# Fair inference on error-prone outcomes

Laura Boeschoten111Shared first authorship. These authors contributed equally
to the current work. Correspondence regarding this article should be sent to
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Erik-Jan van
Kesteren11footnotemark: 1 Ayoub Bagheri  Daniel L. Oberski

(Utrecht University
Department of Methodology & Statistics)

###### Abstract

Fair inference in supervised learning is an important and active area of
research, yielding a range of useful methods to assess and account for
fairness criteria when predicting ground truth targets. As shown in recent
work, however, when target labels are error-prone, potential prediction
unfairness can arise from measurement error. In this paper, we show that, when
an error-prone proxy target is used, existing methods to assess and calibrate
fairness criteria do not extend to the true target variable of interest. To
remedy this problem, we suggest a framework resulting from the combination of
two existing literatures: fair ML methods, such as those found in the
counterfactual fairness literature on the one hand, and, on the other,
measurement models found in the statistical literature. We discuss these
approaches and their connection resulting in our framework. In a healthcare
decision problem, we find that using a latent variable model to account for
measurement error removes the unfairness detected previously. Keywords:
Fairness, Fair machine learning, Measurement error, Algorithmic bias,
Measurement invariance, Differential item functioning, Item bias, Latent
variable model

##  1 Introduction

Supervised learning is used to guide human decisions across a wide range of
different fields. In sensitive areas such as healthcare or criminal justice, a
key issue is that decisions based on such decisions are equitable and fair. To
this end, an active area of research investigates how fairness criteria can be
incorporated into supervised learning [1, 7, 8, 16, 17, 32]. This literature
has focused on supervised learning for a single objective, assumed to be the
target variable of interest. Recently, however, [23] observed that, even when
substantial care has been taken to develop a prediction algorithm, unfairness
in the predictions can still result due to _measurement error_. Intuitively,
calibrating decisions to be fair for an error-prone proxy does not imply the
decision is fair for the true variable of interest. Such effects can be
substantial; for example, [23] demonstrated a large differential in predicted
risk scores between black and white patients with equal values on a new proxy
measurement. However, these authors did not suggest a method for dealing with
this problem. This issue cannot be ignored because fairness is generally
conceptualized on a level more abstract than the proxy label [13]; for
example, it is reasonable to require that fairness in a healthcare need
prediction system should extend to a person’s true health status.

This paper addresses the problem of prediction unfairness arising from
measurement error. By considering the supervised learning problem at the level
of a latent variable of interest, we reformulate the problem as one of
adequate _measurement modeling_. In effect, instead of requiring perfect
measurement to achieve fairness, we propose that researchers developing a
prediction model to be used for decision-making collect several independent,
possibly error-prone, measures of the variable of interest (e.g. health). We
then suggest to combine measurement models from the statistical literature
with techniques from the literature on fair ML to assess and ameliorate the
problem of unfair predictions in the face of measurement error.

Our contributions are as follows:

  * •

We illustrate that existing methods to examine unfairness in error-prone
outcomes are insufficient;

  * •

We suggest a framework, based on the existing measurement modeling literature,
to investigate and ameliorate such issues;

  * •

We perform an example analysis to demonstrate the suggested approach. In an
existing healthcare application, this demonstrates that replacing one proxy
with another does not lead to parity, while our approach does.

In Section 2, we provide a summary of basic concepts in fairness. In Section 3
prior approaches with respect to fair inference are discussed. In Section 4,
the failure of these approaches is discussed when making use of proxies, and
the proposed framework is introduced based on existing measurement models. In
Section 5 the proposed framework is then applied to the exemplary data set
provided by [23].

##  2 Problem definition

We consider probabilistic classification and regression problems with a set of
features $\mathbf{X}$ and true outcome $Y^{*}$. Among the features, there is a
sensitive feature $S\in\mathbf{X}$ (e.g. race, gender), with respect to which
discriminatory predictions are to be avoided. Furthermore, although the
prediction problem is with respect to the true outcome $Y^{*}$ – e.g. “health”
or “crime” – this outcome is not directly observed; instead, we have observed
a set of error-prone proxy variables $\mathbf{Y}$. For example, in practice a
proxy for “health”, $Y\in\mathbf{Y}$, might be the costs of healthcare or the
number of chronic conditions experienced by the patient, whereas, instead of
“crime”, the number of arrests might be measured. Following [22], we represent
the goal of the regression or classification problem as a query on the
(generative) joint distribution $p(Y^{*},\mathbf{X})$, potentially after
conditioning on a set of “fixed” covariates $\mathbf{C}$, i.e. the
(discriminative) conditional joint
$p(Y^{*},\mathbf{X}\setminus\mathbf{C}\mid\mathbf{C})$. Typically, this query
will be the point prediction $\hat{Y}^{*}:=E(Y^{*}\mid\mathbf{X})$.

Following standard social-scientific measurement theory [4], the fact that
$\mathbf{Y}$ is a measurement proxy for $Y^{*}$ is reflected by a _causal
model_ , in the sense of [24, 28], in which $Y^{*}\rightarrow\mathbf{Y}$, i.e.
the true outcome is a common cause of all available proxy variables. Because
$Y^{*}$ is an unobserved latent variable, our causal model will be
identifiable only through additional assumptions of conditional independence;
we discuss these assumptions later. The key point to note here is that,
generally, $E(Y^{*}\mid\mathbf{X})\neq E(Y\in\mathbf{Y}\mid\mathbf{X})$, i.e.
predictions using error-prone proxies as labels, $\hat{Y}$, will, of course,
differ from the $\hat{Y}^{*}$ that would have been obtained had the true
labels been available.

##  3 Related work

A large and growing literature on fairness of predictions for the error-free
outcome $Y^{*}$ exists, with divergent and sometimes mutually exclusive
definitions of the notion of algorithmic fairness. An excellent overview of
this literature can be found in [32], which identified 20 separate
definitions. Broadly, a distinction can be made between statistical metrics,
distance-based measures, and causal reasoning [32].

Statistical metrics define fairness as the presence or absence of a
(conditional) independence in the joint distribution $p(Y^{*},\hat{Y}^{*},S)$.
For example, take a classification problem in which the decision is taken as
$d:=I(\hat{Y}^{*}>\tau)$, where $I$ is the indicator function and $\tau$ is
some threshold on the predicted score. _Statistical parity_ (“group fairness”)
is then defined as $p(d=1\mid S=s)=p(d=1\mid S=s^{\prime})$ for all $s\neq
s^{\prime}$, i.e., the decision should not depend on the sensitive attribute,
whereas _predictive parity_ is defined as $p(Y^{*}=1\mid
d=1,S=s)=p(Y^{*}=1\mid d=1,S=s^{\prime})$ for all $s\neq s^{\prime}$—i.e. the
positive predictive value should not depend on the sensitive attribute.
Further definitions include conditional statistical parity [7], overall
accuracy equality [1], and well calibration [16].

Distance-based measures of fairness account for the non-sensitive predictors
$\mathbf{X}\setminus S$, in addition to the observed and predicted outcomes
and sensitive attribute. The well-known “fairness through awareness” framework
[8] generalises several of the preceding notions, such as statistical parity,
by defining fairness as “similar decisions for similar people”. Consider a
population of potential applicants $P$, and consider any randomised output
from the prediction algorithm, $M(x\in P)$. Fairness is achieved whenever the
distance among the decisions $M$ made for two people is at least as small as
the distance between these people, i.e. when $D(M(x),M(y))\leq d(x,y)$ for any
$x,y\in P$. Here, $D$ and $d$ are arbitrary metrics on the distance between
outputs and people, respectively. Careful choice of these metrics can yield
some of the above definitions as special cases. Since the fairness condition
can be trivially achieved, for example by always outputting a constant
regardless of the input, the prediction model should be trained by minimising
a loss function under the above constraint.

Finally, in recent years, results from the causal modelling literature have
been leveraged to define and achieve “counterfactual” fairness [17, 22]. In
these definitions one first considers a causal model involving $Y$,
$X\setminus S$, and $S$ such as Panel A of Figure 1. This causal model then
induces a counterfactual distribution $p_{do(s)}(\hat{Y}^{*}\mid X)$, i.e. the
distribution we would observe if $S$ were _set_ to the value $s$ [24]. [17]
then defined counterfactual fairness as $p_{do(s)}(\hat{Y}^{*}\mid
X)=p_{do(s^{\prime})}(\hat{Y}^{*}\mid X)$. Note that this definition looks
superficially similar to the definition of statistical parity (group
fairness), but is distinct because it refers to an individual. This definition
has as a disadvantage that _any_ causal effect of the sensitive attribute on
the prediction is deemed illegitimate. Based on the same framework, [22]
suggested a more general definition: some causal _pathways_ originating in $S$
are denoted discriminatory, while others are not. Fairness is then achieved by
performing inference on a distribution $p^{*}(Y^{*},\mathbf{X})$, in which the
“fair world” distribution $p^{*}(Y^{*},\mathbf{X})$ is close in a Kullback-
Leibler sense to the original $p(Y^{*},\mathbf{X})$, but all discriminatory
pathways have been blocked (up to a tolerance) using standard causal inference
techniques. Note that, if all causal pathways originating in $S$ are deemed
discriminatory and the tolerance set to zero, the counterfactual fairness
criterion by [17] will be satisfied.

Figure 1: Graphical representation of causal relations between the sensitive
feature ($S$), the predictors (${\bf X}$), and the error-prone outcome
($\hat{Y}^{*}$) in the naive case (A), in the measurement error framework (B),
and in the measurement error framework with differential item functioning on
the $Y_{1}$ proxy (C). The dotted arrow indicates the discriminatory causal
pathway (as in [22]) which is blocked when performing fair inference,
evaluating $E[Y^{*}\mid{\bf X},S]$ to compute a risk score $\hat{Y}^{*}$.

##  4 Proposed framework

###  4.1 Fair inference in error-prone outcomes

The existing methods from Section 3 do not consider the target $Y^{*}$ to be
error-prone. However, in practice, the target feature $Y\in{\bf Y}$ in the
data set is not a perfect representation of the true underlying outcome
$Y^{*}$. There can be several sources for this imperfect representation. For
example, the true underlying outcome of interest may not be directly
measurable at all (i.e., $Y^{*}\neq Y$ for any possible $Y$). In this case,
the outcome of interest will only partially explain any feature used as its
proxy. For example, in using healthcare costs $Y$ as a proxy for health
$Y^{*}$, the observed value will in part be determined by other factors
besides $Y^{*}$, such as the location of residence of the patient. Then, even
if the outcome of interest were “true healthcare costs” – thus in principle
measurable – the observed feature will in practice still not be an infallible
proxy, because health records are never perfect observations and always
contain some form of noise [6]. Together, such sources of noise in the
observation process are termed “measurement error”, and any outcome $Y^{*}$
containing measurement error can be considered _latent_ [5] and modelled as
such.

Crucially, the presence of measurement error may result in unfair inferences
for the error-prone outcome, even after applying the procedures presented in
Section 3 to account for unfairness. This is shown in a compelling example by
[23], who concluded that commercial algorithms used by insurance companies for
patient referral contain a fundamental racial bias. In the algorithm under
consideration, healthcare costs $Y\in{\bf Y}$ are used as a proxy for health
$Y^{*}$. [23] illustrated that although there is no bias in healthcare costs,
there is strong racial bias in other proxies of health such as whether
patients have chronic conditions. Specifically, in order to be referred to a
primary care physician, the true underlying health status $Y^{*}$ of black
patients was worse than that of white patients.

[23] concluded that fair inference requires selecting a better proxy for
health as the outcome variable $Y^{*}$. Indeed, their analyses were possible
precisely due to the availability of different proxies of health, such as the
number of chronic conditions. However, we note that solving racial bias in a
new proxy does not guarantee the absence of racial bias in other proxies
indicating other aspects of health. Instead, here we suggest incorporating
several proxies, or _indicators_ ${\bf Y}$ in a measurement model for the
unobserved, error-prone outcome $Y^{*}$ [15]. In the next section, we
introduce the existing literature on measurement models and its approach to
fair inference.

###  4.2 Fair inference in measurement models

When outcomes are thought to be error-prone, an existing literature suggests
the use of measurement models [10, 6]. At their core, measurement models
describe the causal relationship between observed scores ${\bf Y}$ and error-
prone unobserved “true scores” $Y^{*}$ as $Y^{*}\to{\bf Y}$. A measurement
model adequately represents the empirical conditions of measurement if
conditional independence can be assumed [2]. More specifically, measurement
models assume that $Y_{1}$ and $Y_{2}$ are conditionally independent given
$Y^{*}$ (i.e., $p(Y_{1},Y_{2}\mid Y^{*})=p(Y_{1}\mid Y^{*})p(Y_{2}\mid
Y^{*})$). A plethora of variations of measurement models assuming conditional
independence have been developed, such as latent class models [19], item
response models [25], mixture models [20], factor models [18], structural
equation models [3], and generalized latent variable models [27].

Measurement models are suggested here as a convenient way to account for a
latent variable’s relationship to sensitive features. The measurement error of
a proxy variable (e.g. $Y_{1}$) is then assumed to differ over different
groups of $S$. To account for group differences in proxy variables, a large
body of literature is available where this issue is known under different
labels. Generally, these approaches are applied within the structural equation
modelling (SEM) framework [14], as SEM explicitly separates the measurement
model $(Y^{*}\to{\bf Y})$ from the structural model $({\bf X}\to Y^{*})$.
Approaches for investigating how features $S$ influence $Y^{*}$ are
investigating item bias [21], Differential Item Functioning (DIF) [12] and
measurement invariance [26]. For an extensive overview of the different
approaches and their benefits and drawbacks, we refer to [9, 26, 29, 31].

###  4.3 Proposed method for fair inference on latent variables

We propose our framework for fair inference on outcomes which are measured
only through error-prone proxies. To clarify the framework and make it more
comparable to earlier work, we use the running example of health risk score
prediction from [23]. Their healthcare data set contains several clinical
features ${\bf X}$ at time point $t-1$ (e.g., age, gender, care utilisation,
biomarker values and comorbidities) which are used to predict healthcare cost
$Y^{*}$ at time $t$. In addition, the patient’s race is the sensitive feature
$S$, coded as $S=b$ for black patients and $S=w$ for white patients. The
relations between these features are shown in panel A of Figure 1.

Based on ${\bf X}$, the expectation of a persons’ healthcare cost is used as a
risk score $\hat{Y}^{*}:=E[Y^{*}\mid{\bf X},S]$. The risk score is used to
make a decision $D$ to refer a patient to their primary care physician to
consider program enrolment. More specifically $d=1$ if $\hat{Y}^{*}$ is above
the $55^{th}$ percentile. In this setting, attributes ${\bf X}$ can be
legitimately controlled. However, conditional on ${\bf X}$ both groups in $S$
should have equal probability of being referred: $\text{P}(d=1\mid{\bf
X}=x,S=b)=\text{P}(d=1\mid{\bf X}=x,S=w)$. As mentioned in Section 4.1 and
shown by [23], this procedure leads to bias in other proxies of $Y^{*}$, such
as a patient’s number of chronic conditions.

Our proposed framework is a SEM implementation of the second and third panels
of Figure 1. The general structure of the model is that of a Multiple
Indicator, Multiple Causes (MIMIC) model: the outcome variable $Y^{*}$ (e.g.,
health) has multiple proxy indicators (e.g., chronic conditions, healthcare
costs, hypertension), and the ${\bf X}$ features predict $Y^{*}$ directly
(thus the proxies only indirectly). A graphical representation of the MIMIC
SEM model is shown in Figure 2. This implementation imposes additional
assumptions on the general causal graphs, most notably linear relationships
between the variables and the multivariate Gaussian residuals.

Figure 2: Structural equation model for the proposed framework on the
healthcare data set. For clarity, residual variances of the endogenous
variables are not drawn in the diagram. For more information on the variables
used in the model, see [23].

Fair inference on $Y^{*}$ can be performed in the following way: during
estimation of the regression parameters (${\bf X}\to Y^{*}$), health is
conditioned on race, but during prediction the path from Race to Health is
blocked by setting $S=b$. Following the notation of [22], this yields a “fair
world” distribution $p^{*}(Y^{*},{\bf X})$. The expectation
$\hat{Y}^{*}=E[Y^{*}\mid{\bf X},S]$ is then computed from this distribution,
meaning for two participants who differ only on $S$ but not on ${\bf X}$, the
risk score $\hat{Y}^{*}$ will be exactly the same. Because the latent outcome
$Y^{*}$ is modelled as a linear combination of the different proxies, the risk
score is a reflection of the underlying health rather than only health cost.

##  5 Experiments

In this section, we evaluate the proposed framework on an application of the
procedures discussed in this paper. We first prepare the data set as provided
by [23] to create a basic risk score based on healthcare cost similar to the
commercial risk score reported in their paper. Then, we illustrate our
argument from Section 4.1: we perform fair inference on the proxy measure for
health (healthcare cost) to show that this does not solve the issue of
unfairness in other proxy measures. This is a reproduction of the results
shown by [23]. Next, we use the SEM framework from Section 4.3 to show how
including a formal measurement model for $Y^{*}$ – as in panel B of Figure 1 –
can largely solve the issue of unfairness in the proxies. Last, we show how
existing differential item functioning (DIF) methods in the SEM framework –
panel C of Figure 1 – can aid in interpreting the extent to which proxy
measures contain unfairness. Fully reproducible R code for this section is
available as supplementary material to this paper at the following DOI:
10.5281/zenodo.3708150.

###  5.1 Data preparation and feature selection

Log-transformations are applied to highly skewed variables at time-point $t$,
such as costs, to meet the assumption of normally distributed residuals in
regression procedures. As an additional normalisation step, the predictors at
time-point $t-1$ are re-scaled to homogenise their levels of variance. The
data set is then split into a training and a test set. In this section,
estimation is always done on the training set and inference is done on the
test set.

To simplify our proposed framework for the purpose of this application, we
select a subset of features at time-point $t-1$ for prediction of the target
of interest at time point $t$, health. We want our procedure to be comparable
to the commercial algorithm which produces the risk scores described in [23].
If the features we select are the same features used by the commercial
algorithm, then our procedure would yield very similar results upon generating
a risk score. Unfortunately, the predicted risk scores used by [23] cannot be
replicated exactly using the provided data set.

To select the subset of predictor features for further use in our procedure,
we performed a LASSO regression [30] where all available features at time-
point $t-1$ are used as predictor variables, and the provided algorithmic risk
score at time-point $t$ is used as a target. Following the guidelines by [11],
we used cross-validation to select the optimal $\lambda$ penalty value. This
yields a set of non-zero predictors which predict the algorithmic risk score
well.

Spearmans rank correlation between the commercial and the replicated risk
score is high $\rho=.82$, indicating that the commercial and replicated risk
scores perform similarly in the rank-based cutoff applied in [23]. The
predictors selected in this model are used as predictors $\boldsymbol{X}$ in
the structural equation models of the following sections.

###  5.2 Fair inference on cost as a proxy of health

Pane A of Figure 1 illustrates conditional statistical parity as defined by
[32]. Here, the outcome $Y^{*}$ is conditioned on sensitive feature $S$ when
estimating the coefficients of the prediction model ($\boldsymbol{X}\to
Y^{*}$), and during prediction all subjects are assumed to have the same level
of $S$, e.g., $S=b$, such that $\text{P}(Y^{*}=y^{*}\mid{\bf
X}=x,S=b)=\text{P}(Y^{*}=y^{*}\mid{\bf X}=x,S=w)$.

Accounting for sensitive feature ‘Race’ by conditioning the outcome
‘Replicated Risk score’ on ‘Race’ when estimating the model and being excluded
during prediction reduces the extent of the problem. Figure 3 illustrates that
although the results improve compared to not including ‘Race’ at all,
conditional statistical parity is still not met. As a consequence, individuals
belonging to $S=b$ will still have a lower health status when being selected
for intervention.

Figure 3: Effect of parity correction in one proxy of health (healthcare cost)
on the race differences in another proxy of health (the number of chronic
conditions). From the replicated risk score to the parity-corrected risk
score, the cross-race difference becomes slightly smaller but does not
disappear.

###  5.3 Fair inference on latent health

A cause for the fact that conditional statistical parity is not met when
following Pane A of Figure 1 can be that $\hat{Y}^{*}$ is a (bad) proxy.
Instead of using one bad proxy, it is better to use multiple (bad) proxies as
indicators of an unobserved latent variable measuring ‘true health’. How such
a model can be specified is illustrated in Pane B of Figure 1. Similarly to
[32], the sensitive feature is excluded during prediction.

Figure 4: Effect of including a measurement model in constructing risk scores.
The first panel shows the uncorrected risk score based on healthcare cost, the
middle panel shows the same risk score but corrected for the sensitive
feature, and the third panel shows the corrected risk score based on the
latent health outcome using a measurement model.

Figure 4 shows the effect of including a measurement model in constructing
risk scores. This figure illustrates that using a measurement model with
multiple imperfect measurements of health as indicators for ‘true health’
substantially improves conditional statistical parity. The improvement has
been more compared to accounting for the sensitive feature. By using this
measurement model, the problem that individuals belonging to $S=b$ had a lower
health status when being selected for intervention is minimised.

###  5.4 Investigating unfairness in proxies

When using a measurement model with multiple imperfect measurements of health
as indicators of ‘true health’, differences in measurement error over the
different groups of the sensitive feature can still be present. Panel C of
Figure 1 illustrates how differences over the sensitive feature groups in the
error prone indicator variables can be incorporated directly when estimating
‘true health’. For example, differences in measurement error of healthcare
cost can be present for the different groups of Race.

Including a DIF parameter $\delta$ on the healthcare cost variable yields a
model which fits significantly better on the test set than the model without
the DIF parameter ($\chi^{2}(1)=50$, $p<0.001$). The value of the DIF
parameter on cost is estimated as $\delta=0.198$ (95% CI = $[0.172,0.225]$).
This means that for the same level of health, the log-healthcare costs of the
white race class in this data set is estimated to be 0.198 higher. This means
that the cost of healthcare for white patients is $(e^{0.198}-1)\cdot
100\%=21.9\%$ higher than that for black patients, _given an equal level of
health_ as measured by the measurement model (95% CI = $[18.7,25.2]$).

Applying the same procedure to the other indicators leads to estimates of DIF
for those indicators. The results are shown in Table 1. This table shows that
some proxies have stronger DIF than others, meaning some proxies are more
unfair than other proxies. Notable, the avoidable healthcare cost and the
renal failure items have low levels of DIF for Race, whereas the healthcare
cost and the number of active chronic conditions have strong DIF.

Table 1: Estimated differential item functioning parameters for each indicator (proxy) of health. $\delta$ parameters should be interpreted as the mean deviation of the black patients compared to the white patients given health.  Indicator | $\delta$ | 2.5% | 97.5%
---|---|---|---
No. active chronic conditions | 0.453 | 0.364 | 0.541
Mean blood pressure | -0.262 | -0.320 | -0.204
Diabetes severity (HbA1c) | -0.343 | -0.391 | -0.296
Anemia severity (hematocrit) | 0.250 | 0.231 | 0.268
Renal failure (creatinine) | -0.019 | -0.025 | -0.014
Cholesterol (mean LDL) | -0.235 | -0.317 | -0.153
Healthcare cost (log) | 0.198 | 0.172 | 0.225
Avoidable healthcare cost (log) | -0.052 | -0.096 | -0.008

##  6 Conclusion

In this paper, we have argued that when measurement error is at play,
performing fair inference on a proxy measure of the outcome is insufficient to
achieve a fair inference on the true outcome. This manifests itself, as shown
in [23], as unfairness in other proxy measures of the outcome of interest.
Alternatively, in this study we proposed to make use of existing measurement
models containing multiple error-prone proxies for the outcome of interest. In
addition, fair inference can be accounted for in each of these proxies
simultaneously if needed by allowing for measurement error in proxies to
differ over groups of a sensitive feature. We provided a framework to perform
these estimations and applied this framework to the exemplary data set
provided by [23]. Here, it was concluded that fair inference was accounted for
when multiple proxies were used in a measurement model instead of a single
proxy. Additionally accounting for differences in measurement error over race
groups was not needed to further improve fairness in predicted risk scores,
although substantive group differences were found for some proxies.

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# Normal form for transverse instability of gZK equation for the line soliton
with nearly critical speed.

Yakine Bahri and Hichem Hajaiej

###### Abstract

In this paper, we study the transverse instability of generalized Zakharov-
Kuznetsov equation for the line soliton with critical speed. We derive and
justify a normal form reduction for a bifurcation problem of the stationary
nonlinear KdV equation on the product space $\mathbb{R}\times\mathbb{T}$.

##  1 Introduction

We consider the generalized Zakharov–Kuznetsov (gZK) equation with a general
power non-linearity

$u_{t}+\partial_{x}(u^{k+1})+u_{xxx}+u_{xyy}=0,\quad(x,y)\in\mathbb{R}\times\mathbb{T},$
(1.1)

where $k\geq 2$. It is well-known that the following quantities are conserved
for smooth solutions of the ZK equation :

$M(u)=\int u^{2}dxdy,$ (1.2)

and

$H(u)=\frac{1}{2}\int\big{(}|\nabla u|^{2}-\frac{1}{k+2}u^{k+2}\big{)}dxdy.$
(1.3)

The gZK equation admits a traveling wave solution. A traveling wave with speed
$c$, is a particular solution of the form $u(t,x,y)=u_{c}(x-ct,y)$ where
$u_{c}$ is a non-trivial solution to the following stationary equation,

$-\partial_{xx}u_{c}-\partial_{yy}u_{c}+cu_{c}-u_{c}^{3}=0\qquad\mbox{in}\quad\mathbb{R}\times\mathbb{T}.$
(1.4)

The main part of our work covers the instability of the line soliton over the
2D flow. We recall the definition of the orbital stability.

######  Definition 1.1.

We say that a solitary wave $Q(x-ct,y)$ is orbitally stable in
$H^{1}\left(\mathbb{R}\times\mathbb{T}_{L}\right)$ if for any $\varepsilon>0$
there exists $\delta>0$ such that for all initial data $u_{0}\in
H^{1}\left(\mathbb{R}\times\mathbb{T}_{L}\right)$ with
$\left\|u_{0}-Q\right\|_{H^{1}}<\delta$, the solution $u(t)$ of $(1.1)$ with
$u(0)=u_{0}$ exists globally in positive time and satisfies

$\sup_{t>0}\inf_{\left(x_{0},y_{0}\right)\in\mathbb{R}\times\mathbb{T}_{L}}\left\|u(t,\cdot,\cdot)-Q\left(\cdot-
x_{0},\cdot-y_{0}\right)\right\|_{H^{1}}<\varepsilon$

Otherwise, we say the solitary wave $Q(x-ct,y)$ is orbitally unstable in
$H^{1}\left(\mathbb{R}\times\mathbb{T}_{L}\right)$.

In the multidimensional case, de Bouard [4] showed the orbital stability of
positive solitary waves of the generalized Zakharov–Kuznetsov equation while
assuming the well-posedness on the energy space. For the two dimensional case,
Côte, Muñoz, Pilod and Simpson [3] have proved the asymptotic stability of
positive solitary waves and multi-solitary waves of the Zakharov–Kuznetsov
equation using the argument of Martel and Merle [6]. A numerical study for
two-dimensional solitary wave interactions and the formation of singularities
in the modified Zakharov–Kuznetsov (mZK) equation was considered in [17]. This
is a generalization of the Korteweg–deVries equation discussed in [15].

Note that, $u_{c}(x,y)=u_{c}(x)$ is also a solution in the energy space. In
this case, $u_{c}$ is the unique positive solution to

$-\partial_{xx}u_{c}+cu_{c}-u_{c}^{k+1}=0\qquad\mbox{in $\mathbb{R}$}.$ (1.5)

We recall that the explicit form of the solution is given by

$u_{c}(x):=c^{\frac{1}{k}}\left(\frac{k+2}{2}\right)^{\frac{1}{k}}\operatorname{sech}^{\frac{2}{k}}\left(\frac{k}{2}\sqrt{c}\
x\right).$ (1.6)

The stability of this traveling wave with respect to the KdV flow (the one-
dimensional version of ZK) has been fully described in the literature. The
orbital stability was proven to occur only for $k<4$ (see [1, 13] for more
details). In addition, Martel and Merle proved in [6] the asymptotic stability
in the energy space. The instability holds for any $k\geq 4$. We refer the
reader to [2] for $k>4$ and to [7, 8, 9, 10] for the $L^{2}$ critical case
$k=4$. We conclude that the line traveling wave $u_{c}$ is unstable under the
ZK flow perturbation for $k\geq 4$. For this, we restrict our study to $2\leq
k<4$.

For $k=1$, Yamazki studied in [18] the orbital and asymptotic transverse
stability. He proved that the results depend on the speed of the traveling
wave. More precisely, he showed that only traveling waves with subcritical as
well as critical speed are asymptotically stable, and are unstable with the
super-critical speed under a $2$D perturbation. He used the method of Evans’
function and the argument of Rousset and Tzvetkov [16]. The asymptotic
stability for orbitally stable line solitary waves of Zakharov–Kuznetsov
equation was proved by using the argument of Martel and Merle [6] and a
Liouville type theorem combined with a modified virial type estimate since the
linearized operator of the stationary equation is degenerate. In [20],
Yamazaki constructed center stable manifolds around unstable line solitary
waves with critical speed. To recover the degeneracy of the linearized
operator around line solitary waves with critical speed, the auther proved the
stability condition of the center stable manifold for critical speed by
applying to the estimate of the 4th order term of a Lyapunov function in [18]
and [19]. In addition, Pelinovsky proved in [14] the same results using a
normal form argument. The normal form was derived by means of symplectic
projections and near-identity transformations. The justification of this
normal form is provided with an energy method. More precisely; the main result
of the work was to derive and to justify the first-order differential equation

$\frac{db}{dt}=\lambda^{\prime}\left(c_{*}\right)\left(c-c_{*}\right)b+\gamma|b|^{2}b,\quad
t>0$ (1.7)

where $\lambda^{\prime}\left(c_{*}\right)>0,\gamma<0$ are real-valued
numerical coefficients, $c_{*}$ is the critical speed of the line soliton,
$c\in\mathbb{R}$ depends on the initial conditions, and
$b(t):\mathbb{R}_{+}\rightarrow\mathbb{C}$ is an amplitude of transverse
perturbation. The above differential equation describes the nonlinear dynamics
of a small transverse perturbation of a fixed period to the line soliton with
a nearly critical speed $c_{*}$ and is referred to as ”normal form for
transverse instability of the line soliton with a nearly critical speed of
propagation”.

In this paper, our main goal is to derive the normal form (1.7) corresponding
to (1.1). We will show that in our case that $\gamma>0$ which implies
instability of the line soliton with a nearly critical speed $c_{*}$. In the
Appendix, we will prove a conjecture stated by Pelinovsky in [14] for the
quadratic case ($k=1$).

###  1.1 Preliminaries

For the sake of simplicity, we will restrict our self to $k=2$ while tackling
the transverse instability. Note that, our argument applies to any
$k\in[2,4)$. Let us consider

$u_{t}+\partial_{x}(u^{3})+u_{xxx}+u_{xyy}=0,\quad(x,y)\in\mathbb{R}\times\mathbb{T},$
(1.8)

The one-dimensional linearized operator around the line soliton is given by

$L_{c}:=-\partial^{2}_{x}+c-3u_{c}^{2}.$ (1.9)

The Schrödinger operator $L_{c}:H^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ is
known [5] to have the kernel spanned by $u^{\prime}_{c}$, the essential
spectrum located on $[c,\infty)$ and a simple isolated negative eigenvalue

$\lambda=-3c,\quad\psi=u_{c}^{2}.$

Let us consider the $2\pi$-periodic transverse perturbation to the line
solitons (1.6). The linearized solution around the soliton
$u(x,t)=u_{c}(\xi)+U(\xi)e^{\lambda t+iky}$ with $\xi:=x-ct$ and
$k\in\mathbb{Z}$ provided the following spectral problem at the linear level

$\partial_{\xi}(L_{c}+k^{2})U=\lambda U,\quad k\in\mathbb{Z},$ (1.10)

where $\partial_{\xi}L_{c}:H^{3}(\mathbb{R})\to L^{2}(\mathbb{R})$ is the
linearized operator for the gKdV equation.

We recall that the spectrum of $\partial_{\xi}L_{c}:H^{3}(\mathbb{R})\to
L^{2}(\mathbb{R})$ consists of a double zero eigenvalue and a continuous
spectrum on $i\mathbb{R}$ (see [12] for more details). The double zero
eigenvalue is associated with the following Jordan block of the operator
$\partial_{\xi}L$:

$\partial_{\xi}L_{c}\partial_{\xi}u_{c}=0,\quad\partial_{\xi}L_{c}\partial_{c}u_{c}=-\partial_{\xi}u_{c},$
(1.11)

where the derivatives of $u_{c}$ in $\xi$ and $c$ are exponentially decaying
functions of $\xi$. The following lemma characterizes the spectral problem
(1.10) for any $k\in\mathbb{N}$.

######  Lemma 1.1.

For any $n\in\mathbb{N}$, the spectral problem (1.10) has a pair of real
eigenvalues $\pm\lambda_{k}(c)$ if $c>c_{n}:=\frac{n^{2}}{3}$. No eigenvalues
with ${\rm Re}(\lambda)\neq 0$ exist if $c\in(0,c_{n})$.

Note that, it is possible to show nonlinear orbital stability of the line
soliton with $c\in(0,c_{*})$ and nonlinear instability of the line soliton
using energy method with $c>c_{*}$, where

$c_{*}:=\min\limits_{n\in\mathbb{N}}c_{n}\equiv\frac{1}{3}.$

In this work, we will consider the critical speed case i.e. $c=c_{*}$. We
first study the spectral problem (1.10) in the exponentially weighted space (
see [12, 13] for more details)

$H^{s}_{\mu}(\mathbb{R})=\left\\{u\in H^{s}_{\rm loc}(\mathbb{R}):\quad
e^{\mu\xi}u\in H^{s}(\mathbb{R})\right\\},\quad s\geq 0,\quad\mu>0.$ (1.12)

The following lemma reviews the spectral property of (1.10).

######  Lemma 1.2.

There is $\mu_{0}>0$ such that for every $\mu\in(0,\mu_{0})$, the spectral
problem (1.10) with $k=1$ and $c=c_{*}$ considered in
$L^{2}_{\mu}(\mathbb{R})$ admits a simple zero eigenvalue with the
eigenfunction $\psi_{*}\in H^{3}_{\mu}(\mathbb{R})$ and the adjoint
eigenfunction $\eta_{*}\in H^{3}_{-\mu}(\mathbb{R})$, where

$\psi_{*}(\xi)=2c_{*}{\rm
sech}^{2}(\sqrt{c_{*}}\xi),\quad\eta_{*}(\xi)=2c_{*}\int_{-\infty}^{\xi}{\rm
sech}^{2}(\sqrt{c_{*}}\xi^{\prime})d\xi^{\prime},$ (1.13)

hence,

$\langle\eta_{*},\psi_{*}\rangle_{L^{2}}=\frac{(2c_{*})^{2}}{2}\left(\int_{\mathbb{R}}{\rm
sech}^{2}(\sqrt{c_{*}}\xi)d\xi\right)^{2}=8c_{*}.$ (1.14)

Moreover, for a given $\mu\in(0,\mu_{0})$, there exists an interval
$(c_{-},c_{+})$ with $c_{-}<c_{*}<c_{+}$ such that the spectral problem (1.10)
with $k=1$ and $c\in(c_{-},c_{+})$ considered in $L^{2}_{\mu}(\mathbb{R})$
admits a small eigenvalue $\lambda(c)$, where the mapping $c\mapsto\lambda$ is
smooth and is given by

$\lambda(c)=\lambda^{\prime}(c_{*})(c-c_{*})+\mathcal{O}((c-c_{*})^{2})\quad\mbox{\rm
as}\quad c\to c_{*},$ (1.15)

with

$\lambda^{\prime}(c_{*})=2\sqrt{c_{*}}.$ (1.16)

###  1.2 Instability result

We use a modulation argument to construct a translation parameter $a(t)$ and a
speed parameter $c(t)$ of the line solitons as well as its perturbation
$\tilde{u}(t)$ defined in $H^{1}(\mathbb{R}\times\mathbb{T})\cap
H_{\mu}^{1}(\mathbb{R}\times\mathbb{T})$ for every $t\in[0,T)$. This allows us
to consider the traveling coordinate $\xi=x-a(t)$ and use the decomposition

$u(x,y,t)=u_{c(t)}(\xi)+\tilde{u}(\xi,y,t),\quad\xi=x-4a(t)$

The time evolution of the varying parameters $a(t)$ and $c(t)$ and the
perturbation term $\tilde{u}(t)$ are to be found from the evolution problem

$\tilde{u}_{t}=\partial_{\xi}\left(L_{c}-\partial_{y}^{2}+4(\dot{a}-c)\right)\tilde{u}+4(\dot{a}-c)\partial_{\xi}u_{c}-\dot{c}\partial_{c}u_{c}-6\partial_{\xi}\tilde{u}^{2}$

where the differential expression for $L_{c}$ is the linearized operator
around the line soliton (see (1.9) for more details).

The following theorem represents the normal form for transverse instability of
the line soliton with a nearly critical speed of propagation.

######  Theorem 1.1.

Consider the Cauchy problem for the evolution equation (1.1) with

$\tilde{u}(0)\in H^{1}(\mathbb{R}\times\mathbb{T})\cap
H_{\mu}^{1}(\mathbb{R}\times\mathbb{T}),$

where $\mu>0$ is sufficiently small. There exist $\varepsilon_{0}>0$ and
$C_{0}>0$ such that if the initial data satisfy the bound

$\left\|\tilde{u}(0)-2\varepsilon\cos(y)\psi_{*}\right\|_{H^{1}(\mathbb{R}\times\mathbb{T})\cap
H_{\mu}^{1}(\mathbb{R}\times\mathbb{T})}+\left|c(0)-c_{*}\right|\leqslant\varepsilon^{2},$

for some $\varepsilon\in\left(0,\varepsilon_{0}\right)$, then there exist
unique functions $a,b,c\in C^{1}\left(\mathbb{R}_{+}\right),$ and the unique
solution

$\tilde{u}(t)\in C\left(\mathbb{R}_{+};H^{1}(\mathbb{R}\times\mathbb{T})\cap
H_{\mu}^{1}(\mathbb{R}\times\mathbb{T})\right)$

of the evolution equation (1.1) satisfying the bound

$\left\|\tilde{u}(t)-\left(b(t)e^{iy}+\bar{b}(t)e^{-iy}\right)\psi_{*}\right\|_{H^{1}(\mathbb{R}\times\mathbb{T})}+\left|c(t)-c_{*}\right|+|\dot{a}(t)-c(t)|\leqslant
C_{0}\varepsilon^{2},\quad t\in\mathbb{R}_{+}.$

Furthermore, the function $b(t)$ satisfies the normal form

$\dot{b}=\lambda^{\prime}\left(c_{*}\right)\left(c_{+}-c_{*}\right)b+\gamma|b|^{2}b,\quad
t\in\mathbb{R}_{+},$

with $b(0)=\varepsilon$ and $c_{+}\in\mathbb{R}$ satisfying
$\left|c_{+}-c_{*}\right|\leqslant C_{0}\varepsilon^{2}$, where
$\lambda^{\prime}\left(c_{*}\right)>0$ is given by (2.13) and $\gamma>0$ is a
specific numerical coefficient given by (2.51) below. Consequently,
$|b(t)|>\frac{C_{0}}{\varepsilon}$ for every $t\in\mathbb{R}_{+}$.

##  2 Transverse Instability

###  2.1 Transversely modulated solitary waves

In this section, we construct a solution to (1.4) which bifurcates from the
line soliton with critical speed. We define the space of even functions both
in $\xi$ and $y$:

$H^{s}_{\rm even}=\left\\{u\in H^{s}(\mathbb{R}\times\mathbb{T}):\quad
u(-\xi,y)=u(\xi,y)=u(\xi,-y)\right\\},\quad s\geq 0.$ (2.1)

The following lemma describes this bifurcation.

######  Lemma 2.1.

There exists $c_{+}>c_{*}$ such that for every $c\in(c_{*},c_{+})$, the
nonlinear elliptic problem (1.4) has a nontrivial solution $u_{b}$ in
$H^{2}_{\rm even}$ in addition to the line soliton (1.6). The solution $u_{b}$
is expressed by the expansion

$u_{b}(\xi,y)=u_{c_{*}}(\xi)+b\cos(y)\psi_{*}(\xi)+\tilde{u}_{b}(\xi,y),$
(2.2)

where $b\in\mathbb{R}$ is a nonzero root of the algebraic equation

$\alpha(c-c_{*})b+\beta b^{3}=0$ (2.3)

and $\tilde{u}_{b}\in H^{2}_{\rm even}$ satisfies the bound
$\|\tilde{u}_{b}\|_{H^{2}}\leq Ab^{2}$ for a positive constant $A$
independently of $b$ and $c$. Here

$\alpha=8(c_{*})^{\frac{3}{2}}>0$

and $\beta<0$ is a numerical coefficient given by (2.16) below.

###### Proof.

The proof is similar to the one of Lemma 1.3 in [14] and relies on the method
of Lyapunov–Schmidt reduction [11]. We denote by
$v_{1}(\xi,y):=\cos(y)\psi_{*}(\xi)$ the eigenfunction of the kernel of
$L_{c_{*}}-\partial_{y}^{2}:H^{2}_{\rm even}\to L^{2}_{\rm even}$. Let
$c=c_{*}+\delta$ with $\delta\in\mathbb{R}$ being sufficiently small. We plug
the decomposition (2.2) in (1.4), to obtain the projection equations of the
Lyapunov–Schmidt reduction method:

$(L_{c_{*}}-\partial_{y}^{2}+\delta)\tilde{u}_{b}=\Pi\tilde{F},$ (2.4)

with

$\tilde{F}:=-\delta u_{c_{*}}-\delta
bv_{1}+(bv_{1}+\tilde{u}_{b})^{3}+3u_{c_{*}}(bv_{1}+\tilde{u}_{b})^{2}$ (2.5)

and $\Pi$ is an orthogonal projection operator in
$L^{2}(\mathbb{R}\times\mathbb{T})$ on the subspace spanned by $v_{1}$. This
means that

$\frac{1}{2\pi}\langle
v_{1},\tilde{F}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}=0,$

which yields

$-\frac{\delta}{2}b\|\psi_{*}\|^{2}_{L^{2}(\mathbb{R})}+\frac{3}{8}b^{3}\|\psi_{*}\|^{4}_{L^{4}(\mathbb{R})}+\frac{3}{\pi}b\langle
v_{1}^{2},u_{c_{*}}\tilde{u}_{b}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}+\mathcal{O}(b^{5})=0.$
(2.6)

Since the kernel of the linear operator $L_{c_{*}}-\partial_{y}^{2}:H^{2}_{\rm
even}\to L^{2}_{\rm even}$ is one-dimensional, we know that there are $B>0$
and $\delta_{0}>0$ such that

$\|\Pi(L_{c_{*}}-\partial_{y}^{2}+4\delta)^{-1}\Pi\|_{L^{2}_{\rm even}\to
L^{2}_{\rm even}}\leq B,\quad\forall|\delta|<\delta_{0}.$ (2.7)

Thus, for every small $b\in\mathbb{R}$ and small $\delta\in\mathbb{R}$, we
apply the fixed-point argument to solve equation (2.4) with (2.5) in
$H^{2}_{\rm even}$ and to obtain a unique $\tilde{u}_{b}\in H^{2}_{\rm even}$
satisfying the bound

$\|\tilde{u}_{b}\|_{H^{2}}\leq A(|\delta|+b^{2}),$ (2.8)

where the positive constant $A$ is independent of $\delta$ and $b$. Now, we
shall derive the algebraic equation (2.3). To do so, we perform a near-
identity transformation

$\tilde{u}_{b}(\xi,y)=b^{2}\cos(2y)w_{2}(\xi)+b^{2}w_{0}(\xi)+\delta\partial_{c}u_{c_{*}}(\xi)+\tilde{w}_{b}(\xi,y),$
(2.9)

where $w_{0}$ and $w_{2}$ are given by

$L_{c_{*}}w_{0}=\frac{3}{2}u_{c_{*}}\psi_{*}^{2}$ (2.10)

and

$(L_{c_{*}}+4)w_{2}=\frac{3}{2}u_{c_{*}}\psi_{*}^{2}.$ (2.11)

On the other hand, $\tilde{w}_{b}$ satisfies the transformed equation

$\displaystyle(L_{c_{*}}-\partial_{y}^{2}+\delta)\tilde{w}_{b}=\Pi\tilde{G},$
(2.12)

with

$\displaystyle\tilde{G}:=-\delta^{2}\partial_{c}u_{c_{*}}-\delta
b^{2}\cos(2y)w_{2}-\delta b^{2}w_{0}-\delta
bv_{1}+6bu_{c_{*}}v_{1}\tilde{u}_{b}+3u_{c_{*}}\tilde{u}_{b}^{2}++(bv_{1}+\tilde{u}_{b})^{3}$

for $\tilde{u}_{b}$ is defined by (2.9). Similarly to the above competition,
for every small $b\in\mathbb{R}$ and $\delta\in\mathbb{R}$, there exists a
unique solution $\tilde{w}\in H^{2}_{\rm even}$ of equation (2.12) satisfying
the bound

$\|\tilde{w}\|_{H^{2}}\leq A(\delta^{2}+|\delta||b|+|b|^{3}),$ (2.13)

where the positive constant $A$ is independent of $\delta$ and $b$.

Substituting the near-identity transformation (2.9) into the bifurcation
equation (2.6) and using the bound (2.13) for the component $\tilde{w}\in
H^{2}_{\rm even}$, we rewrite (2.6) in the equivalent form

$\alpha\delta b+\beta b^{3}+\mathcal{O}(\delta b^{2},b^{4})=0,$ (2.14)

where we have introduced numerical coefficients $\alpha$ and $\beta$ as
follows:

$2\alpha:=-\|\psi_{*}\|^{2}_{L^{2}}+6\langle\psi_{*}^{2},u_{c_{*}}\partial_{c}u_{c_{*}}\rangle_{L^{2}}$
(2.15)

and

$\beta:=\frac{3}{4}\langle\psi_{*}^{2}u_{c_{*}},2w_{0}+w_{2}\rangle_{L^{2}}+\frac{3}{8}\|\psi_{*}\|^{4}_{L^{4}(\mathbb{R})}.$
(2.16)

Plugging (1.6) and (1.13) into (2.15), we get

$\alpha=-\frac{1}{2}\langle\psi_{*},L_{c_{*}}^{\prime}\psi_{*}\rangle_{L^{2}}=8(c_{*})^{\frac{3}{2}},$

where $L_{c_{*}}^{\prime}:=1-6u_{c_{*}}\partial_{c}u_{c_{*}}$. On the other
hand, we are not able to provide an explicit form to $\beta$. This is why we
will determine a bound to it. We first compute $w_{0}$ using (2.10). More
precisely, we have

$w_{0}(\xi)=\frac{1}{2}\left(u_{c_{*}}^{3}-4c_{*}u_{c_{*}}\right).$ (2.17)

For $w_{2}$, since $L_{c_{*}}+4:H^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ is
strictly positive, we just write equation (2.11) as
$w_{2}=\frac{3}{2}(L_{c_{*}}+4)^{-1}(u_{c_{*}}\psi_{*}^{2}).$ . Thus, using
the bound $\|(L_{c_{*}}+4)^{-1}\|\leq\frac{1}{3}$, we obtain

$\beta\leq\frac{3}{2}\langle\psi_{*}^{2}u_{c_{*}},w_{0}\rangle_{L^{2}}+\frac{3}{8}\|\psi_{*}\|^{4}_{L^{4}(\mathbb{R})}+\frac{1}{4}\|\psi_{*}^{2}u_{c_{*}}\|^{2}_{L^{2}}=\frac{2^{6}}{315}(c_{*})^{\frac{7}{2}}\left(-45+\frac{32}{3}\right)<0.$
(2.18)

Here, we have used

$\displaystyle\|\psi_{*}\|^{4}_{L^{4}}=\frac{2^{9}}{35}(c_{*})^{\frac{7}{2}},\quad\langle\psi_{*}^{2}u_{c_{*}},w_{0}\rangle_{L^{2}}=-\frac{2^{10}}{105}(c_{*})^{\frac{7}{2}}\quad\|\psi_{*}^{2}u_{c_{*}}\|^{2}_{L^{2}}=\frac{2^{13}}{315}(c_{*})^{\frac{9}{2}}.$

This finishes the proof of this lemma. ∎

###  2.2 Derivation and justification of the normal form

In this section, we derive and justify the normal form. We follow the steps of
the proof in [14].

####  2.2.1 Modulation equations for parameters $a$ and $c$

We start by stating the modulation theory in the following lemma.

######  Lemma 2.2.

Let $T>0$. There exists $\varepsilon_{0}>0$, $\mu_{0}>0$, and $C_{0}>0$ such
that if $u\in C([0,T),H^{1}(\mathbb{R}\times\mathbb{T})\cap
H^{1}_{\mu}(\mathbb{R}\times\mathbb{T}))$ with $\mu\in(0,\mu_{0})$ is a
solution to the ZK equation (1.8) satisfying

$\varepsilon:=\inf_{a\in\mathbb{R}}\|u(x+a,y,t)-u_{c_{*}}(x)\|_{H^{1}(\mathbb{R}\times\mathbb{T})\cap
H^{1}_{\mu}(\mathbb{R}\times\mathbb{T})}\leq\varepsilon_{0},\quad
t\in\mathbb{R}_{+},$ (2.19)

then there exist $a,c\in C([0,T))$ and $\tilde{u}\in
C([0,T),H^{1}(\mathbb{R}\times\mathbb{T})\cap
H^{1}_{\mu}(\mathbb{R}\times\mathbb{T}))$ such that the decomposition

$u(x,y,t)=u_{c(t)}(\xi)+\tilde{u}(\xi,y,t),\quad\xi=x-a(t)$ (2.20)

holds with $\tilde{u}(t)\in[X_{c(t)}^{*}]^{\perp}$ for every
$t\in\mathbb{R}_{+}$, where

$[X_{c(t)}^{*}]^{\perp}=\left\\{\tilde{u}\in
L^{2}_{\mu}(\mathbb{R}\times\mathbb{T}):\quad\langle
u_{c(t)},\tilde{u}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}=\langle\partial_{\xi}^{-1}\partial_{c}u_{c(t)},\tilde{u}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}=0\right\\}.$
(2.21)

Note that the modulation parameters $a,c\in C^{1}(\mathbb{R}_{+})$ satisfy the
following system

$\displaystyle S\left[\begin{array}[]{c}\dot{c}\\\
\dot{a}-c\end{array}\right]=\frac{1}{2\pi}\left[\begin{array}[]{c}\langle\partial_{c}u_{c},\tilde{u}^{3}+3u_{c}\tilde{u}^{2}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}\\\
\langle\partial_{\xi}u_{c},\tilde{u}^{3}+3u_{c}\tilde{u}^{2}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}\end{array}\right]$
(2.26)

with the coefficient matrix

$S:=\left[\begin{array}[]{cc}\frac{1}{2}(M^{\prime}(c))^{2}-\frac{1}{2\pi}\langle\partial_{\xi}^{-1}\partial^{2}_{c}u_{c},\tilde{u}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}&P^{\prime}(c)+\frac{1}{2\pi}\langle\partial_{c}u_{c},\tilde{u}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}\\\
P^{\prime}(c)-\frac{1}{2\pi}\langle\partial_{c}u_{c},\tilde{u}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}&\frac{1}{2\pi}\langle\partial_{\xi}u_{c},\tilde{u}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}\end{array}\right],$
(2.27)

where $M(c)=\int_{\mathbb{R}}u_{c}(\xi)d\xi$ and
$P(c)=\frac{1}{2}\int_{\mathbb{R}}u_{c}^{2}(\xi)d\xi$. From the expression
(1.6), we obtain $M^{\prime}(c)=0$ and $P^{\prime}(c)=1/\sqrt{c}$.

####  2.2.2 A secondary decomposition for $c=c_{*}$

By Lemma 1.16, if $c=c_{*}=\frac{1}{3}$ and $\mu>0$ is sufficiently small,
then $Y_{c_{*}}={\rm span}\\{\psi_{*}\\}$ is an invariant subspace of
$L^{2}_{\mu}(\mathbb{R})$ for the simple zero eigenvalue of the linearized
operator $\partial_{\xi}(L_{c_{*}}+1):H^{3}_{\mu}(\mathbb{R})\to
L^{2}_{\mu}(\mathbb{R})$. Similarly, $Y_{c_{*}}^{*}={\rm span}\\{\eta_{*}\\}$
is an invariant subspace of $L^{2}_{-\mu}(\mathbb{R})$ for the simple zero
eigenvalue of the adjoint operator
$-(L_{c_{*}}+1)\partial_{\xi}:H^{3}_{-\mu}(\mathbb{R})\to
L^{2}_{-\mu}(\mathbb{R})$. Let us point out the double degeneracy of the
Fourier harmonics $e^{iy}$ and $e^{-iy}$ when general transverse perturbations
are considered.

The following lemma states the secondary decomposition of the solution
$\tilde{u}$ defined in the primary decomposition (2.20).

######  Lemma 2.3.

Under the assumptions of Lemma 2.2, let $\tilde{u}\in
C(\mathbb{R}_{+},H^{1}(\mathbb{R}\times\mathbb{T})\cap
H^{1}_{\mu}(\mathbb{R}\times\mathbb{T}))$ be given by the decomposition (2.20)
and (2.21). There exist $b\in C(\mathbb{R}_{+})$ and $v\in
C(\mathbb{R}_{+},H^{1}(\mathbb{R}\times\mathbb{T})\cap
H^{1}_{\mu}(\mathbb{R}\times\mathbb{T}))$ such that the decomposition

$\tilde{u}(\xi,y,t)=\left(b(t)e^{iy}+\bar{b}(t)e^{-iy}\right)\psi_{*}(\xi)+v(\xi,y,t),$
(2.28)

holds with $v(t)\in[Y_{c(t)}^{*}]^{\perp}$ for every $t\in\mathbb{R}_{+}$,
where

$[Y_{c(t)}^{*}]^{\perp}=\left\\{v\in[X_{c(t)}^{*}]^{\perp}:\quad\langle\eta_{*}e^{iy},v\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}=\langle\eta_{*}e^{-iy},v\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}=0\right\\}.$
(2.29)

We decompose the modulation parameters as

$a(t)=\int_{0}^{t}c(t^{\prime})dt^{\prime}+h(t),\quad c(t)=c_{*}+\delta(t).$
(2.30)

We denote $L_{c}=L_{c_{*}}+\Delta L_{c}$, with $\|\Delta
L_{c}\|_{L^{\infty}}\leq A|c-c_{*}|$ for $|c-c_{*}|$ sufficiently small and
$A$ is a positive constant independent of $c$.

The correction term $v$ in the decomposition (2.28) satisfies the time
evolution equation

$\displaystyle v_{t}$ $\displaystyle=$
$\displaystyle\partial_{\xi}(L_{c_{*}}-\partial_{y}^{2}+\dot{h}+\Delta
L_{c})v+\dot{h}\partial_{\xi}u_{c_{*}+\delta}-\dot{\delta}\partial_{c}u_{c_{*}+\delta}-(\dot{b}e^{iy}+\dot{\bar{b}}e^{-iy})\psi_{*}$
(2.31) $\displaystyle+\partial_{\xi}\left(\dot{h}+\Delta
L_{c}\right)(be^{iy}+\bar{b}e^{-iy})\psi_{*}$
$\displaystyle-\partial_{\xi}\left(\left((be^{iy}+\bar{b}e^{-iy})\psi_{*}+v\right)^{3}+3u_{c}\left((be^{iy}+\bar{b}e^{-iy})\psi_{*}+v\right)^{2}\right).$

For $b\in C^{1}(\mathbb{R}_{+})$ the modulation equation is given by:

$\displaystyle\dot{b}$
$\displaystyle\langle\eta_{*},\psi_{*}\rangle_{L^{2}(\mathbb{R})}+b\langle\psi_{*},(\dot{h}+\Delta
L_{c})\psi_{*}\rangle_{L^{2}(\mathbb{R})}+\frac{1}{2\pi}\langle\psi_{*}e^{iy},\Delta
L_{c}v\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}$
$\displaystyle=\frac{1}{2\pi}\langle\psi_{*}e^{iy},\left[(be^{iy}+\bar{b}e^{-iy})\psi_{*}+v\right]^{3}+3u_{c}\left[(be^{iy}+\bar{b}e^{-iy})\psi_{*}+v\right]^{2}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}.$
(2.32)

On the other hand, plugging (2.28) and (2.30) into the system (2.26), we write
the equations for $h$ and $\delta$:

$\displaystyle S\left[\begin{array}[]{c}\dot{\delta}\\\
\dot{h}\end{array}\right]=\frac{1}{2\pi}\left[\begin{array}[]{c}\langle\partial_{c}u_{c},\left[(be^{iy}+\bar{b}e^{-iy})\psi_{*}+v\right]^{3}+3u_{c}\left[(be^{iy}+\bar{b}e^{-iy})\psi_{*}+v\right]^{2}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}\\\
\langle\partial_{\xi}u_{c},\left[(be^{iy}+\bar{b}e^{-iy})\psi_{*}+v\right]^{3}+3u_{c}\left[(be^{iy}+\bar{b}e^{-iy})\psi_{*}+v\right]^{2}\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}\end{array}\right],$
(2.37)

where $S$ in (2.27) is written as

$S:=\left[\begin{array}[]{cc}-\frac{1}{2\pi}\langle\partial_{\xi}^{-1}\partial^{2}_{c}u_{c},v\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}&P^{\prime}(c)+\frac{1}{2\pi}\langle\partial_{c}u_{c},v\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}\\\
P^{\prime}(c)-\frac{1}{2\pi}\langle\partial_{c}u_{c},v\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}&\frac{1}{2\pi}\langle\partial_{\xi}u_{c},v\rangle_{L^{2}(\mathbb{R}\times\mathbb{T})}\end{array}\right].$
(2.38)

####  2.2.3 Near-identity transformations

Now, we write the order $\mathcal{O}(|b|^{2})$ in the correction term $v$ in
the decomposition (2.28) as for (2.9):

$v(\xi,y,t)=\left(b(t)^{2}e^{2iy}+\bar{b}(t)^{2}e^{-2iy}\right)w_{2}(\xi)+|b(t)|^{2}w_{0}(\xi)+w(\xi,y,t),$
(2.39)

where $w_{0}$ and $w_{2}$ are the same solutions of the linear inhomogeneous
equations (2.10) and (2.11), whereas $w$ satisfies the transformed evolution
equation

$\displaystyle w_{t}=$
$\displaystyle\partial_{\xi}(L_{c_{*}}-\partial_{y}^{2}+\dot{h}+\Delta
L_{c})w+\dot{h}\partial_{\xi}u_{c_{*}+\delta}-\dot{\delta}\partial_{c}u_{c_{*}+\delta}$
(2.40)
$\displaystyle-(\dot{b}e^{iy}+\dot{\bar{b}}e^{-iy})\psi_{*}-(2b\dot{b}e^{2iy}+2\bar{b}\dot{\bar{b}}e^{-2iy})w_{2}-(\bar{b}\dot{b}+b\dot{\bar{b}})w_{0}$
$\displaystyle+\partial_{\xi}\left(\dot{h}+\Delta
L_{c}\right)\left[(be^{iy}+\bar{b}e^{-iy})\psi_{*}+(b^{2}e^{2iy}+\bar{b}^{2}e^{-2iy})w_{2}+|b|^{2}w_{0}\right]$
$\displaystyle-\partial_{\xi}\left(\left((be^{iy}+\bar{b}e^{-iy})\psi_{*}+v\right)^{3}+3u_{c}\left((be^{iy}+\bar{b}e^{-iy})\psi_{*}+v\right)^{2}\right).$

On the other hand, we need to write the expansion of $\dot{h}$ with respect to
$b$. This is given by the first equation in the system (2.37) which implies
that

$P^{\prime}(c_{*})\dot{h}=6|b|^{2}\langle
u_{c_{*}}\partial_{c}u_{c_{*}},\psi_{*}^{2}\rangle_{L^{2}}+\mathcal{O}(|b|^{4})=\frac{64}{3}(c_{*})^{\frac{3}{2}}|b|^{2}+\mathcal{O}(|b|^{4}).$
(2.41)

Since $P^{\prime}(c_{*})=1/\sqrt{c_{*}}$ and $c_{*}=\frac{1}{3}$, we obtain

$\dot{h}=\frac{64}{27}|b|^{2}+\mathcal{O}(|b|^{4})$ (2.42)

and

$w(\xi,y,t)=\frac{64}{27}|b|^{2}\partial_{c}u_{c_{*}}(\xi)+\tilde{w}(\xi,y,t),$
(2.43)

where $\tilde{w}$ satisfied a transformed evolution equation without the
$\mathcal{O}(|b|^{2})$ terms in the right-hand side of (2.40). Substituting
(2.39), (2.42), and (2.43) into the modulation equation (2.32) yields to:

$\displaystyle\dot{b}\langle\eta_{*},\psi_{*}\rangle_{L^{2}}$ $\displaystyle=$
$\displaystyle
6|b|^{2}b\langle\psi_{*}^{2}u_{c_{*}},w_{0}+w_{2}\rangle_{L^{2}}+\frac{128}{9}|b|^{2}b\langle\psi_{*}^{2},u_{c_{*}}\partial_{c}u_{c_{*}}\rangle_{L^{2}}+3b|b|^{2}\|\psi_{*}\|^{4}_{L^{4}}$
(2.44)
$\displaystyle-\frac{64}{27}b|b|^{2}\|\psi_{*}\|^{2}_{L^{2}}-b\delta\langle\psi_{*},L_{c_{*}}^{\prime}\psi_{*}\rangle_{L^{2}}+\mathcal{O}(\delta^{2}|b|+|b|^{5}),$

where $L_{c_{*}}^{\prime}:=1-6u_{c_{*}}\partial_{c}u_{c_{*}}$ and we have used
$\Delta L_{c}=L_{c_{*}}^{\prime}\delta+\mathcal{O}(\delta^{2})$ for
$\delta=c-c_{*}$. Thus, the equation (2.44) takes the form

$\dot{b}\langle\eta_{*},\psi_{*}\rangle_{L^{2}}=\frac{73}{105}2^{9}(c_{*})^{\frac{7}{2}}|b|^{2}b+6|b|^{2}b\langle\psi_{*}^{2},w_{2}\rangle_{L^{2}}+16(c_{*})^{\frac{3}{2}}b\delta+\mathcal{O}(\delta^{2}|b|+|b|^{5}).$
(2.45)

Here, we have used the following identities

$\displaystyle\|\psi_{*}\|^{2}_{L^{2}}$
$\displaystyle=\frac{16}{3}(c_{*})^{\frac{3}{2}},\qquad\|\psi_{*}\|^{4}_{L^{4}}=\frac{2^{9}}{35}(c_{*})^{\frac{7}{2}},\quad
6\langle\psi_{*}^{2},u_{c_{*}}\partial_{c}u_{c_{*}}\rangle_{L^{2}}=\frac{64}{3}(c_{*})^{\frac{3}{2}}$
$\displaystyle\langle\psi_{*}^{2}u_{c_{*}},w_{0}\rangle_{L^{2}}$
$\displaystyle=-\frac{2^{10}}{105}(c_{*})^{\frac{7}{2}}\quad\langle\psi_{*},L_{c_{*}}^{\prime}\psi_{*}\rangle_{L^{2}}=-16(c_{*})^{\frac{3}{2}}.$

At this stage, we still need to write the expansion of $\delta$ with respect
to $b$. To do so, we include (2.20), (2.28), (2.30), (2.39), and (2.43) into
the momentum

$Q(u):=\frac{1}{2}\int_{\mathbb{R}\times\mathbb{T}}u^{2}(x,y)\,dxdy,$

in order to obtain

$Q(u)=2\pi\left[P(c_{*}+\delta)+|b|^{2}\|\psi_{*}\|^{2}_{L^{2}}+16(c_{*})^{\frac{3}{2}}|b|^{2}+\mathcal{O}(|\delta||b|^{2}+|b|^{4})\right].$
(2.46)

We have used the fact that:

$\langle u_{c_{*}},w_{0}\rangle_{L^{2}}+\frac{64}{27}\langle
u_{c_{*}},\partial_{c}u_{c_{*}}\rangle_{L^{2}}=-16(c_{*})^{\frac{3}{2}}$

By the momentum conservation, we get

$Q(u_{0})=2\pi\left[P(c_{*})+\frac{\delta}{\sqrt{c_{*}}}+\frac{56}{3}(c_{*})^{\frac{3}{2}}|b|^{2}+\mathcal{O}(\delta^{2}+|b|^{4})\right],$
(2.47)

Hence, since $c_{*}=\frac{1}{3}$, we obtain

$\delta=\delta_{0}-\frac{56}{27}|b|^{2}+\mathcal{O}(\delta_{0}^{2}+|b|^{4}),$
(2.48)

where $\delta_{0}$ is a constant in $t$ determined by the initial data.

Substituting equation (2.48) into the modulation equation (2.45) yields to:

$\dot{b}\langle\eta_{*},\psi_{*}\rangle_{L^{2}}=6|b|^{2}b\langle\psi_{*}^{2}u_{c_{*}},w_{2}\rangle_{L^{2}}+16(c_{*})^{\frac{3}{2}}b\delta_{0}+\frac{47}{105}(c_{*})^{\frac{3}{2}}|b|^{2}b+\mathcal{O}(\delta_{0}^{2}|b|+|b|^{5}).$
(2.49)

Defining $\delta_{0}:=c_{+}-c_{*}$, using the explicit expression (1.14) and
(1.16), and truncating (2.49) we obtain the normal form

$\dot{b}=\lambda^{\prime}(c_{*})(c_{+}-c_{*})b+\gamma|b|^{2}b,\quad
t\in\mathbb{R}_{+},$ (2.50)

with

$\gamma:=\frac{1}{8c_{*}}\left(\langle\psi_{*}^{2}u_{c_{*}},w_{2}\rangle_{L^{2}}+\frac{47}{105}(c_{*})^{\frac{3}{2}}\right).$
(2.51)

We recall that equation (2.11) can be written as
$w_{2}=\frac{3}{2}(L_{c_{*}}+4)^{-1}(u_{c_{*}}\psi_{*}^{2}).$ Thus

$\langle\psi_{*}^{2}u_{c_{*}},w_{2}\rangle_{L^{2}}=\frac{3}{2}\langle\psi_{*}^{2}u_{c_{*}},(L_{c_{*}}+4)^{-1}(u_{c_{*}}\psi_{*}^{2})\rangle_{L^{2}}.$

Since $L_{c_{*}}+4:H^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ is strictly
positive, we infer that

$\langle\psi_{*}^{2}u_{c_{*}},w_{2}\rangle_{L^{2}}>0.$

Hence, we deduce that $\gamma>0$.

######  Remark 2.1.

Since $\gamma>0$, the zero solution to (2.50) is unstable.

##  Appendix A The quadratic case.

For k=1, Pelinovsky used in [14] numerical simulations in order to show that
$\gamma<0$. We provide a complete proof of that claim. First, we recall that
in the quadratic case $\gamma$ is given by

$\gamma:=12\langle\psi_{*}^{2},w_{0}+w_{2}\rangle_{L^{2}}+144\langle\psi_{*}^{2},\partial_{c}u_{c_{*}}\rangle_{L^{2}}-48\|\psi_{*}\|^{2}_{L^{2}},$

where $w_{0}$ and $w_{2}$ are solutions to the following linear inhomogeneous
equations:

$L_{c_{*}}w_{0}=12\psi_{*}^{2}$

and

$\left(L_{c_{*}}+4\right)w_{2}=6\psi_{*}^{2}$

with

$L_{c}:=-\partial^{2}_{x}+c-2u_{c},\quad\psi_{*}(\xi)=\operatorname{sech}^{3}\left(\sqrt{c_{*}}\xi\right)\quad{\rm
and}\quad c_{*}=\frac{1}{5}.$

From [14], we recall that

$\langle\psi_{*}^{2},w_{0}\rangle_{L^{2}}=\frac{-160}{21\sqrt{c}_{*}},\quad\langle\psi_{*}^{2},\partial_{c}u_{c_{*}}\rangle_{L^{2}}=\frac{4}{5\sqrt{c}_{*}},\quad{\rm
and}\quad\|\psi_{*}\|^{2}_{L^{2}}=\frac{16}{15\sqrt{c}_{*}}.$

On the other hand, we know that

$w_{2}=6(L_{c_{*}}+4)^{-1}(\psi_{*}^{2}).$

Thus, from the bound $\|(L_{c_{*}}+4)^{-1}\|\leq 1/3,$ we have

$\langle\psi_{*}^{2},w_{2}\rangle_{L^{2}}\leq
2\|\psi_{*}^{2}\|^{2}_{L^{2}}=\frac{1024}{693\sqrt{c}_{*}}.$

Finally, we conclude that

$\gamma\leq
12\left(-\frac{160}{21\sqrt{c}_{*}}+\frac{1024}{693\sqrt{c}_{*}}\right)+\frac{576}{5\sqrt{c}_{*}}-\frac{256}{5\sqrt{c}_{*}}=-12\frac{4256}{693\sqrt{c}_{*}}+\frac{64}{\sqrt{c}_{*}}=-\frac{6720}{693\sqrt{c}_{*}}<0.$

Declarations:

Ethical Approval: The work is original and has not been submitted elsewhere.
Competing interests: The authors declare that they have no competing
interests.
Author contribution: The authors declare that the study was realized in
collaboration with the same contribution.
All authors read and approved the final manuscript.
Funding is Not applicable.
Availability of data and materials: Not applicable.

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Yakine Bahri
Department of Mathematics and Statistics
University of Victoria
3800 Finnerty Road, Victoria, B.C., Canada V8P 5C2
E-mail<EMAIL_ADDRESS>
Hichem Hajaiej
Department of Mathematics
California State University, Los Angeles, USA
5151 State Drive, 5151 Los Angeles, 90331 California, USA
<EMAIL_ADDRESS>