A Riemann–Hilbert Approach for the Novikov Equation A Riemann–Hilbert Approach
for the Novikov Equation\star\star$\star$This paper is a contribution to the
Special Issue on Asymptotics and Universality in Random Matrices, Random
Growth Processes, Integrable Systems and Statistical Physics in honor of Percy
Deift and Craig Tracy. The full collection is available at
http://www.emis.de/journals/SIGMA/Deift-Tracy.html

Anne BOUTET DE MONVEL †, Dmitry SHEPELSKY ‡ and Lech ZIELINSKI § A. Boutet de
Monvel, D. Shepelsky and L. Zielinski

† Institut de Mathématiques de Jussieu-PRG, Université Paris Diderot,
† 75205 Paris Cedex 13, France<EMAIL_ADDRESS>https://webusers.imj-prg.fr/~anne.boutet-de-monvel/

‡ Mathematical Division, Institute for Low Temperature Physics,
‡ 47 Nauki Avenue, 61103 Kharkiv, Ukraine<EMAIL_ADDRESS>
§ LMPA, Université du Littoral Côte d’Opale,
§ 50 rue F. Buisson, CS 80699, 62228 Calais, France<EMAIL_ADDRESS>littoral.fr

Received June 08, 2016, in final form September 14, 2016; Published online
September 24, 2016

We develop the inverse scattering transform method for the Novikov equation
$u_{t}-u_{txx}+4u^{2}u_{x}=3uu_{x}u_{xx}+u^{2}u_{xxx}$ considered on the line
$x\in(-\infty,\infty)$ in the case of non-zero constant background. The
approach is based on the analysis of an associated Riemann–Hilbert (RH)
problem, which in this case is a $3\times 3$ matrix problem. The structure of
this RH problem shares many common features with the case of the
Degasperis–Procesi (DP) equation having quadratic nonlinear terms (see [Boutet
de Monvel A., Shepelsky D., Nonlinearity 26 (2013), 2081–2107,
arXiv:1107.5995]) and thus the Novikov equation can be viewed as a “modified
DP equation”, in analogy with the relationship between the Korteweg–de Vries
(KdV) equation and the modified Korteweg–de Vries (mKdV) equation. We present
parametric formulas giving the solution of the Cauchy problem for the Novikov
equation in terms of the solution of the RH problem and discuss the
possibilities to use the developed formalism for further studying of the
Novikov equation.

Novikov equation; Degasperis–Procesi equation; Camassa–Holm equation; inverse
scattering transform; Riemann–Hilbert problem

35Q53; 37K15; 35Q15; 35B40; 35Q51; 37K40

Dedicated to Percy Deift and Craig Tracy
on the occasion of their 70th birthdays.

##  1 Introduction

1.1. In this paper we present an inverse scattering approach, based on an
appropriate Riemann–Hilbert problem formulation, for the initial value problem
for the Novikov equation [29, 37, 39]

$\displaystyle u_{t}-u_{txx}+4u^{2}u_{x}=3uu_{x}u_{xx}+u^{2}u_{xxx},\qquad$
$\displaystyle-\infty<x<+\infty,\quad t>0,$ (1.1) $\displaystyle
u(x,0)=u_{0}(x),$ $\displaystyle-\infty<x<+\infty,$

where $u_{0}(x)$ is assumed to decay to a non-zero constant:

$\displaystyle u_{0}(x)\to\varkappa>0,\qquad x\to\pm\infty.$

The solution $u(x,t)$ is also assumed to decay to $\varkappa$ for all $t>0$:

$\displaystyle u(x,t)\to\varkappa,\qquad x\to\pm\infty.$

Introducing the momentum variable

$\displaystyle m\coloneqq u-u_{xx},$

the Novikov equation (1.1) can be written as

$\displaystyle m_{t}+(m_{x}u+3mu_{x})u=0$ (1.2)

or, equivalently,

$\displaystyle\big{(}m^{\frac{2}{3}}\big{)}_{t}+\big{(}u^{2}m^{\frac{2}{3}}\big{)}_{x}=0.$
(1.3)

1.2. The Novikov equation (1.1) was obtained in the search for a
classification of integrable generalized Camassa–Holm equations of the form

$\displaystyle\big{(}1-\partial_{x}^{2}\big{)}u_{t}=F(u,u_{x},u_{xx},u_{xxx},\dots),\qquad
u=u(x,t),\qquad\partial_{x}=\partial/\partial x$

possessing infinite hierarchies of higher symmetries [39] (see also [37]).
Equation (1.1) listed by Novikov in [39, equation (31)] attracted further
considerable attention in the literature, first of all, as an example of
nonlinear equation admitting, like the Camassa–Holm (CH) [13, 14] and the
Degasperis–Procesi (DP) [22, 23] equations, peaked solutions (peakons), but
having cubic (rather than quadratic) nonlinear terms. Another integrable
Camassa–Holm type equation with cubic nonlinearities was discovered by Fokas
[25] and Qiao [41, 42].

Hone, Lundmark, and Szmigielski [28] obtained explicit formulas for
multipeakon solutions of (1.1). Some smooth and nonsmooth soliton solutions
were presented by Pan and Yi in [40]. For studies concerned with blow-up
phenomenon and the Cauchy problem for (1.1) we refer the reader to [16, 17,
26, 27, 30, 32, 38, 43, 44].

1.3. In [39] Novikov presented a scalar Lax pair for (1.1), which involves the
third order derivative with respect to $x$. Hone and Wang [29] proposed a
$3\times 3$ matrix Lax pair for (1.1), which allowed presenting explicit
formulas for peakon solutions [28, 29] on zero background. Recently, Matsuno
[36] presented parametric representations of smooth multisoliton solutions (as
well as singular solitons with single cusp and double peaks) of (1.1) on a
constant (non-zero) background, using a Hirota-type, purely algebraic
procedure. He also demonstrated that a smooth soliton converges to a peakon in
the limit where the constant background tends to $0$ while the velocity of the
soliton is fixed. Furthermore, he performed the asymptotic analysis of pure
multisoliton solutions and noticed that the formulas for the phase shifts of
the solitons as well as their peakon limits coincide with those for the
Degasperis–Procesi (DP) equation [34, 35]

$\displaystyle u_{t}-u_{txx}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx},$ (1.4)

which, in terms of $m$ reads

$\displaystyle\big{(}m^{\frac{1}{3}}\big{)}_{t}+\big{(}um^{\frac{1}{3}}\big{)}_{x}=0.$
(1.5)

Comparing with the Degasperis–Procesi equation, it is natural to view (at
least formally) the Novikov equation as a “modified DP equation”, in analogy
with the relationship between the Korteweg–de Vries (KdV) equation

$\displaystyle u_{t}+6uu_{x}+u_{xxx}=0$

and the modified Korteweg–de Vries (mKdV) equation

$\displaystyle u_{t}+6u^{2}u_{x}+u_{xxx}=0.$

The subsequent analysis presented in the paper supports this point of view.
Indeed, as we will show below, the implementation of the inverse scattering
transform method involves a Riemann–Hilbert problem of the same structure as
in the case of the DP equation (recall that this is true when comparing the
KdV and the mKdV equations).

1.4. Recall that for the DP equation, the transformation
$u(x,t)\mapsto\tilde{u}(x-\varkappa t,t)+\varkappa$ reduces (1.4) and (1.5) to

$\displaystyle\tilde{u}_{t}-\tilde{u}_{txx}+3\varkappa\tilde{u}_{x}+4\tilde{u}\tilde{u}_{x}=3\tilde{u}_{x}\tilde{u}_{xx}+\tilde{u}\tilde{u}_{xxx}$
(1.6)

and

$\displaystyle\bigl{(}\tilde{m}^{\frac{1}{3}}\bigr{)}_{t}+\bigl{(}\tilde{u}\tilde{m}^{\frac{1}{3}}\bigr{)}_{x}=0,$

respectively. Here $\tilde{m}\coloneqq\tilde{u}-\tilde{u}_{xx}+\varkappa$ and
thus $\tilde{u}\to 0$ as $x\to\pm\infty$ provided $u\to\varkappa$ as
$x\to\pm\infty$. Therefore, the study of the Cauchy problem for the DP
equation in the form (1.4) in the case of non-zero constant background is
equivalent to the study of the Cauchy problem on zero background for the DP
equation in the form (1.6), i.e., for the DP equation with _non-zero linear
dispersion term_. For the latter problem, the Riemann–Hilbert approach, which
is a variant of the inverse scattering transform method, has been applied in
[10], which allows obtaining a useful representation of the solution in a form
suitable for the analysis of its long time behavior. Notice that the further
transformation
$\tilde{u}\mapsto\tilde{\tilde{u}}\colon\tilde{u}(x,t)=\varkappa\tilde{\tilde{u}}(x,\varkappa
t)$, which preserves the zero background, allows reducing the study of (1.6)
with any $\varkappa>0$ to the case of (1.6) with $\varkappa=1$.

Similar arguments for the Novikov equation lead to the following: The
transformations $u(x,t)=\tilde{u}(x-\varkappa^{2}t,t)+\varkappa$ and
$u(x,t)=\varkappa\tilde{\tilde{u}}(x-\varkappa^{2}t,\varkappa^{2}t)+\varkappa$
reduce the Cauchy problem for (1.1) (or (1.3)) on a non-zero constant
background $u\to\varkappa$ as $x\to\pm\infty$ to the Cauchy problem on zero
background ($\tilde{u}\to 0$ and $\tilde{\tilde{u}}\to 0$ as $x\to\pm\infty$)
for the equations

$\displaystyle\bigl{(}\tilde{m}^{\frac{2}{3}}\bigr{)}_{t}+\bigl{(}\big{(}\tilde{u}^{2}+2\varkappa\tilde{u}\big{)}\tilde{m}^{\frac{2}{3}}\bigr{)}_{x}=0,$
(1.7)

and

$\displaystyle\bigl{(}\tilde{\tilde{m}}^{\frac{2}{3}}\bigr{)}_{t}+\bigl{(}\big{(}\tilde{\tilde{u}}^{2}+2\tilde{\tilde{u}}\big{)}\tilde{\tilde{m}}^{\frac{2}{3}}\bigr{)}_{x}=0,$
(1.8)

respectively. Here $\tilde{m}\coloneqq\tilde{u}-\tilde{u}_{xx}+\varkappa$ and
$\tilde{\tilde{m}}=\tilde{\tilde{u}}-\tilde{\tilde{u}}_{xx}+1$.

1.5. Henceforth, we consider the Cauchy problem for equation (1.8) on zero
background, which, to simplify notations, will be written as

$\displaystyle\bigl{(}\hat{m}^{2/3}\bigr{)}_{t}+\bigl{(}\big{(}u^{2}+2u\big{)}\hat{m}^{2/3}\bigr{)}_{x}=0,\qquad$
$\displaystyle-\infty<x<\infty,\quad t>0,$ (1.9a) $\displaystyle\hat{m}\equiv
m+1=u-u_{xx}+1,$ (1.9b) $\displaystyle u(x,0)=u_{0}(x),$
$\displaystyle-\infty<x<\infty,$ (1.9c) where $u_{0}(x)$ is sufficiently
smooth and decays fast as $x\to\pm\infty$. Moreover, we assume that $u_{0}(x)$
satisfies the sign condition $\displaystyle u_{0}(x)-(u_{0})_{xx}(x)+1>0.$
(1.9d)

Then there exists [32] a unique global solution $u(x,t)$ of (1.9), such that
$u(x,t)\to 0$ as $x\to\pm\infty$ for all $t$.

Notice that the Novikov equation (1.9a), being written in terms of $u$ only,
contains linear as well as quadratic dispersion terms:

$\displaystyle
u_{t}-u_{txx}+4u^{2}u_{x}+8uu_{x}+3u_{x}=3uu_{x}u_{xx}+u^{2}u_{xxx}+3u_{x}u_{xx}+2uu_{xxx}.$

1.6. The analysis of Camassa–Holm-type equations by using the inverse
scattering approach was initiated in [18, 21, 31, 33] for the Camassa–Holm
equation itself:

$\displaystyle u_{t}-u_{txx}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}.$

A version of the inverse scattering method for the CH equation based on a
Riemann–Hilbert (RH) factorization problem was proposed in [5, 8] (another RH
formulation of the inverse scattering transform is presented in [19]). The RH
approach has proved its efficiency in the study of the long-time behavior of
solutions of both initial value problems [2, 3, 7] and initial boundary value
problems [9] for the CH equation. In [4, 10] it has been adapted to the study
of the Degasperis–Procesi equation.

In the present paper we develop the RH approach to the Novikov equation in the
form (1.9a) on zero background, following the main ideas developed in [8, 10].
To the best of our knowledge, no equations of the Camassa–Holm type with
_cubic_ nonlinearity have been treated before by the inverse scattering method
in the form of a RH problem.

A major difference between the implementations of the RH method to the CH
equation, on one hand, and to the DP as well as Novikov equation, on the other
hand, is that in the latter cases, the spatial equations of the associated Lax
pairs are of the third order, which implies that when rewriting them in matrix
form, one has to deal with $3\times 3$ matrix-valued equations, while in the
case of the CH equation, they have a $2\times 2$ matrix structure, as in the
cases of the most known integrable equations (KdV, mKdV, nonlinear
Schrödinger, sine-Gordon, etc.). Hence, the construction and analysis of the
associated RH problem become considerably more complicated.

In our approach, we propose (Section 3) an associated RH problem and give
(Theorem 4.1) a representation of the solution $u(x,t)$ of the initial value
problem (1.9) in terms of the solution of this RH problem evaluated at a
distinguished point of the plane of the spectral parameter. Remarkably, the
formulas for $u(x,t)$ obtained in this way have the same structure as the
parametric formulas obtained in [36] for pure multisoliton solutions.

##  2 Lax pairs and eigenfunctions

######  Assumptions.

Recall that we assume that the initial function $u_{0}(x)$ in (1.9c) is
sufficiently smooth with fast decay at $\pm\infty$ and satisfies the sign
condition (1.9d).

Then, similarly to the case of the CH equation (see, e.g., [18]), the solution
$\hat{m}(x,t)$ of (1.9) satisfies the sign condition $\hat{m}(x,t)>0$ for all
$x\in\mathbb{R}$ and all $t>0$.

###  2.1 Lax pairs

####  2.1.1 A first Lax pair

The Lax pair found by Hone and Wang [29] for the Novikov equation in the form
(1.1) (or (1.2)) reads

$\displaystyle\partial_{x}\begin{pmatrix}\psi_{1}\\\ \psi_{2}\\\
\psi_{3}\end{pmatrix}=\begin{pmatrix}0&zm&1\\\ 0&0&zm\\\
1&0&0\end{pmatrix}\begin{pmatrix}\psi_{1}\\\ \psi_{2}\\\
\psi_{3}\end{pmatrix},\qquad m\coloneqq u-u_{xx},$ (2.1a)
$\displaystyle\partial_{t}\begin{pmatrix}\psi_{1}\\\ \psi_{2}\\\
\psi_{3}\end{pmatrix}=\begin{pmatrix}-uu_{x}&\frac{u_{x}}{z}-u^{2}mz&u_{x}^{2}\vspace{1mm}\\\
\frac{u}{z}&-\frac{1}{z^{2}}&-\frac{u_{x}}{z}-u^{2}mz\vspace{1mm}\\\
-u^{2}&\frac{u}{z}&uu_{x}\end{pmatrix}\begin{pmatrix}\psi_{1}\\\ \psi_{2}\\\
\psi_{3}\end{pmatrix},$ (2.1b)

where $z$ is the _spectral parameter_.

####  2.1.2 A modified Lax pair

For the Novikov equation in the form (1.9a), the Lax pair (2.1) has to be
appropriately modified. While for the Camassa–Holm and Degasperis–Procesi
equations the corresponding modification (when passing from the equation with
zero linear dispersive term to that with a non-zero one) consists simply in
replacing $m$ by $\hat{m}=m+1$, the modification for the Novikov equation
turns out to be more involved.

######  Lemma 2.1.

The Novikov equation (1.9a) admits as Lax pair the system

$\displaystyle\Phi_{x}=U\Phi,\qquad\Phi_{t}=V\Phi,$ (2.2a) where
$\Phi\equiv\Phi(x,t;z)$ and $\displaystyle
U(x,t;z)=\begin{pmatrix}0&z\hat{m}&1\\\ 0&0&z\hat{m}\\\ 1&0&0\end{pmatrix},$
(2.2b) $\displaystyle
V(x,t;z)=\begin{pmatrix}-(u+1)u_{x}+\frac{1}{3z^{2}}&\frac{u_{x}}{z}-(u^{2}+2u)\hat{m}z&u_{x}^{2}+1\\\\[2.84526pt]
\frac{u+1}{z}&-\frac{2}{3z^{2}}&-\frac{u_{x}}{z}-(u^{2}+2u)\hat{m}z\\\\[2.84526pt]
-u^{2}-2u&\frac{u+1}{z}&(u+1)u_{x}+\frac{1}{3z^{2}}\end{pmatrix}.$ (2.2c)

######  Remark 2.2.

The freedom in adding to $V$ a constant (independent of $(x,t)$) term $c\cdot
I$, where $I$ is the $3\times 3$ identity matrix, has been used in (2.2c) in
order to make $V$ traceless, which provides that the determinant of a matrix
solution to the equation $\Phi_{t}=V\Phi$ is independent of $t$. The same
property holds for the equation $\Phi_{x}=U\Phi$ whose coefficient $U$ is
obviously traceless.

The coefficient matrices $U$ and $V$ in (2.2) have singularities (in the
extended complex $z$-plane) at $z=0$ and at $z=\infty$. In order to control
the behavior of solutions to (2.2) as functions of the spectral parameter $z$
(which is crucial for the Riemann–Hilbert method), we follow a strategy
similar to that adopted for the CH equation [5, 8] and the DP equation [10].

####  2.1.3 A Lax pair appropriate for large $\boldsymbol{z}$

In order to control the large $z$ behavior of the solutions of (2.2), we will
transform this Lax pair as follows (cf. [1]):

######  Lemma 2.3.

The Lax pair (2.2) can be transformed into a new Lax pair

$\displaystyle\hat{\Phi}_{x}-Q_{x}\hat{\Phi}=\hat{U}\hat{\Phi},\qquad\hat{\Phi}_{t}-Q_{t}\hat{\Phi}=\hat{V}\hat{\Phi},$
(2.3)

whose coefficients $Q(x,t;z)$, $\hat{U}(x,t;z)$, and $\hat{V}(x,t;z)$ have the
following properties:

  1. (i)

$Q$ is diagonal;

  2. (ii)

$\hat{U}=\mathrm{O}(1)$ and $\hat{V}=\mathrm{O}(1)$ as $z\to\infty$, whereas
$Q_{x}$ is growing;

  3. (iii)

the diagonal parts of $\hat{U}$ and $\hat{V}$ decay as $z\to\infty$;

  4. (iv)

$\hat{U}\to 0$ and $\hat{V}\to 0$ as $x\to\pm\infty$.

###### Proof.

As in the case of the DP equation [10], we perform this transformation into
two steps:

  * •

We transform (2.2) into a system where the leading terms are represented as
products of $(x,t)$-independent (matrix-valued) and $(x,t)$-dependent (scalar)
factors.

  * •

We diagonalize the $(x,t)$-independent factors.

First step. Introducing $\tilde{\Phi}\equiv\tilde{\Phi}(x,t;z)$ by

$\displaystyle\tilde{\Phi}=D^{-1}\Phi,$

where $D(x,t)=\operatorname{diag}\big{\\{}q(x,t),1,q^{-1}(x,t)\big{\\}}$ and
($\hat{m}$ is as in (1.9b))

$\displaystyle q=q(x,t)\coloneqq\hat{m}^{1/3}(x,t),$ (2.4)

transforms (2.2) into the new Lax pair

$\displaystyle\tilde{\Phi}_{x}=\tilde{U}\tilde{\Phi},\qquad\tilde{\Phi}_{t}=\tilde{V}\tilde{\Phi},$
(2.5a) where $\displaystyle\tilde{U}(x,t;z)=q^{2}(x,t)\begin{pmatrix}0&z&1\\\
0&0&z\\\
1&0&0\end{pmatrix}+\begin{pmatrix}-\frac{q_{x}}{q}&0&\frac{1}{q^{2}}-q^{2}\\\
0&0&0\\\ 0&0&\frac{q_{x}}{q}\end{pmatrix}$
$\displaystyle\hphantom{\tilde{U}(x,t;z)}{}\equiv
q^{2}(x,t)U_{\infty}(z)+\tilde{U}^{(1)}(x,t)$ (2.5b)
$\displaystyle\tilde{V}(x,t;z)=-\big{(}u^{2}+2u\big{)}q^{2}\begin{pmatrix}0&z&1\\\
0&0&z\\\ 1&0&0\end{pmatrix}+\begin{pmatrix}\frac{1}{3z^{2}}&0&1\vspace{1mm}\\\
\frac{1}{z}&-\frac{2}{3z^{2}}&0\vspace{1mm}\\\
0&\frac{1}{z}&\frac{1}{3z^{2}}\end{pmatrix}$
$\displaystyle\hphantom{\tilde{V}(x,t;z)=}{}+\begin{pmatrix}(u^{2}+2u)\frac{q_{x}}{q}&0&(u^{2}+2u)q^{2}+\frac{u_{x}^{2}+1}{q^{2}}-1\\\
0&0&0\\\ 0&0&-(u^{2}+2u)\frac{q_{x}}{q}\end{pmatrix}$
$\displaystyle\hphantom{\tilde{V}(x,t;z)=}{}+\frac{1}{z}\begin{pmatrix}0&\frac{u_{x}}{q}&0\\\
(u+1)q-1&0&-\frac{u_{x}}{q}\\\ 0&(u+1)q-1&0\end{pmatrix}$
$\displaystyle\hphantom{\tilde{V}(x,t;z)}{}\equiv-\big{(}u^{2}(x,t)+2u(x,t)\big{)}q^{2}(x,t)U_{\infty}(z)+V_{\infty}(z)+\tilde{V}^{(1)}(x,t)+\frac{1}{z}\tilde{V}^{(2)}(x,t).$
(2.5c)

Second step. The commutator of $U_{\infty}$ and $V_{\infty}$ vanishes
identically, i.e., $[U_{\infty},V_{\infty}]\equiv 0$, which allows
simultaneous diagonalization of $U_{\infty}$ and $V_{\infty}$. Indeed, we have
(for $z\neq 0$)

$\displaystyle P^{-1}(z)U_{\infty}(z)P(z)=\Lambda(z),\qquad
P^{-1}(z)V_{\infty}(z)P(z)=A(z),$

where

$\displaystyle\Lambda(z)=\begin{pmatrix}\lambda_{1}(z)&0&0\\\
0&\lambda_{2}(z)&0\\\ 0&0&\lambda_{3}(z)\end{pmatrix},$ (2.6a) $\displaystyle
A(z)=\frac{1}{3z^{2}}I+\Lambda^{-1}(z)\equiv\begin{pmatrix}A_{1}(z)&0&0\\\
0&A_{2}(z)&0\\\ 0&0&A_{3}(z)\end{pmatrix},$ (2.6b) $\displaystyle
P(z)=\begin{pmatrix}\lambda_{1}^{2}(z)&\lambda_{2}^{2}(z)&\lambda_{3}^{2}(z)\\\
z&z&z\\\ \lambda_{1}(z)&\lambda_{2}(z)&\lambda_{3}(z)\end{pmatrix},$ (2.6c)
$\displaystyle
P^{-1}(z)=\left(\begin{matrix}(3\lambda_{1}^{2}(z)-1)^{-1}\\!&0&0\\\
0&(3\lambda_{2}^{2}(z)-1)^{-1}&0\\\
0&0&\\!(3\lambda_{3}^{2}(z)-1)^{-1}\end{matrix}\right)\\!\left(\begin{matrix}1&\frac{z}{\lambda_{1}(z)}&\lambda_{1}(z)\\\
1&\frac{z}{\lambda_{2}(z)}&\lambda_{2}(z)\\\
1&\frac{z}{\lambda_{3}(z)}&\lambda_{3}(z)\end{matrix}\right).\\!\\!\\!$ (2.6d)

Here $\lambda_{1}(z)$, $\lambda_{2}(z)$, and $\lambda_{3}(z)$ are the
solutions of the algebraic equation

$\displaystyle\lambda^{3}-\lambda=z^{2},$

enumerated in such a way that $\lambda_{j}(z)\sim\omega^{j}z^{2/3}$ as
$z\to\infty$, where $\omega=\mathrm{e}^{\frac{2\mathrm{i}\pi}{3}}$.

Now, introducing $\hat{\Phi}\equiv\hat{\Phi}(x,t;z)$ by

$\displaystyle\hat{\Phi}=P^{-1}\tilde{\Phi}$

transforms the Lax pair (2.5) into the new Lax pair:

$\displaystyle\hat{\Phi}_{x}-q^{2}\Lambda(z)\hat{\Phi}=\hat{U}\hat{\Phi},$
(2.7a)
$\displaystyle\hat{\Phi}_{t}+\left(\big{(}u^{2}+2u\big{)}q^{2}\Lambda(z)-A(z)\right)\hat{\Phi}=\hat{V}\hat{\Phi},$
(2.7b) where $\displaystyle\hat{U}(x,t;z)=P^{-1}(z)\tilde{U}^{(1)}(x,t)P(z),$
(2.7c)
$\displaystyle\hat{V}(x,t;z)=P^{-1}(z)\left(\tilde{V}^{(1)}(x,t)+\frac{1}{z}\tilde{V}^{(2)}(x,t)\right)P(z).$
(2.7d)

Here $\hat{U}(x,t;z)=\mathrm{O}(1)$ and $\hat{V}(x,t;z)=\mathrm{O}(1)$ as
$z\to\infty$ due to the fact that $\tilde{U}^{(1)}$ and $\tilde{V}^{(1)}$ are
upper triangular matrices. Moreover, the fact that $\tilde{U}^{(1)}$ and
$\tilde{V}^{(1)}$ are traceless implies that the diagonal entries of
$\hat{U}(x,t;z)$ and $\hat{V}(x,t;z)$ are $\mathrm{O}(z^{-2/3})$ as
$z\to\infty$.

Indeed, we can write $\hat{U}=\hat{U}^{(1)}\hat{U}^{(2)}$, where

$\displaystyle\hat{U}^{(1)}=\left(\begin{matrix}\frac{1}{3\lambda_{1}^{2}-1}&0&0\\\
0&\frac{1}{3\lambda_{2}^{2}-1}&0\\\
0&0&\frac{1}{3\lambda_{3}^{2}-1}\end{matrix}\right),$ (2.8a)
$\displaystyle\hat{U}^{(2)}=\left(\begin{matrix}c_{2}\lambda_{1}&c_{1}(\lambda_{1}\lambda_{2}-\lambda_{2}^{2})+c_{2}\lambda_{2}&c_{1}(\lambda_{1}\lambda_{3}-\lambda_{3}^{2})+c_{2}\lambda_{3}\vspace{1mm}\\\
c_{1}(\lambda_{1}\lambda_{2}-\lambda_{1}^{2})+c_{2}\lambda_{1}&c_{2}\lambda_{2}&c_{1}(\lambda_{1}\lambda_{3}-\lambda_{3}^{2})+c_{2}\lambda_{3}\vspace{1mm}\\\
c_{1}(\lambda_{1}\lambda_{3}-\lambda_{1}^{2})+c_{2}\lambda_{1}&c_{1}(\lambda_{2}\lambda_{3}-\lambda_{2}^{2})+c_{2}\lambda_{2}&c_{2}\lambda_{3}\end{matrix}\right)$
(2.8b) with $c_{1}=q_{x}/q$ and $c_{2}=q^{-2}-q^{2}$. Notice that
$\hat{U}(x,t;z)$ has a finite limit at $z=0$.

We can also write
$\hat{V}=\hat{U}^{(1)}\big{(}\hat{V}^{(1)}+\hat{V}^{(2)}\Lambda\big{)}$ where
$\hat{V}^{(1)}$ has the form of $\hat{U}^{(2)}$ with $c_{1}$ and $c_{2}$
replaced by $c_{3}=-(u^{2}+2u)\frac{q_{x}}{q}$ and
$c_{4}=(u^{2}+2u)q^{2}+\frac{u_{x}^{2}+1}{q^{2}}-1$, respectively, and

$\displaystyle\hat{V}^{(2)}=\left(\begin{matrix}2c_{6}&\begin{matrix}c_{5}\left(\frac{1}{\lambda_{2}}-\frac{1}{\lambda_{1}}\right)\\\
{}+c_{6}\left(\frac{\lambda_{1}}{\lambda_{2}}+\frac{\lambda_{2}}{\lambda_{1}}\right)\end{matrix}&\begin{matrix}c_{5}\left(\frac{1}{\lambda_{3}}-\frac{1}{\lambda_{1}}\right)\\\
{}+c_{6}\left(\frac{\lambda_{1}}{\lambda_{3}}+\frac{\lambda_{3}}{\lambda_{1}}\right)\end{matrix}\vspace{1mm}\\\
\begin{matrix}c_{5}\left(\frac{1}{\lambda_{1}}-\frac{1}{\lambda_{2}}\right)\\\
{}+c_{6}\left(\frac{\lambda_{1}}{\lambda_{2}}+\frac{\lambda_{2}}{\lambda_{1}}\right)\end{matrix}&2c_{6}&\begin{matrix}c_{5}\left(\frac{1}{\lambda_{3}}-\frac{1}{\lambda_{2}}\right)\\\
{}+c_{6}\left(\frac{\lambda_{2}}{\lambda_{3}}+\frac{\lambda_{3}}{\lambda_{2}}\right)\end{matrix}\vspace{1mm}\\\
\begin{matrix}c_{5}\left(\frac{1}{\lambda_{1}}-\frac{1}{\lambda_{3}}\right)\\\
{}+c_{6}\left(\frac{\lambda_{1}}{\lambda_{3}}+\frac{\lambda_{3}}{\lambda_{1}}\right)\end{matrix}&\begin{matrix}c_{5}\left(\frac{1}{\lambda_{2}}-\frac{1}{\lambda_{3}}\right)\\\
{}+c_{6}\left(\frac{\lambda_{2}}{\lambda_{3}}+\frac{\lambda_{3}}{\lambda_{2}}\right)\end{matrix}&2c_{6}\end{matrix}\right)$
(2.8c)

with $c_{5}=\frac{u_{x}}{q}$, $c_{6}=(u+1)q-1$.

Finally, in order to write (2.7) in the desired form (2.3) it suffices to find
a solution of the system

$\displaystyle Q_{x}=q^{2}\Lambda(z),$ (2.9a) $\displaystyle
Q_{t}=-\left(\big{(}u^{2}+2u\big{)}q^{2}\right)\Lambda(z)+A(z).$ (2.9b)

We first notice that both equations are consistent. This follows directly from
the equation

$\displaystyle(q^{2})_{t}+\big{(}\big{(}u^{2}+2u\big{)}q^{2}\big{)}_{x}=0,$
(2.10)

which is just another form of the Novikov equation (1.9a). We actually find
that a solution $Q(x,t;z)$ of the system (2.9) is given by the $3\times 3$
diagonal function

$\displaystyle Q(x,t;z)=y(x,t)\Lambda(z)+tA(z)$ (2.11)

with

$\displaystyle y(x,t)\coloneqq
x-\int_{x}^{\infty}(q^{2}(\xi,t)-1)\mathrm{d}\xi.$ (2.12)

This solution $Q(x,t;z)$ is normalized in such a way that

$\displaystyle Q(x,t;z)\sim x\Lambda(z)+tA(z)\qquad\text{as}\quad
x\to+\infty.$ ∎

###  2.2 Eigenfunctions

####  2.2.1 Fredholm integral equations

Introducing the $3\times 3$ matrix-valued function $M\equiv M(x,t;z)$ by

$\displaystyle M=\hat{\Phi}\mathrm{e}^{-Q}$

reduces the Lax pair (2.3) to the system

$\displaystyle M_{x}-[Q_{x},M]=\hat{U}M,\qquad M_{t}-[Q_{t},M]=\hat{V}M.$
(2.13)

Assume that the coefficients in (2.13), which are expressed in terms of
$u(x,t)$, are given. Then particular solutions of (2.13) having well-
controlled properties as functions of the spectral parameter $z$ can be
constructed as solutions of the Fredholm integral equation (cf. [1])

$\displaystyle M(x,t;z)=I$
$\displaystyle{}+\int_{(x^{*},t^{*})}^{(x,t)}\mathrm{e}^{Q(x,t;z)-Q(\xi,\tau;z)}\big{(}\hat{U}M(\xi,\tau;z)\mathrm{d}\xi+\hat{V}M(\xi,\tau;z)\mathrm{d}\tau\big{)}\mathrm{e}^{-Q(x,t;z)+Q(\xi,\tau;z)},$
(2.14)

where the initial points of integration $(x^{*},t^{*})$ can be chosen
differently for different matrix entries of the equation. $Q$ being diagonal,
(2.14) must be seen as the collection of scalar integral equations ($1\leq
j,l\leq 3$)

$\displaystyle M_{jl}(x,t;z)=I_{jl}$
$\displaystyle{}+\int_{(x_{jl}^{*},t_{jl}^{*})}^{(x,t)}\mathrm{e}^{Q_{jj}(x,t;z)-Q_{jj}(\xi,\tau;z)}\bigl{(}(\hat{U}M)_{jl}(\xi,\tau;z)\mathrm{d}\xi+(\hat{V}M)_{jl}(\xi,\tau;z)\mathrm{d}\tau\bigr{)}\mathrm{e}^{-Q_{ll}(x,t;z)+Q_{ll}(\xi,\tau;z)}.$

Notice that choosing the $(x_{jl}^{*},t_{jl}^{*})$ appropriately allows
obtaining eigenfunctions which are piecewise analytic w.r.t. the spectral
parameter $z$ and thus can be used in the construction of Riemann–Hilbert
problems associated with initial value problems [8] as well as initial
boundary value problems [9].

Figure 1: Paths of integration. Left: $(x^{*},t^{*})=(-\infty,t)$. Right:
$(x^{*},t^{*})=(+\infty,t)$.

In particular, for the Cauchy problem considered in the present paper, it is
reasonable to choose these points to be $(-\infty,0)$ or $(+\infty,0)$ thus
reducing the integration in (2.14) to paths parallel to the $x$-axis (see Fig.
1) provided the integrals $\int_{(-\infty,0)}^{(-\infty,t)}$ and
$\int_{(\infty,0)}^{(\infty,t)}$ vanish:

$\displaystyle
M(x,t;z)=I+\int_{(\pm)\infty}^{x}\mathrm{e}^{Q(x,t;z)-Q(\xi,t;z)}[\hat{U}M(\xi,t;z)]\mathrm{e}^{-Q(x,t;z)+Q(\xi,t;z)}\mathrm{d}\xi,$
(2.15)

or, in view of (2.11) and (2.9a),

$\displaystyle
M(x,t;z)=I+\int_{(\pm)\infty}^{x}\mathrm{e}^{-\left(\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta\right)\Lambda(z)}[\hat{U}M(\xi,t;z)]\mathrm{e}^{\big{(}\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta\big{)}\Lambda(z)}\mathrm{d}\xi.$
(2.16)

Since $q^{2}>0$, the domains (in the complex $z$-plane), where the exponential
factors in (2.15) are bounded, are determined by the signs of
$\operatorname{Re}\lambda_{j}(z)-\operatorname{Re}\lambda_{l}(z)$, $1\leq
j\neq l\leq 3$.

####  2.2.2 A new spectral parameter

As in the case of the Degasperis–Procesi equation (see [10, 20] and also [15])
it is convenient to introduce a new spectral parameter $k$ such that

$\displaystyle
z^{2}(k)=\frac{1}{3\sqrt{3}}\left(k^{3}+\frac{1}{k^{3}}\right).$ (2.17)

We thus have

$\displaystyle\lambda_{j}=\lambda_{j}(z(k))=\frac{1}{\sqrt{3}}\left(\omega^{j}k+\frac{1}{\omega^{j}k}\right),\qquad\text{where}\quad\omega=\mathrm{e}^{\frac{2\mathrm{i}\pi}{3}}.$
(2.18)

In what follows, we will work in the complex $k$-plane only. So, by a slight
abuse of notation, we will write $\lambda_{j}(k)$ for $\lambda_{j}(z(k))$, and
similarly for other functions of $z(k)$, e.g., $M(x,t;k)$ for $M(x,t;z(k))$
and $\Lambda(k)$ for $\Lambda(z(k))$.

Figure 2: Rays $l_{\nu}$, domains $\Omega_{\nu}$, and points
$\varkappa_{\nu}$, $\kappa_{l}$ in the $k$-plane.

The $\lambda_{j}$’s are the same, as functions of $k$, as in the case of the
DP equation, see [10]. Thus, the contour
$\Sigma=\\{k\,|\,\operatorname{Re}\lambda_{j}(k)=\operatorname{Re}\lambda_{l}(k)\text{
for some }j\neq l\\}$ is also the same; it consists of six rays

$\displaystyle
l_{\nu}=\mathbb{R}_{+}\mathrm{e}^{\frac{\mathrm{i}\pi}{3}(\nu-1)}=\mathbb{R}_{+}\varkappa_{\nu},\qquad\nu=1,\dots,6$

dividing the $k$-plane into six sectors

$\displaystyle\Omega_{\nu}=\left\\{k\,\Big{|}\,\frac{\pi}{3}(\nu-1)<\arg
k<\frac{\pi}{3}\nu\right\\},\qquad\nu=1,\dots,6.$

In order that (2.16) have a (matrix-valued) solution that is analytic, as a
function of $k\in\mathbb{C}\setminus\Sigma$, the initial points of integration
$\infty_{jl}$ are specified as follows for each matrix entry $(j,l)$, $1\leq
j,l\leq 3$:

$\displaystyle\infty_{j,l}=\begin{cases}+\infty,&\text{if
}\operatorname{Re}\lambda_{j}(k)\geq\operatorname{Re}\lambda_{l}(k),\\\
-\infty,&\text{if
}\operatorname{Re}\lambda_{j}(k)<\operatorname{Re}\lambda_{l}(k).\end{cases}$
(2.19)

This means that we consider the system of scalar Fredholm integral equations,
$1\leq j,\,l\leq 3$,

$\displaystyle
M_{jl}(x,t;k)=I_{jl}+\int_{\infty_{j,l}}^{x}\mathrm{e}^{-\lambda_{j}(k)\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta}[(\hat{U}M)_{jl}(\xi,t;k)]\mathrm{e}^{\lambda_{l}(k)\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta}\mathrm{d}\xi,$
(2.20)

where $k\in\mathbb{C}$ and $I$ denotes the $3\times 3$ identity matrix.

######  Remark 2.4.

In spite of the fact that some of the coefficients in (2.7) seemingly depend
on the first order of $z$ (for instance, the coefficient
$\frac{1}{z}P^{-1}(z)\tilde{V}^{(2)}P(z)$), which, as a function of $k$, is
not rational, direct calculations show that $\hat{U}$ in (2.20) as well as
$\hat{V}$ depends on $k$ rationally, through the
$\lambda_{j}=\lambda_{j}(k)$’s, see (2.8) and (2.18).

######  Proposition 2.5 (analyticity).

Let $M(x,t;k)$ be the $($unique$)$ solution of the system of integral
equations (2.20), where the limits of integration $\infty_{j,l}$ are chosen
according to (2.19). Then

  1. (i)

$M$ is piecewise meromorphic with respect to $\Sigma$, as function of the
spectral parameter $k$;

  2. (ii)

$M(x,t;k)\to I$ as $k\to\infty$, where $I$ is the $3\times 3$ identity matrix;

  3. (iii)

for $k\in\mathbb{C}\setminus\Sigma$, $M$ is bounded as $x\to-\infty$ and $M\to
I$ as $x\to+\infty$;

  4. (iv)

$\det M\equiv 1$.

###### Proof.

The proof follows the same lines as in [1]. Notice that in order to have (ii),
it is important that the diagonal part of $\hat{U}$ vanish as $k\to\infty$.
Notice also that (iv) follows from the fact that the coefficient matrices in
(2.7) are traceless and from (ii). ∎

######  Proposition 2.6 (symmetries).

The solution $M(x,t;k)$ of (2.20) satisfies the symmetry relations:

  1. (S1)

$\Gamma_{1}\overline{M(x,t;\bar{k})}\Gamma_{1}=M(x,t;k)$, where
$\Gamma_{1}=\left(\begin{smallmatrix}0&1&0\\\ 1&0&0\\\
0&0&1\end{smallmatrix}\right)$;

  2. (S2)

$\Gamma_{2}\overline{M(x,t;\bar{k}\omega^{2})}\Gamma_{2}=M(x,t;k)$, where
$\Gamma_{2}=\left(\begin{smallmatrix}0&0&1\\\ 0&1&0\\\
1&0&0\end{smallmatrix}\right)$;

  3. (S3)

$\Gamma_{3}\overline{M(x,t;\bar{k}\omega)}\Gamma_{3}=M(x,t;k)$, where
$\Gamma_{3}=\left(\begin{smallmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{smallmatrix}\right)$;

  4. (S4)

$M(x,t;\frac{1}{k})=\overline{M(x,t;\bar{k})}$.

###### Proof.

Indeed, the diagonal entries of the matrix
$\Lambda(k)=\operatorname{diag}(\lambda_{j}(k))$ satisfy the following
relations:

$\displaystyle\overline{\lambda_{1}(\bar{k})}=\lambda_{2}(k),\quad$
$\displaystyle\overline{\lambda_{3}(\bar{k})}=\lambda_{3}(k),$
$\displaystyle\overline{\lambda_{2}(\bar{k}\omega)}=\lambda_{3}(k),\qquad$
$\displaystyle\overline{\lambda_{1}(\bar{k}\omega)}=\lambda_{1}(k),$
$\displaystyle\overline{\lambda_{1}(\bar{k}\omega^{2})}=\lambda_{3}(k),\qquad$
$\displaystyle\overline{\lambda_{2}(\bar{k}\omega^{2})}=\lambda_{2}(k),\qquad$
$\displaystyle\lambda_{j}(1/k)=\overline{\lambda_{j}(\bar{k})},\quad j=1,2,3.$
∎

######  Remark 2.7.

From (S1)–(S3) it follows that the values of $M$ at $k$ and at $\omega k$ are
related by

$\displaystyle M(x,t;k\omega)=C^{-1}M(x,t;k)C,\qquad\text{where}\quad
C=\left(\begin{matrix}0&0&1\\\ 1&0&0\\\ 0&1&0\end{matrix}\right).$

If $\lambda_{j}(k)=\lambda_{l}(k)$, $j\neq l$ for some value of the spectral
parameter $k$, then $P$ at this value becomes degenerate (see (2.6d)), which
in turn leads to a singularity for $\hat{U}$ and, consequently, for
$\hat{\Phi}$ and $M$. These particular values of the spectral parameter are
$\varkappa_{\nu}=\mathrm{e}^{\frac{\mathrm{i}\pi}{3}(\nu-1)}$,
$\nu=1,\dots,6$. Taking into account the symmetries described in Proposition
2.6 leads to the following proposition.

######  Proposition 2.8 (singularities at $\varkappa_{\nu}$).

The limiting values of $M(x,t;k)$ as $k$ approaches one of the points
$\varkappa_{\nu}=\mathrm{e}^{\frac{\mathrm{i}\pi}{3}(\nu-1)}$, $\nu=1,\dots,6$
have pole singularities with leading terms of a specific matrix structure.

  1. (i)

As $k\to\varkappa_{1}=1$,

$\displaystyle M(x,t;k)=\frac{1}{k-1}\left(\begin{matrix}1&1&1\\\ -1&-1&-1\\\
0&0&0\end{matrix}\right)\left(\begin{matrix}\alpha(x,t)&0&0\\\
0&\alpha(x,t)&0\\\ 0&0&\beta(x,t)\end{matrix}\right)$
$\displaystyle\hphantom{M(x,t;k)=}{}+\tilde{M}+\mathrm{O}(k-1),$ (2.21a)
$\displaystyle M^{-1}(x,t;k)=\frac{1}{k-1}\left(\begin{matrix}1&1&1\\\
-1&-1&-1\\\ 0&0&0\end{matrix}\right)\left(\begin{matrix}\alpha_{1}(x,t)&0&0\\\
0&\alpha_{1}(x,t)&0\\\ 0&0&\beta_{1}(x,t)\end{matrix}\right)$
$\displaystyle\hphantom{M^{-1}(x,t;k)=}{}+\tilde{M}^{(1)}+\mathrm{O}(k-1),$
(2.21b)

where $\alpha=-\overline{\alpha}$, $\beta=-\overline{\beta}$,
$\alpha_{1}=-\overline{\alpha_{1}}$, $\beta_{1}=-\overline{\beta_{1}}$.
Moreover, $(\alpha,\beta)\neq(0,0)$ iff $(\alpha_{1},\beta_{1})\neq(0,0)$ and
in this case, the entries of $\tilde{M}(x,t)$ and $\tilde{M}^{(1)}(x,t)$
satisfy the relations

$\displaystyle\tilde{M}_{31}=\tilde{M}_{32},\qquad$
$\displaystyle\tilde{M}_{11}+\tilde{M}_{21}=\tilde{M}_{12}+\tilde{M}_{22},$
(2.22a) $\displaystyle\tilde{M}_{31}^{(1)}=\tilde{M}_{32}^{(1)},\qquad$
$\displaystyle\tilde{M}_{11}^{(1)}+\tilde{M}_{21}^{(1)}=\tilde{M}_{12}^{(1)}+\tilde{M}_{22}^{(1)}.$
(2.22b)

  2. (ii)

As $k\to\varkappa_{2}=\mathrm{e}^{\frac{\mathrm{i}\pi}{3}}$, properties (2.21)
hold with $k-1$ replaced by $k-\mathrm{e}^{\frac{\mathrm{i}\pi}{3}}$.
Relations (2.22) also hold provided we replace the indices $1$, $2$, $3$ of
the matrix entries by $2$, $3$, $1$, respectively.

  3. (iii)

$M$ and $M^{-1}$ have similar leading terms at the other polar singularities
$\varkappa_{3},\dots,\varkappa_{6}$ in accordance with the symmetry conditions
stated in Proposition 2.6.

###### Proof.

(i-1) Let $\check{M}\coloneqq P(k)M(x,t;k)P^{-1}(k)$. By (2.20) this function
satisfies the integral equation

$\displaystyle\check{M}(x,t;k)=I$
$\displaystyle{}+\int_{(\pm)\infty}^{x}P(k)\mathrm{e}^{-\Lambda(k)\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta}P^{-1}(k)[(\tilde{U}^{(1)}\check{M})(\xi,t;k)]P(k)\mathrm{e}^{\Lambda(k)\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta}P^{-1}(k)\mathrm{d}\xi.$

We first show that, in spite of the singularity of $P^{-1}(k)$ at $k=1$,
$\check{M}$ is regular at this point. It suffices to show that
$P(k)\mathrm{e}^{\pm\Lambda(k)\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta}P^{-1}(k)$
is non-singular at $k=1$. As $k\to 1$,

$\displaystyle\lambda_{1}(k)=-\frac{1}{\sqrt{3}}\big{(}1-\mathrm{i}\sqrt{3}(k-1)\big{)}+\mathrm{O}\big{(}(k-1)^{2}\big{)}\qquad\text{and}$
$\displaystyle\lambda_{2}(k)=-\frac{1}{\sqrt{3}}\big{(}1+\mathrm{i}\sqrt{3}(k-1)\big{)}+\mathrm{O}\big{(}(k-1)^{2}\big{)},$

hence $(3\lambda_{1}^{2}(k)-1)^{-1}=c/(k-1)+\mathrm{O}(1)$ and
$(3\lambda_{2}^{2}(k)-1)^{-1}=-c/(k-1)+\mathrm{O}(1)$ with
$c=\frac{\mathrm{i}}{2\sqrt{3}}$ , whereas $\lambda_{3}(1)=2/\sqrt{3}$ and
$(3\lambda_{3}^{2}(1)-1)^{-1}=1/3$. Thus, according to (2.6d) we find that

$\displaystyle P^{-1}(k)=\frac{c}{k-1}\begin{pmatrix}a_{1}&a_{2}&a_{3}\\\
-a_{1}&-a_{2}&-a_{3}\\\ 0&0&0\end{pmatrix}+\mathrm{O}(1)\qquad\text{as}\quad
k\to 1$

with $a_{1}=1$, $a_{2}=-\sqrt{2}/\sqrt[4]{3}$, and $a_{3}=-1/\sqrt{3}$. On the
other hand, since $\lambda_{1}(1)=\lambda_{2}(1)$ the first two columns of
$P(1)$ are equal (see (2.6c)) and the first two diagonal entries of the
diagonal matrix
$\mathrm{e}^{\pm\Lambda(1)\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta}$ are
the same. Then, the first two columns of the product
$P(1)\mathrm{e}^{\pm\Lambda(1)\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta}$
are the same, and
$P(k)\mathrm{e}^{\pm\Lambda(k)\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta}P^{-1}(k)$
is regular at $k=1$ since its polar part vanishes:

$\displaystyle\frac{c}{k-1}P(1)\mathrm{e}^{\pm\Lambda(1)\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta}\begin{pmatrix}a_{1}&a_{2}&a_{3}\\\
-a_{1}&-a_{2}&-a_{3}\\\ 0&0&0\end{pmatrix}\equiv 0.$

Thus, $\check{M}$ is also regular at $k=1$.

Since $M(x,t;k)=P^{-1}(k)\check{M}(x,t;k)P(k)$ where the last two factors are
regular at $k=1$, the leading term of $M$ at $k=1$ is

$\displaystyle\frac{1}{k-1}\begin{pmatrix}1&1&1\\\ -1&-1&-1\\\
0&0&0\end{pmatrix}\begin{pmatrix}ca_{1}&0&0\\\ 0&ca_{2}&0\\\
0&0&ca_{3}\end{pmatrix}\check{M}(x,t;1)P(1).$

The first two columns of $P(1)$ being equal, it is the same for the product
$R(x,t)$ of the last three factors. For multiplication by the first factor we
can replace $R$ by the diagonal matrix whose diagonal entries are the sums
$R_{1j}+R_{2j}+R_{3j}$. Thus we arrive at (2.21a) with some $\alpha(x,t)$ and
$\beta(x,t)$. The relations $\alpha=-\bar{\alpha}$ and $\beta=-\bar{\beta}$
come from the symmetry (S1) stated in Proposition 2.6.

(i-2) Similarly, $\check{M}^{-1}=PM^{-1}P^{-1}$ also satisfies an integral
equation with non-singular coefficients:

$\displaystyle\check{M}^{-1}(x,t;k)=I$
$\displaystyle{}-\int_{(\pm)\infty}^{x}P(k)\mathrm{e}^{-\Lambda(k)\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta}P^{-1}(k)[\big{(}\check{M}^{-1}\tilde{U}^{(1)}\big{)}(\xi,t;k)]P(k)\mathrm{e}^{\Lambda(k)\int_{x}^{\xi}q^{2}(\zeta,t)\mathrm{d}\zeta}P^{-1}(k)\mathrm{d}\xi,$

which gives (2.21b).

(i-3) Using the fact that $\det M\equiv 1$, we calculate $M^{-1}$ starting
from (2.21a), then, comparing the result with (2.21b) we get different
expressions of $\alpha_{1}$ and $\beta_{1}$ as linear combinations of $\alpha$
and $\beta$. Thus, $\alpha=\beta=0$ implies $\alpha_{1}=\beta_{1}=0$.
Moreover, by comparison of these different expressions of $\alpha_{1}$ and
$\beta_{1}$ we get the relations (2.22a) for $\tilde{M}$ provided $\alpha$ or
$\beta$ is $\neq 0$. Similarly, starting from $\tilde{M}^{-1}$ instead of
$\tilde{M}$ we get the relations (2.22b) for $\tilde{M}^{(1)}$ provided
$\alpha_{1}$ or $\beta_{1}$ is $\neq 0$.

(ii)–(iii) Similar arguments apply at $k=\varkappa_{2},\dots,\varkappa_{6}$. ∎

##  3 How the solution $\boldsymbol{u(x,t)}$ can be recovered from
eigenfunctions

Starting from a solution $u(x,t)$ of the Novikov equation we have introduced

  1. (a)

a $3\times 3$ matrix-valued function $M(x,t;k)$, solution of the system of
integral equations (2.20),

  2. (b)

a new variable $y(x,t)$ defined by (2.12).

In this section, introducing $\hat{M}(y,t;k)\coloneqq M(x(y,t),t;k)$ through
(a) and (b), we will show that $u(x,t)$ can be recovered in terms of
$\hat{M}(y,t;k)$ evaluated at $k=\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}$.

###  3.1 A Lax pair appropriate for small $\boldsymbol{z}$

Coming back to the original Lax pair (2.2), let us introduce another Lax pair,
whose solutions are well-controlled at
$k=\kappa_{l}\coloneqq\mathrm{e}^{\frac{\mathrm{i}\pi}{6}+\frac{\mathrm{i}\pi(l-1)}{3}}$,
$l=1,\dots,6$; here $\\{\kappa_{l}\\}_{l=1}^{6}$ are characterized by the
property $z(\kappa_{l})=0$. Define
$\tilde{\Phi}^{(0)}\equiv\tilde{\Phi}^{(0)}(x,t;k)$ (for
$k\neq\kappa_{1},\dots,\kappa_{6}$) by

$\displaystyle\tilde{\Phi}^{(0)}=P^{-1}(k)\Phi.$

This reduces the Lax pair (2.2) to the new one

$\displaystyle\tilde{\Phi}^{(0)}_{x}-\Lambda(k)\tilde{\Phi}^{(0)}=\tilde{U}^{(0)}\tilde{\Phi}^{(0)},$
(3.1a)
$\displaystyle\tilde{\Phi}^{(0)}_{t}-A(k)\tilde{\Phi}^{(0)}=\tilde{V}^{(0)}\tilde{\Phi}^{(0)},$
(3.1b) where
$\displaystyle\tilde{U}^{(0)}(x,t;k)=P^{-1}(k)\left(U(x,t;k)-U_{\infty}(k)\right)P(k)$
$\displaystyle{}=z^{2}(k)m(x,t)\begin{pmatrix}\frac{2\lambda_{1}(k)}{(3\lambda_{1}^{2}(k)-1)\lambda_{1}(k)}&\frac{\lambda_{1}(k)+\lambda_{2}(k)}{(3\lambda_{1}^{2}(k)-1)\lambda_{1}(k)}&\frac{\lambda_{1}(k)+\lambda_{3}(k)}{(3\lambda_{1}^{2}(k)-1)\lambda_{1}(k)}\vspace{1mm}\\\
\frac{\lambda_{2}(k)+\lambda_{1}(k)}{(3\lambda_{2}^{2}(k)-1)\lambda_{2}(k)}&\frac{2\lambda_{2}(k)}{(3\lambda_{2}^{2}(k)-1)\lambda_{2}(k)}&\frac{\lambda_{2}(k)+\lambda_{3}(k)}{(3\lambda_{2}^{2}(k)-1)\lambda_{2}(k)}\vspace{1mm}\\\
\frac{\lambda_{3}(k)+\lambda_{1}(k)}{(3\lambda_{3}^{2}(k)-1)\lambda_{3}(k)}&\frac{\lambda_{3}(k)+\lambda_{2}(k)}{(3\lambda_{3}^{2}(k)-1)\lambda_{3}(k)}&\frac{2\lambda_{3}(k)}{(3\lambda_{3}^{2}(k)-1)\lambda_{3}(k)}\end{pmatrix},$
(3.1c)
$\displaystyle\tilde{V}^{(0)}(x,t;k)=P^{-1}(k)\left(V(x,t;k)-V_{\infty}(k)\right)P(k)$
$\displaystyle{}=P^{-1}(k)\begin{pmatrix}-(u+1)u_{x}&\frac{u_{x}}{z(k)}-(u^{2}+2u)\hat{m}z(k)&u_{x}^{2}\vspace{1mm}\\\
\frac{u}{z(k)}&0&-\frac{u_{x}}{z(k)}-(u^{2}+2u)\hat{m}z(k)\vspace{1mm}\\\
-u^{2}-2u&\frac{u}{z(k)}&(u+1)u_{x}\end{pmatrix}P(k).$ (3.1d)

######  Remark 3.1.

Notice that $\tilde{U}^{(0)}(x,t;k)$ has a finite limit at $k=\kappa_{l}$,
$l=1,\dots,6$. Indeed, $\lambda_{j}(\kappa_{l})=0$ for one, and only one,
value of $j=1$, $2$ or $3$. For example, for
$k=\kappa_{1}=\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}$ we have
$\lambda_{2}(\kappa_{1})=0$ whereas $\lambda_{1}(\kappa_{1})=-1$ and
$\lambda_{3}(\kappa_{1})=1$. Moreover,
$\lim_{k\to\kappa_{1}}\frac{z^{2}(k)}{\lambda_{2}(k)}=\lambda_{1}(\kappa_{1})\lambda_{3}(\kappa_{1})=-1$
and thus

$\displaystyle\lim_{k\to\kappa_{1}}\tilde{U}^{(0)}(x,t;k)=m(x,t)\begin{pmatrix}0&0&0\\\
-1&0&1\\\ 0&0&0\end{pmatrix}\not\equiv 0.$

Similarly if $k\to\kappa_{l}$, $l=2,\dots,6$.

We emphasize that this is different from the case of the DP equation, where,
on the contrary, the equality $\tilde{U}^{(0)}(x,t;\kappa_{1})=0$ holds for
all $x$ and $t$.

###  3.2 Eigenfunctions

####  3.2.1 Eigenfunctions appropriate for small $\boldsymbol{z}$

We introduce $M^{(0)}\equiv M^{(0)}(x,t;k)$ by

$\displaystyle M^{(0)}=\tilde{\Phi}^{(0)}\mathrm{e}^{-x\Lambda-tA},$

and determine $M^{(0)}$ as the solution of a system of integral equations
similar to the system (2.20) determining $M$:

$\displaystyle
M^{(0)}_{jl}(x,t;k)=I_{jl}+\int_{\infty_{j,l}}^{x}\mathrm{e}^{-\lambda_{j}(k)(\xi-x)}[(\tilde{U}^{(0)}M^{(0)})_{jl}(\xi,t;k)]\mathrm{e}^{\lambda_{l}(k)(\xi-x)}\mathrm{d}\xi.$
(3.2)

In the case of the DP equation, $M^{(0)}(x,t;z(k))\big{|}_{z=0}\equiv I$. For
the Novikov equation, this is not true; but, since
$\tilde{U}^{(0)}(\xi,t;\kappa_{1})\Delta\tilde{U}^{(0)}(x,t;\kappa_{1})\equiv
0$ for any $\xi$ and $x$, and any diagonal matrix $\Delta$, the solution of
(3.2) for $k=\kappa_{1}=\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}$ can be written
explicitly:

$\displaystyle
M_{jl}^{(0)}(x,t;\kappa_{1})=I+\int_{\infty_{jl}}^{x}\mathrm{e}^{-\lambda_{j}(\kappa_{1})(\xi-x)}\tilde{U}_{jl}^{(0)}(\xi,t;\kappa_{1})\mathrm{e}^{\lambda_{l}(\kappa_{1})(\xi-x)}\mathrm{d}\xi\equiv
I_{jl}+L_{jl}(x,t),$

where

$\displaystyle L(x,t)=\begin{pmatrix}0&0&0\\\
\int_{x}^{\infty}m(\xi,t)\mathrm{e}^{x-\xi}\mathrm{d}\xi&0&\int_{-\infty}^{x}m(\xi,t)\mathrm{e}^{\xi-x}\mathrm{d}\xi\\\
0&0&0\end{pmatrix}.$

Similarly for $k=\kappa_{2},\dots,\kappa_{6}$. Using that $m=u-u_{xx}$, we see
that the non-zero entries $L_{21}$ and $L_{23}$ reduce to

$\displaystyle L_{21}=u+u_{x},\qquad L_{23}=u-u_{x},$

and thus $M^{(0)}\big{(}x,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}$ can
be explicitly expressed in terms of $u(x,t)$:

$\displaystyle
M^{(0)}\big{(}x,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}=\begin{pmatrix}1&0&0\\\
u+u_{x}&1&u-u_{x}\\\ 0&0&1\end{pmatrix}(x,t).$ (3.3)

####  3.2.2 Comparison of eigenfunctions

We will get the value of $M$ at $k=\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}$ from
that of $M^{(0)}$ by using that $M$ and $M^{(0)}$ are related. We indeed have

$\displaystyle M=P^{-1}D^{-1}\Phi\mathrm{e}^{-y\Lambda-tA},\qquad
M^{(0)}=P^{-1}\Phi^{(0)}\mathrm{e}^{-x\Lambda-tA},$

where $\Phi$ and $\Phi^{(0)}$ are solutions of the same system of linear
differential equations (2.2). They are then related by $\Phi=\Phi^{(0)}C$
where $C\equiv C(k)$ is independent of $(x,t)$. Thus, $M$ and $M^{(0)}$ are
related by

$\displaystyle
M(x,t;k)=P^{-1}(k)D^{-1}(x,t)P(k)M^{(0)}(x,t;k)\mathrm{e}^{x\Lambda(k)+tA(k)}C(k)\mathrm{e}^{-y(x,t)\Lambda(k)-tA(k)}.$

Now, since for $k\not\in\Sigma$, $M^{(0)}$ and $M$ are bounded as
$x\to-\infty$ and have the same limit as $x\to+\infty$:

$\displaystyle M,\,M^{(0)}\xrightarrow[x\to+\infty]{}I,$

it follows that $C(k)\equiv I$. Finally, we get

$\displaystyle
M(x,t;k)=P^{-1}(k)D^{-1}(x,t)P(k)M^{(0)}(x,t;k)\mathrm{e}^{(x-y(x,t))\Lambda(k)},$

where $y$ is as in (2.12):

$\displaystyle
y(x,t)=x-\int_{x}^{\infty}\big{(}q^{2}(\xi,t)-1\big{)}\mathrm{d}\xi.$

In particular, at $k=\kappa_{1}\equiv\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}$ we
have

$\displaystyle
P^{-1}(k)D^{-1}(x,t)P(k)\big{|}_{k=\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}}=\begin{pmatrix}\frac{q^{-1}+q}{2}&0&\frac{q^{-1}-q}{2}\\\
1-q^{-1}&1&1-q^{-1}\\\ \frac{q^{-1}-q}{2}&0&\frac{q^{-1}+q}{2}\end{pmatrix}$

and thus, using (3.3),

$\displaystyle
M\big{(}x,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}=\begin{pmatrix}\frac{q^{-1}+q}{2}\mathrm{e}^{y-x}&0&\frac{q^{-1}-q}{2}\mathrm{e}^{x-y}\\\
(1-q^{-1}+u+u_{x})\mathrm{e}^{y-x}&1&(1-q^{-1}+u-u_{x})\mathrm{e}^{x-y}\\\
\frac{q^{-1}-q}{2}\mathrm{e}^{y-x}&0&\frac{q^{-1}+q}{2}\mathrm{e}^{x-y}\end{pmatrix}.$
(3.4)

###  3.3 Recovering $\boldsymbol{u(x,t)}$ from eigenfunctions

######  Proposition 3.2.

Let $u(x,t)$ be the solution of the Cauchy problem (1.9) for the Novikov
equation. Let $\hat{M}(y,t;k)\coloneqq M(x(y,t),t;k)$ where $M(x,t;k)$ is the
solution of the system (2.20) and $y(x,t)$ is defined by (2.12), both built on
$u(x,t)$.

Then we can recover $u(x,t)$ from values of the eigenfunctions
$\hat{M}_{jl}(y,t;k)$ at $k=\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}$. We indeed
have the following parametric representation:

$\displaystyle u(x,t)=\hat{u}(y(x,t),t),$ (3.5a) where $x(y,t)$ and
$\hat{u}(y,t)$ are given by $\displaystyle
x(y,t)=y+\frac{1}{2}\ln\frac{\hat{M}_{33}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}}{\hat{M}_{11}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}},$
(3.5b)
$\displaystyle\hat{u}(y,t)=\frac{1}{2}\hat{N}_{1}(y,t)\left(\frac{\hat{M}_{33}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}}{\hat{M}_{11}(y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}})}\right)^{1/2}+\frac{1}{2}\hat{N}_{3}(y,t)\left(\frac{\hat{M}_{33}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}}{\hat{M}_{11}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}}\right)^{-1/2}-1,$
(3.5c) where
$\displaystyle\hat{N}_{k}(y,t)\coloneqq\sum_{j=1}^{3}\hat{M}_{jk}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)},\qquad
k=1,2,3.$ (3.5d)

###### Proof.

(3.5b) follows from (3.4) and from the definition (2.12) of $y(x,t)$. Further,
multiplying (3.4) by the row vector $(\begin{matrix}1&1&1\end{matrix})$ from
the left and introducing

$\displaystyle
N(x,t)\equiv\begin{pmatrix}N_{1}(x,t)&N_{2}(x,t)&N_{3}(x,t)\end{pmatrix}\coloneqq\begin{pmatrix}1&1&1\end{pmatrix}M\big{(}x,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}$

we have

$\displaystyle N_{1}(x,t)=(1+u(x,t)+u_{x}(x,t))\mathrm{e}^{y-x},$
$\displaystyle N_{2}(x,t)=1,$ $\displaystyle
N_{3}(x,t)=(1+u(x,t)-u_{x}(x,t))\mathrm{e}^{x-y}$

and thus $u(x,t)$ can be obtained in terms of $N(x,t)$ as follows:

$\displaystyle
u(x,t)=\frac{1}{2}N_{1}(x,t)\mathrm{e}^{x-y}+\frac{1}{2}N_{3}(x,t)\mathrm{e}^{y-x}-1.$
(3.6)

Now notice that (3.5b) reads

$\displaystyle\mathrm{e}^{2(x-y)}=\frac{\hat{M}_{33}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}}{\hat{M}_{11}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}}.$
(3.7)

Finally, introducing $\hat{N}_{k}$ as in (3.5d), using (3.7), and writing
(3.6) in the variables $(y,t)$ (so that $N(x,t)=\hat{N}(y,t)$), we arrive at
(3.5c). ∎

######  Remark 3.3.

From (2.12) and (2.10), another expression for $\hat{u}(y,t)$ follows:

$\displaystyle(\hat{u}(y,t)+1)^{2}=\partial_{t}\,x(y,t)+1.$ (3.8)

Notice that the analogous expression in the case of the DP equation looks
differently [10]:

$\displaystyle\hat{u}_{\mathrm{DP}}(y,t)=\partial_{t}\,x(y,t)+1.$

######  Remark 3.4.

The structure of (3.5) coincides with that for the multisoliton solution in
[36] (see formulas (3.3a) and (3.3b) in [36]), taking into account the
relationship between solutions of the Novikov equation (1.1) on a non-zero
constant background and solutions of (1.9a) on the zero background, presented
in the Introduction.

##  4 Riemann–Hilbert problem

In Section 3 we have shown how we can express the solution $u$ of the Cauchy
problem for the Novikov equation by evaluating certain eigenfunctions –
solutions of the Lax pair equations. Notice they were defined using the
solution $u$ itself.

In the framework of the Riemann–Hilbert approach to the Cauchy problem for an
integrable nonlinear equation, one is looking for obtaining these
eigenfunctions in terms of the solution of an appropriate factorization
problem, of Riemann–Hilbert type. The factorization problem is formulated in
the complex plane of a spectral parameter $k$ whereas $x$ and $t$ play the
role of parameters, and the data for this problem are uniquely determined, in
spectral terms, by the initial data for the Cauchy problem.

###  4.1 RH problem satisfied by $\boldsymbol{\hat{M}}$

Let $M(x,t;k)$ be as in Section 3, solution of the system of integral
equations (2.20). The key observation is that the limiting values
$M_{\pm}(x,t;k)$ (on $l_{\nu}$) of $M(x,t;k^{\prime})$ as $k^{\prime}\to k$
from the positive or negative side of $l_{\nu}$, $\nu=1,\dots,6$ are related
as follows:

$\displaystyle
M_{+}(x,t;k)=M_{-}(x,t;k)\mathrm{e}^{Q(x,t;k)}S_{0}(k)\mathrm{e}^{-Q(x,t;k)},\qquad
k\in l_{\nu}.$ (4.1)

We indeed have $M_{\pm}=\hat{\Phi}_{\pm}\mathrm{e}^{-Q}$ where
$\hat{\Phi}_{\pm}$ are two solutions of the system of ordinary differential
equations (2.7). They are then related by $\hat{\Phi}_{+}=\hat{\Phi}_{-}S_{0}$
where $S_{0}$ is a matrix independent of $(x,t)$. Considering (4.1) at $t=0$
we see that $S_{0}(k)$ is completely determined by $u(x,0)$, i.e., by the
initial data for the Cauchy problem (1.9), via the solution $M(x,0;k)$ of the
system (2.20) whose coefficients are determined by $u(x,0)$:

$\displaystyle
S_{0}(k)=\mathrm{e}^{-Q(x,0;k)}M_{-}^{-1}(x,0;k)M_{+}(0,0;k)\mathrm{e}^{Q(x,0;k)}$

(recall that
$Q(x,0;k)=\left(x-\int_{x}^{\infty}(q^{2}(\xi,0)-1)\mathrm{d}\xi\right)\Lambda(k)$
and thus $Q(x,0;k)$ is also determined by $u(x,0)$).

Moreover, $S_{0}(k)$ has a special matrix structure: for $k\in l_{1}\cup
l_{4}\equiv\mathbb{R}$,

$\displaystyle S_{0}(k)=\begin{pmatrix}1&0&0\\\ -r(k)&1&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}1&\overline{r(k)}&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},$ (4.2)

where $r(k)\in L^{\infty}(\mathbb{R})$ and $r(k)=\mathrm{O}(k^{-1})$ as
$k\to\pm\infty$ (this structure follows from the analysis of the behavior of
$M_{\pm}(x,0;k)$ as $x\to\pm\infty$; for the details, see [10, Section 3.1]),
whereas the expression of $S_{0}(k)$ for the other parts of $\Sigma$ follows
from the symmetries stated in Proposition 2.6. Thus the jump matrices on all
parts of the contour are determined in terms of a single scalar function, the
_reflection coefficient_ $r(k)$.

On the other hand, in order to have explicit exponentials in the r.h.s. of
(4.1), one replaces the pair of parameters $(x,t)$ by $(y,t)$, where
$y=y(x,t)$ was introduced in (2.12), see also (2.11). Following this
observation, (4.1) can be rewritten as

$\displaystyle\hat{M}_{+}(y,t;k)=\hat{M}_{-}(y,t;k)\mathrm{e}^{y\Lambda(k)+tA(k)}S_{0}(k)\mathrm{e}^{-y\Lambda(k)-tA(k)},\qquad
k\in\Sigma$ (4.3)

where $\Sigma=l_{1}\cup\dots\cup l_{6}$ and

$\displaystyle M(x,t;k)=\hat{M}(y(x,t),t;k).$ (4.4)

Now (4.3) says that $\hat{M}$ – as in (4.4) – is solution of a factorization
problem:

######  Factorization problem.

Given $S_{0}(k)$, $k\in\Sigma$, find a piece-wise (w.r.t. $\Sigma$)
meromorphic (in $k$), $3\times 3$ matrix-valued function $\hat{M}(y,t;k)$ ($y$
and $t$ are parameters), whose limiting values satisfy the jump condition
(4.3).

This condition, being supplemented by conditions at possible poles of
$\hat{M}(y,t;k)$, by a normalization condition, and by certain structural
conditions, will constitute the RH problem satisfied by $\hat{M}$.

######  Additional conditions.

The dependence on $k$ of the eigenvalues $\lambda_{j}(k)$ in (2.20) is exactly
the same as in the case of the DP equation, which implies that most conditions
involved in the RH problem for the Novikov equation have exactly the same form
as for the Degasperis–Procesi equation, cf. [10].

  1. 1.

For the residue conditions at poles (if any)
$k_{n}\in\mathbb{C}\setminus\Sigma$ of $\hat{M}(y,t;k)$, see [10, Section 3.2]
and [11, Section 2.3.5]. Recall the following result ([1]): Generically, there
are at most a finite number of poles $k_{n}$ lying in
$\mathbb{C}\setminus\Sigma$, each of them being simple, with residue
conditions of a special matrix form. In particular,

$\displaystyle\operatorname{Res}_{k=k_{n}}\hat{M}(y,t;k)=\lim_{k\to
k_{n}}\hat{M}(y,t;k)\mathrm{e}^{y\Lambda(k)+tA(k)}v_{n}\mathrm{e}^{-y\Lambda(k)-tA(k)},$

where $v_{n}$ is some constant $3\times 3$ matrix with only one non-zero entry
at a position depending on the sector of $\mathbb{C}\setminus\Sigma$ to which
$k_{n}$ belongs.

  2. 2.

The normalization condition $\hat{M}(y,t;\infty)=I$ is Proposition 2.5(ii).

  3. 3.

The structural conditions on the polar parts of $\hat{M}$ at
$\varkappa_{1},\dots,\varkappa_{6}$ have exactly the same form as for the DP
equation, cf. [10].

  4. 4.

On the other hand, the structural conditions at
$k=\kappa_{1},\dots,\kappa_{6}$, see (3.3) and (3.4), are different from those
for the DP equation (cf. (3.12) and (3.13b) in [10]).

Summarizing, we get that the eigenfunctions $\hat{M}(y,t;k)$ – defined as in
Section 3 – satisfy the following Riemann–Hilbert problem:

######  Main RH problem (for the Novikov equation).

Given $r(k)\in L^{\infty}(\mathbb{R})$ such that $r(k)=\mathrm{O}(k^{-1})$ as
$k\to\pm\infty$ and $\\{k_{n},v_{n}\\}_{n=1}^{N}$, find a piece-wise (w.r.t.
$\Sigma=l_{1}\cup\dots\cup l_{6}$) meromorphic (in the complex $k$-plane),
$3\times 3$ matrix-valued function $\hat{M}(y,t;k)$ satisfying the following
conditions:

  1. (1)

_Jump conditions_ :

$\displaystyle\hat{M}_{+}(y,t;k)=\hat{M}_{-}(y,t;k)S(y,t;k),\qquad
k\in\Sigma\setminus\\{\varkappa_{1},\dots,\varkappa_{6}\\},$

where $\varkappa_{\nu}\coloneqq\mathrm{e}^{\frac{\mathrm{i}\pi}{3}(\nu-1)}$
and

$\displaystyle
S(y,t;k)=\mathrm{e}^{y\Lambda(k)+tA(k)}S_{0}(k)\mathrm{e}^{-y\Lambda(k)-tA(k)},$

where $S_{0}(k)$ is given in terms of $r(k)$ by (4.2) for $k\in l_{1}\cup
l_{4}$ and then by the respective symmetries for the other parts of $\Sigma$.
Here $\Lambda(k)$ and $A(k)$ are diagonal $3\times 3$ matrices with diagonal
entries $\lambda_{i}(k)$ and $A_{i}(k)$, respectively, see (2.6a) and (2.6b),
where we have performed the change of parameter $z=z(k)$, see (2.17) and
(2.18).

  2. (2)

_Residue conditions_ (generic): For some $k_{n}\in\mathbb{C}\setminus\Sigma$,
$n=1,\dots,N$,

$\displaystyle\operatorname{Res}_{k=k_{n}}\hat{M}(y,t;k)=\lim_{k\to
k_{n}}\hat{M}(y,t;k)\mathrm{e}^{y\Lambda(k)+tA(k)}v_{n}\mathrm{e}^{-y\Lambda(k)-tA(k)},$
(4.5)

where $v_{n}$ is some constant $3\times 3$ matrix with only one non-zero
entry, whose position depends on the sector of $\mathbb{C}\setminus\Sigma$
containing $k_{n}$ (dictated by the order of $\operatorname{Re}\lambda_{j}(k)$
in the sector, see [1]). For example, if $k_{n}\in\Omega_{1}$, the non-zero
entry of $v_{n}$ can be either $(v_{n})_{12}$ or $(v_{n})_{23}$. Then the
positions (as well as the values) of the non-zero entries of $v_{n}$ in the
other sectors of $\mathbb{C}\setminus\Sigma$ are determined by the symmetries
(S1)–(S3) from Proposition 2.6.

  3. (3)

_Normalization condition_ :

$\displaystyle\hat{M}(y,t;k)=I+\mathrm{o}(1)\qquad\text{as}\quad k\to\infty.$

  4. (4)

_Boundedness condition_ : $\hat{M}(y,t;k)$ and $\hat{M}^{-1}(y,t;k)$ are
bounded in the closure of each component of $\mathbb{C}\setminus\Sigma$ apart
from vicinities of $k_{1},\dots,k_{N}$ and
$\varkappa_{1},\dots,\varkappa_{6}$.

  5. (5)

_Structural conditions_ at $k=\varkappa_{1},\dots,\varkappa_{6}$, where
$\varkappa_{\nu}\coloneqq\mathrm{e}^{\frac{\mathrm{i}\pi}{3}(\nu-1)}$:

    1. i)

The singular behavior of $\hat{M}$ and $\hat{M}^{-1}$ at $k=\varkappa_{1}=1$
is described by the structural conditions (2.21), (2.22) (where $\alpha$,
$\beta$, $\alpha_{1}$, $\beta_{1},\dots$ are not specified).

    2. ii)

At $k=\varkappa_{2},\dots,\varkappa_{6}$ we have similar structural conditions
on the singularities of $\hat{M}$ and $\hat{M}^{-1}$ (see Proposition 2.8,
(ii)–(iii)).

  6. (6)

_Structural conditions_ at $k=\kappa_{1},\dots,\kappa_{6}$, where
$\kappa_{l}\coloneqq\mathrm{e}^{\frac{\mathrm{i}\pi}{6}+\frac{\mathrm{i}\pi}{3}(l-1)}$:

    1. i)

For $k=\kappa_{1}=\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}$,

$\displaystyle\hat{M}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}=\begin{pmatrix}\frac{\hat{q}^{-1}+\hat{q}}{2}&0&\frac{\hat{q}^{-1}-\hat{q}}{2}\\\
p_{1}&1&p_{2}\\\
\frac{\hat{q}^{-1}-\hat{q}}{2}&0&\frac{\hat{q}^{-1}+\hat{q}}{2}\end{pmatrix}\begin{pmatrix}f&0&0\\\
0&1&0\\\ 0&0&f^{-1}\end{pmatrix}$ (4.6)

with some (unspecified) $\hat{q}(y,t)>0$, $f(y,t)>0$,
$p_{1}(y,t)\in\mathbb{R}$ and $p_{2}(y,t)\in\mathbb{R}$ (see (3.4)).

    2. ii)

For $k=\kappa_{2},\dots,\kappa_{6}$, the corresponding structure of
$\hat{M}(y,t;\kappa_{l})$ follows from (4.6) taking into account the
symmetries of Proposition 2.6.

Now we observe that this RH problem can be defined using the initial data of
the Cauchy problem.

######  Properties (of the main RH problem).

Let $u_{0}(x)$ be given satisfying the assumptions made for the Cauchy problem
(1.9).

  1. i)

$u_{0}(x)$ defines the main RH problem as follows.

    1. ia)

We first define the jump matrix $S_{0}(k)$ by

$\displaystyle
S_{0}(k)=\mathrm{e}^{-Q(0,0;k)}M_{-}^{-1}(0,0;k)M_{+}(0,0;k)\mathrm{e}^{Q(0,0;k)},$

where $M(x,0;k)$ and $Q(x,0;k)$ can be completely determined by $u_{0}(x)$.
Notice that $S_{0}(k)$ has necessarily the structure (4.2) with some $r(k)$.

    2. ib)

The $k_{n}$’s in $\mathbb{C}\setminus\Sigma$ and the $v_{n}$’s in (4.5) are
determined by $u_{0}(x)$.

  2. ii)

This RH problem has a solution.

  3. iii)

The solution is unique.

###### Proof.

ia) The expression giving $S_{0}(k)$ derives from (4.1). Further, $M(x,0;k)$
is the solution of the system of integral equations (2.20) written for $t=0$,
and this system is completely determined by $u_{0}(x)$ through
$\hat{m}_{0}(x)\coloneqq u_{0}(x)-u_{0}^{\prime\prime}(x)+1$. By (2.4) we
indeed have $q^{2}(x,0)=\hat{m}_{0}^{2/3}(x)$, and in the definition of
$\hat{U}(x,0;k)$ (see (2.7c) and (2.5b)) the term $\hat{m}(x,0)$ is also
$\hat{m}_{0}(x)$. Similarly for $Q(x,0;k)=y(x,0)\Lambda(k)$ where $y(x,0)$ is
determined by $q^{2}(x,0)$.

ib) The $k_{n}$’s are determined by $M(x,0;k)$ and $v_{n}$ is determined by
(4.5) considered at $t=0$, hence also by $u_{0}(x)$.

ii) Under the assumptions made on $u_{0}$ there exists a solution $u(x,t)$ of
the Cauchy problem (1.9). The main RH problem has then as solution the
associated eigenfunction $\hat{M}(y,t;k)$.

iii) The normalization condition supplemented by the structural conditions
(4.6) and (2.21), (2.22) provides uniqueness of the solution (see [6,
Proposition 3.2]). ∎

###  4.2 Main theorem

Starting directly from the main RH problem we show how its solution gives a
representation of the solution of the Cauchy problem for the Novikov equation.

######  Theorem 4.1.

Let $u_{0}(x)$ satisfy assumptions made for the Cauchy problem (1.9) for the
Novikov equation. Let $\hat{M}(y,t;k)$ be the solution of the associated main
RH problem (1)–(6). By evaluating $\hat{M}$ at
$k=\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}$ we get the parametric representation
(3.5) for the solution $u(x,t)$ of the Cauchy problem (1.9):

$\displaystyle u(x,t)=\hat{u}(y(x,t),t),$

where $x(y,t)$ and $\hat{u}(y,t)$ are given by

$\displaystyle
x(y,t)=y+\frac{1}{2}\ln\frac{\hat{M}_{33}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}}{\hat{M}_{11}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}},$
$\displaystyle\hat{u}(y,t)=\frac{1}{2}\hat{N}_{1}(y,t)\left(\frac{\hat{M}_{33}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}}{\hat{M}_{11}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}}\right)^{1/2}+\frac{1}{2}\hat{N}_{3}(y,t)\left(\frac{\hat{M}_{33}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}}{\hat{M}_{11}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}}\right)^{-1/2}-1,$

with
$\hat{N}_{k}(y,t):=\sum\limits_{j=1}^{3}\hat{M}_{jk}\big{(}y,t;\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}\big{)}$,
$k=1,3$.

######  Remark 4.2.

The solution of the DP equation is extracted from the solution of the
associated RH problem in a different way (see [10, equation (3.15)]).

###  4.3 Applications

The representation (3.5) of the solution of the Cauchy problem (1.9) for the
Novikov equation (1.9a) in terms of the solution of the associated main RH
problem (1)–(6) (given in Theorem 4.1) can be used in principle, as for other
integrable peakon equations (see, e.g., [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]),
to construct soliton-type solutions and also to analyse the long time behavior
of the solution.

####  4.3.1 Soliton-type solutions

In the case $r(k)\equiv 0$, solving the RH problem reduces to solving a system
of linear algebraic equations associated with the residue conditions.
Following this way, one can describe not only smooth classical solutions, but
also singular and/or multivalued solutions associated with breaks of
bijectivity in the change of variables $x\mapsto y$ (see, e.g., [11, 12]).

####  4.3.2 Long-time asymptotics

Noticing that the form and the $k$-dependence of the jump matrix, see (4.3)
and (4.2), are the same as in the case of the DP equation, a reasonable
conjecture is that the long-time analysis for the Novikov equation can be done
following the same steps as for the DP equation [10], by adapting the
nonlinear steepest descent method by Deift and Zhou [24].

But there is a problem that prevents from transferring the long-time analysis
literally from the case of the DP equation to the Novikov equation. Indeed, in
[4, 10] the long-time analysis was actually done for a matrix RH problem
regular at the points $\varkappa_{1},\dots,\varkappa_{6}$ whereas the original
RH problem, as in the case of the Novikov equation, was singular at these
points. This discrepancy disappeared in [4, 10] after left multiplying by the
row vector $(\begin{matrix}1&1&1\end{matrix})$ and thus passing from the
matrix to a (regular) row vector RH problem and exploiting the uniqueness of
the solution of the latter problem. In turn, the solution of the row vector RH
problem evaluated at $\kappa_{1},\dots,\kappa_{6}$, and particularly, at
$k=\kappa_{1}=\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}$, gave, in a simple way,
the quantity $\mathrm{e}^{x(y,t)-y}$, from which it was straightforward to
deduce a parametric representation for $u(x,t)$, see (3.5).

Notice that a similar situation takes place for other Camassa–Holm-type
equations including the Camassa–Holm equation itself, see [5, 8]. On the
contrary, in the case of the Novikov equation, the situation is different.
Indeed, the solution of the similar row vector RH problem cannot be directly
used to recover $\mathrm{e}^{x(y,t)-y}$ since, being evaluated at
$k=\mathrm{e}^{\frac{\mathrm{i}\pi}{6}}$, it has the form

$\displaystyle\begin{pmatrix}(1+u+u_{x})\mathrm{e}^{y-x}&1&(1+u-u_{x})\mathrm{e}^{x-y}\end{pmatrix},$

left multiplying (3.4) by the row vector $(\begin{matrix}1&1&1\end{matrix})$.
This issue has forced us to use instead the solution of the matrix RH problem
(3.4) in (3.5). But now there is an (open) problem of relating the solution of
the original (singular) problem to the solution of the limiting (as
$t\to\infty$) regular RH problem. In [4, 10], it was the latter problem that
was used for obtaining the main asymptotic term of $u(x,t)$ in an explicit
form, with parameters determined by the initial data in terms of the
associated reflection coefficient.

####  4.3.3 Initial boundary value problem

Eigenfunctions associated with the Lax pair equations (2.13) and (3.1) via
integral Fredholm equations of type (2.14) with an appropriate choice of
$(x^{*},t^{*})$ allows formulating a RH problem suitable for analyzing initial
boundary value (or half-line) problems following the procedure presented in
[4] in the case of the DP equation.

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