch rate tensor $\mathbf{Z}$
stands for the axial stretch along the direction of frame axis; (c) The in-
plane tensor $\bm{\Psi}$ orthogonal to the frame axis: $\varDelta>0$,
$\bm{\Psi}$ stands for the normal rotation in the plane, $\varDelta<0$,
$\bm{\Psi}$ stands for the stretch in the plane; (d) The shear rate tensor
$\mathbf{S}$ stands for simple shear.

Now we analyze the relationship between the $\mathbf{A},\bm{\Omega}$ in the
usual triple decomposition and the $\mathbf{Z},\bm{\Psi},\mathbf{S}$ in the
quadruple decomposition Eq. 15

Firstly, the $\mathbf{S}$ is decomposed into a symmetric simple shear
$\mathbf{S}_{S}$ and an anti-symmetric simple shear $\mathbf{S}_{A}$ with

$\displaystyle\mathbf{S}_{S}=\frac{1}{2}(\mathbf{S}+\mathbf{S}^{\mathsf{T}}),$
$\displaystyle\mathbf{S}_{A}=\frac{1}{2}(\mathbf{S}-\mathbf{S}^{\mathsf{T}}),$
(16)

where $\mathbf{S}^{\mathsf{T}}$ is the transpose of $\mathbf{S}$.

When $\bm{V}$ has three real eigenvalues,

$\displaystyle\mathbf{A}=$ $\displaystyle\mathbf{Z}+\bm{\Psi}+\mathbf{S}_{S},$
$\displaystyle\bm{\Omega}=$ $\displaystyle\mathbf{S}_{A}.$ (17)

The stretch of the fluid element represented by the tensor $\mathbf{A}$
consists of three parts: the axial stretch $\mathbf{Z}$, the in-plane stretch
$\bm{\Psi}$, and the symmetric simple shear $\mathbf{S}_{S}$. The mean
rotation represented by $\bm{\Omega}$ is just the anti-symmetric simple shear,
which is not genuine rotation.

When $\nabla\bm{V}$ has complex conjugate eigenvalues,

$\displaystyle\mathbf{A}=$ $\displaystyle\mathbf{Z}+\mathbf{S}_{S},$
$\displaystyle\bm{\Omega}=$ $\displaystyle\bm{\Psi}+\mathbf{S}_{A}.$ (18)

The stretch of the fluid element represented by the tensor $\mathbf{A}$
consists of two parts: the axial stretch $\mathbf{Z}$ and the symmetric simple
shear $\mathbf{S}_{S}$. The mean rotation represented by $\bm{\Omega}$
consists of the normal rotation $\bm{\Psi}$ and the anti-symmetric simple
shear $\mathbf{S}_{A}$. That is to say that the normal rotation is the mean
rotation with the part of simple shear $\mathbf{S}_{A}$ excluded.

Therefore, no matter in which case of the eigenvalues of $\nabla\bm{V}$, the
stretch rate tensor $\mathbf{A}$ and vorticity tensor $\bm{\Omega}$ consist of
parts of the simple shear tensor $\mathbf{S}$.

Since the quadruple decomposition separates the simple shear $\mathbf{S}$, the
physical interpretation of the decomposition of the fluid element becomes
clear.

According to the triple decomposition of $\nabla\bm{V}$, the $Q$ introduced by
[4] and [6] is defined as the second principal invariant $I_{2}$ in both
compressible and incompressible flows,

$\displaystyle
Q=I_{2}=\frac{\vartheta^{2}}{3}-\frac{\|\mathbf{A}\|_{\mathrm{F}}^{2}}{2}+\frac{\omega}{4}.$

Actually, $I_{2}$ is affected by the dilation $\vartheta=\nabla\cdot\bm{V}$ of
a fluid element. Even for the compressible flow with uniform dilation, $Q$ is
positive, which is obviously unreasonable. Therefore, the relationship between
$Q$ and $I_{2}$ should be (modified to)

$\displaystyle Q=I_{2}-\frac{\vartheta^{2}}{3}.$ (19)

Then, for both compressible and incompressible flows, $Q$ has a unified
expression

$\displaystyle
Q=\frac{1}{4}\left(\omega^{2}-2\|\mathbf{A}\|_{\mathrm{F}}^{2}\right)=\frac{1}{2}\left(\|\bm{\Omega}\|_{\mathrm{F}}^{2}-\|\mathbf{A}\|_{\mathrm{F}}^{2}\right).$
(20)

When $\nabla\bm{V}$ has three real eigenvalues, plug Eq. 17 into Eq. 20, we
get

$\displaystyle Q=-\left(\frac{1}{3}\varepsilon^{2}+\psi^{2}\right)\leq 0.$
(21)

We can see that $Q$ is a measure of the negative sum of the axial stretch and
in-plane stretch of the fluid element.

In the definition Eq. 3 of $\varDelta$, replace the principal invariants with
eigenvalues, then considering the relationships among
$\vartheta,\psi,\varepsilon$ in Eq. 8, we can get

$\displaystyle\varDelta=-\frac{1}{27}\psi^{2}(\varepsilon^{2}-\psi^{2})^{2}\leq
0.$ (22)

We can see that $\varDelta$ is a measure of the difference between the axial
stretch and in-plane stretch of the fluid element.

When $\nabla\bm{V}$ has complex conjugate eigenvalues, plug Eq. 18 into Eq.
20, we get

$\displaystyle Q=\psi(\psi+\gamma)-\frac{1}{3}\varepsilon^{2},$

or

$\displaystyle Q=\lambda_{ci}^{2}-\frac{1}{3}\varepsilon^{2}.$ (23)

The Eq. 23 makes the meaning of $Q$ more significant, which is the difference
between the net rotation around the direction of the eigenvector associated
with the real eigenvalue $\lambda_{r}$ and the axial stretch. In the region
satisfying $Q>0$, there must be $\lambda_{ci}\neq 0$ ($\delta>0$), i.e., the
net rotation exists. But in the region of $\lambda_{ci}\neq 0$, the condition
$Q>0$ may not be satisfied because the sign of $Q$ is affected by the axial
stretch $\varepsilon$. Furthermore, Eq. 23 indicates that the sign of $Q$ is
not affected by the existence of simple shear $\mathbf{S}$.

Let’s look at an example of the Burgers vortex, which is axisymmetric and has
axial stretching. The radial velocity and axial velocity indicate that the
flow has axial stretching. The Fig. 3 shows the profiles of the axial velocity
$V_{z}$, the radial velocity $V_{r}$ and the azimuth velocity $V_{\theta}$ of
a Burgers vortex with strong axial stretching. The Fig. 4 shows the comparison
of the four vortex criteria on this flow.

Figure 3: Profiles of velocities of Burgers vortex.

$\displaystyle 0.00<r/\sqrt{\frac{\nu}{2a}}<1.38,$ $\displaystyle\quad
Q>0,\quad\Delta>0,\quad\lambda_{ci}\neq 0,\quad\omega\neq
0,\quad\lambda_{2}<0;$ $\displaystyle 1.38<r/\sqrt{\frac{\nu}{2a}}<1.63,$
$\displaystyle\quad Q<0,\quad\Delta>0,\quad\lambda_{ci}\neq 0,\quad\omega\neq
0,\quad\lambda_{2}<0;$ $\displaystyle 1.63<r/\sqrt{\frac{\nu}{2a}}<2.20,$
$\displaystyle\quad Q<0,\quad\Delta>0,\quad\lambda_{ci}\neq 0,\quad\omega\neq
0,\quad\lambda_{2}\geq 0;$ $\displaystyle 2.20<r/\sqrt{\frac{\nu}{2a}}<6.00,$
$\displaystyle\quad Q<0,\quad\Delta<0,\quad\lambda_{ci}\,\text{doesn't
exist},\quad\omega\neq 0,\quad\lambda_{2}\geq 0.$

Figure 4: Comparison of size of regions in Burgers vortex determined by the
four vortex criteria.

When $\nabla\bm{V}$ has complex conjugate eigenvalues, plug the
$\vartheta,\varepsilon$ in Eq. 14 and their relationships with eigenvalues
into Eq. 11, we get

$\displaystyle\varDelta=\frac{1}{27}\lambda_{ci}^{2}\left(\varepsilon^{2}+\lambda_{ci}^{2}\right)^{2}>0.$
(24)

The Eq. 24 makes the physical interpretation of $\varDelta$ more significant
too. It measures the sum of the net rotation about the eigenvector associated
with the real eigenvalue $\lambda_{r}$ and the axial stretch, where the axial
stretch $\varepsilon$ does not affect the validity of $\varDelta>0$. Hence the
$\varDelta$-criterion is suitable for determining the qualitative structure of
local streamlines. However, as a measure of the absolute strength of net
rotation, it is affected by the axial stretch.

The discriminant obtained from Eq. 9 should be identical to Eq. 11, which
gives

$\displaystyle\lambda_{ci}^{2}=\psi(\psi+\gamma).$ (25)

The existence of $\lambda_{ci}$ indicates most clearly that there is net
rotation within the invariant plane of a fluid element. Its absolute value or
square can be used as a measure of the absolute strength of net rotation. From
Eq. 25 we can see that the net rotation represented by $\lambda_{ci}$ is a
total effect of the normal rotation $\psi$ within its invariant plane and the
simple shear $\gamma$.

According to Eq. 18, the normal rotation $\psi$ is the mean rotation with part
of simple rotation removed, and it is the most basic rotation. It is contained
in the four vortex criteria that appeared in this paper. In theory, it can be
used as the most basic local vortex criterion. But its computation is a little
more complicated than the four popular vortex criteria.

Although the $\lambda_{2}$-criterion is motivated by dynamical considerations,
its definition is completely a kinematical quantity. The kinematical meaning
of the $\lambda_{2}$-criterion is far less significant than the $\varDelta$\-
and $\lambda_{ci}$-criteria. The second principal invariant of
$\mathbf{D}^{2}+\bm{\Omega}^{2}$ consists of the coupled dilation, stretch,
and mean rotation of a fluid element. Hence the phenomenon described by
$\lambda_{2}$ is completely three-dimensional, whose complexity prevents a
decomposition into several parts like the other four vortex criteria. It is a
compound of all parts. A deeper discussion about the $\lambda_{2}$-criterion
will be left to future study.

##  4 Conclusions

By decomposing the velocity gradient tensor $\nabla\bm{V}$ into the isotropic
dilation rate tensor $\mathbf{E}$, the axial stretch tensor $\mathbf{Z}$, the
in-plane tensor $\bm{\Psi}$ and the simple shear tensor $\mathbf{S}$, we can
see that the mean rotation represented by the vorticity of the fluid element
always consists of components of simple shear $\mathbf{S}$. Therefore, the
condition of nonzero vorticity can not distinguish parallel shear flow. The
value of $Q>0$ represents the strength of net rotation relative to axial
stretch, but it is only a sufficient and non-necessary condition for the
existence of net rotation within the invariant plane of $\nabla\bm{V}$. The
sign of $\varDelta$ determines directly the meaning of in-plane tensor
$\bm{\Psi}$: whether it is in-plane stretch or normal rotation in the
invariant plane of $\bm{\Psi}$. Also, it can exactly identify the existence of
net rotation within the invariant plane of $\nabla\bm{V}$. However, its value
is not the strength of net rotation due to the influence of axial stretch. The
$\lambda_{ci}$ is a measure of the absolute strength of net rotation within
the invariant plane of $\nabla\bm{V}$ of a fluid element, which is a
combination of normal rotation within its invariant plane and simple shear.
Normal rotation is the mean rotation with part of the simple shear removed. It
is a basic rotation, but its computation is relatively complicated as a vortex
criterion. The sizes of regions determined by the four vortex criteria are
generally different. Their values reflect more about different kinematical
features of the determined regions. The meanings of vortex patterns
illustrated by level sets of different vortex criteria are also different. The
$\lambda_{ci}$-criterion is relatively simple. The $\lambda_{2}$-criterion is
initiated based on an analysis of dynamics, but its definition is a
kinematical quantity. Due to its complex characteristics, it can not be
decomposed into several parts in a simple way for analysis.

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