A Riemann-Hilbert Approach for the Novikov Equation

We develop the inverse scattering transform method for the Novikov equation $u_t-u_{txx}+4u^2u_x=3u u_xu_{xx}+u^2u_{xxx}$ 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.

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 × 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 (nonzero) 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] u t − u txx + 4uu x = 3u x u xx + uu xxx , (1.4) which, in terms of m reads m 1 3 and the modified Korteweg-de Vries (mKdV) equation 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) →ũ(x − κt, t) + κ reduces (1.4) and (1.5) tõ u t −ũ txx + 3κũ x + 4ũũ x = 3ũ xũxx +ũũ xxx (1.6) and m 1 3 t + ũm 1 3 x = 0, respectively. Herem :=ũ −ũ xx + κ and thusũ → 0 as x → ±∞ provided u → κ as x → ±∞. 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 transformationũ →ũ :ũ(x, t) = κũ(x, κt), which preserves the zero background, allows reducing the study of (1.6) with any κ > 0 to the case of (1.6) with κ = 1. Similar arguments for the Novikov equation lead to the following: The transformations u(x, t) =ũ(x − κ 2 t, t) + κ and u(x, t) = κũ(x − κ 2 t, κ 2 t) + κ reduce the Cauchy problem for (1.1) (or (1.3)) on a non-zero constant background u → κ as x → ±∞ to the Cauchy problem on zero background (ũ → 0 andũ → 0 as x → ±∞) for the equations m 2 3 t + ũ 2 + 2κũ m respectively. Herem :=ũ −ũ xx + κ andm =ũ −ũ 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 where u 0 (x) is sufficiently smooth and decays fast as x → ±∞. Moreover, we assume that u 0 (x) satisfies the sign condition Then there exists [32] a unique global solution u(x, t) of (1.9), such that u(x, t) → 0 as x → ±∞ 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: 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: 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 × 3 matrix-valued equations, while in the case of the CH equation, they have a 2 × 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.

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 ±∞ and satisfies the sign condition (1.9d).
Then, similarly to the case of the CH equation (see, e.g., [18]), the solutionm(x, t) of (1.9) satisfies the sign conditionm(x, t) > 0 for all x ∈ R and all t > 0.

A f irst Lax pair
The Lax pair found by Hone and Wang [29] for the Novikov equation in the form (1.1) (or (1.2)) reads where z is the spectral parameter.

A modif ied 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 bym = m + 1, the modification for the Novikov equation turns out to be more involved.
Remark 2.2. The freedom in adding to V a constant (independent of (x, t)) term c · I, where I is the 3 × 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 Φ t = V Φ is independent of t. The same property holds for the equation Φ x = U Φ 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 = ∞. 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].
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.
Here λ 1 (z), λ 2 (z), and λ 3 (z) are the solutions of the algebraic equation transforms the Lax pair (2.5) into the new Lax pair: Indeed, we can writeÛ =Û (1)Û (2) , wherê Notice thatÛ (x, t; z) has a finite limit at z = 0. We can also writeV =Û (1) V (1) +V (2) Λ whereV (1) has the form ofÛ (2) with c 1 and c 2 replaced by c 3 = −(u 2 + 2u) qx q and c 4 = (u 2 + 2u)q 2 + u 2 x +1 q 2 − 1, respectively, and Finally, in order to write (2.7) in the desired form (2.3) it suffices to find a solution of the system We first notice that both equations are consistent. This follows directly from the equation 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 × 3 diagonal function

Fredholm integral equations
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]) 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 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]. In particular, for the Cauchy problem considered in the present paper, it is reasonable to choose these points to be (−∞, 0) or (+∞, 0) thus reducing the integration in (2.14) to paths parallel to the x-axis (see Fig. 1) provided the integrals (2.15) or, in view of (2.11) and (2.9a), 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 Re λ j (z) − Re λ l (z), 1 ≤ j = l ≤ 3.

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 We thus have In what follows, we will work in the complex k-plane only. So, by a slight abuse of notation, we will write λ j (k) for λ j (z(k)), and similarly for other functions of z(k), e.g., M (x, t; k) for M (x, t; z(k)) and Λ(k) for Λ(z(k)). The λ j 's are the same, as functions of k, as in the case of the DP equation, see [10]. Thus, the contour Σ = {k | Re λ j (k) = Re λ l (k) for some j = l} is also the same; it consists of six rays dividing the k-plane into six sectors Figure 2. Rays l ν , domains Ω ν , and points κ ν , κ l in the k-plane.
In order that (2.16) have a (matrix-valued) solution that is analytic, as a function of k ∈ C \ Σ, the initial points of integration ∞ jl are specified as follows for each matrix entry (j, l), 1 ≤ j, l ≤ 3: This means that we consider the system of scalar Fredholm integral equations, 1 ≤ j, l ≤ 3, where k ∈ C and I denotes the 3 × 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 1 z P −1 (z)Ṽ (2) P (z)), which, as a function of k, is not rational, direct calculations show thatÛ in (2.20) as well asV depends on k rationally, through the λ j = λ j (k)'s, see (2.8) and (2.18).  (ii) M (x, t; k) → I as k → ∞, where I is the 3 × 3 identity matrix; (iii) for k ∈ C \ Σ, M is bounded as x → −∞ and M → I as x → +∞; (iv) det M ≡ 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 ofÛ vanish as k → ∞. 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: , t;k). Proof . Indeed, the diagonal entries of the matrix Λ(k) = diag(λ j (k)) satisfy the following relations: Remark 2.7. From (S1)-(S3) it follows that the values of M at k and at ωk are related by If λ j (k) = λ l (k), j = 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 forÛ and, consequently, forΦ and M . These particular values of the spectral parameter are κ ν = e iπ 3 (ν−1) , ν = 1, . . . , 6. Taking into account the symmetries described in Proposition 2.6 leads to the following proposition. (i) As k → κ 1 = 1,  Proof . (i-1) LetM := P (k)M (x, t; k)P −1 (k). By (2.20) this function satisfies the integral equatioň We first show that, in spite of the singularity of P −1 (k) at k = 1,M is regular at this point. It suffices to show that P (k)e ±Λ(k) ξ x q 2 (ζ,t)dζ P −1 (k) is non-singular at k = 1. As k → 1, 3 , whereas λ 3 (1) = 2/ √ 3 and (3λ 2 3 (1) − 1) −1 = 1/3. Thus, according to (2.6d) we find that On the other hand, since λ 1 (1) = λ 2 (1) the first two columns of P (1) are equal (see (2.6c)) and the first two diagonal entries of the diagonal matrix e ±Λ(1) ξ x q 2 (ζ,t)dζ are the same. Then, the first two columns of the product P (1)e ±Λ(1) ξ x q 2 (ζ,t)dζ are the same, and P (k)e ±Λ(k) ξ x q 2 (ζ,t)dζ P −1 (k) is regular at k = 1 since its polar part vanishes: Thus,M is also regular at k = 1.
Since M (x, t; k) = P −1 (k)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 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 α(x, t) and β(x, t). The relations α = −ᾱ and β = −β come from the symmetry (S1) stated in Proposition 2.6.
We emphasize that this is different from the case of the DP equation, where, on the contrary, the equalityŨ (0) (x, t; κ 1 ) = 0 holds for all x and t.

Eigenfunctions appropriate for small z
and determine M (0) as the solution of a system of integral equations similar to the system (2.20) determining M : In the case of the DP equation, M (0) (x, t; z(k)) z=0 ≡ I. For the Novikov equation, this is not true; but, sinceŨ (0) (ξ, t; κ 1 )∆Ũ (0) (x, t; κ 1 ) ≡ 0 for any ξ and x, and any diagonal matrix ∆, the solution of (3.2) for k = κ 1 = e iπ 6 can be written explicitly: Similarly for k = κ 2 , . . . , κ 6 . Using that m = u − u xx , we see that the non-zero entries L 21 and L 23 reduce to and thus M (0) x, t; e iπ 6 can be explicitly expressed in terms of u(x, t):

Comparison of eigenfunctions
We will get the value of M at k = e iπ 6 from that of M (0) by using that M and M (0) are related. We indeed have where Φ and Φ (0) are solutions of the same system of linear differential equations (2.2). They are then related by Φ = Φ (0) C where C ≡ C(k) is independent of (x, t). Thus, M and M (0) are related by M (x, t; k) = P −1 (k)D −1 (x, t)P (k)M (0) (x, t; k)e xΛ(k)+tA(k) C(k)e −y(x,t)Λ(k)−tA(k) . Now, since for k ∈ Σ, M (0) and M are bounded as x → −∞ and have the same limit as x → +∞: it follows that C(k) ≡ I. Finally, we get where y is as in (2.12): In particular, at k = κ 1 ≡ e iπ 6 we have and thus, using (3.3), Then we can recover u(x, t) from values of the eigenfunctionsM jl (y, t; k) at k = e iπ 6 . We indeed have the following parametric representation:

Recovering u(x, t) from eigenfunctions
where x(y, t) andû(y, t) are given by x(y, t) = y + 1 2 lnM 33 y, t; e we have t))e x−y and thus u(x, t) can be obtained in terms of N (x, t) as follows: Finally, introducingN k as in (3.5d), using (3.7), and writing (3.6) in the variables (y, t) (so that N (x, t) =N (y, t)), we arrive at (3.5c).
Remark 3.3. From (2.12) and (2.10), another expression forû(y, t) follows: Notice that the analogous expression in the case of the DP equation looks differently [10]: 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.

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.

RH problem satisf ied byM
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 ± (x, t; k) (on l ν ) of M (x, t; k ) as k → k from the positive or negative side of l ν , ν = 1, . . . , 6 are related as follows: We indeed have M ± =Φ ± e −Q whereΦ ± are two solutions of the system of ordinary differential equations (2.7). They are then related byΦ + =Φ − 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): x (q 2 (ξ, 0) − 1)dξ Λ(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 ∈ l 1 ∪ l 4 ≡ R, where r(k) ∈ L ∞ (R) and r(k) = O(k −1 ) as k → ±∞ (this structure follows from the analysis of the behavior of M ± (x, 0; k) as x → ±∞; for the details, see [10, Section 3.1]), whereas the expression of S 0 (k) for the other parts of Σ 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).
This condition, being supplemented by conditions at possible poles ofM (y, t; k), by a normalization condition, and by certain structural conditions, will constitute the RH problem satisfied byM .
Additional conditions. The dependence on k of the eigenvalues λ 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]. where v n is some constant 3×3 matrix with only one non-zero entry at a position depending on the sector of C \ Σ to which k n belongs.
Summarizing, we get that the eigenfunctionsM (y, t; k) -defined as in Section 3 -satisfy the following Riemann-Hilbert problem: Main RH problem (for the Novikov equation). Given r(k) ∈ L ∞ (R) such that r(k) = O(k −1 ) as k → ±∞ and {k n , v n } N n=1 , find a piece-wise (w.r.t. Σ = l 1 ∪ · · · ∪ l 6 ) meromorphic (in the complex k-plane), 3 × 3 matrix-valued functionM (y, t; k) satisfying the following conditions: where v n is some constant 3 × 3 matrix with only one non-zero entry, whose position depends on the sector of C \ Σ containing k n (dictated by the order of Re λ j (k) in the sector, see [1]). For example, if k n ∈ Ω 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 C \ Σ are determined by the symmetries (S1)-(S3) from Proposition 2.6.  with some (unspecified)q(y, t) > 0, f (y, t) > 0, p 1 (y, t) ∈ R and p 2 (y, t) ∈ R (see (3.4)). ii) For k = κ 2 , . . . , κ 6 , the corresponding structure ofM (y, t; κ 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). i) u 0 (x) defines the main RH problem as follows.
ia) We first define the jump matrix S 0 (k) by 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). ib) The k n 's in C \ Σ and the v n 's in (4.5) are determined by u 0 (x).
ii) This RH problem has a solution.
iii) The solution is unique.
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 eigenfunctionM (y, t; k).

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. u(x, t) =û(y(x, t), t), where x(y, t) andû(y, t) are given by x(y, t) = y + 1 2 lnM 33 y, t; e left multiplying (3.4) by the row vector (1 1 1). 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 → ∞) 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.

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.