{'fraction_non_alphanumeric': 0.06968835651202923, 'fraction_numerical': 0.04103644300071059, 'mean_word_length': 3.9398270026325686, 'pattern_counts': {'":': 0, '<': 16, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 30, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background

We address the most general periodic travelling wave of the modified Korteweg-de Vries (mKdV) equation written as a rational function of Jacobian elliptic functions. By applying an algebraic method which relates the periodic travelling waves and the squared periodic eigenfunctions of the Lax operators, we characterize explicitly the location of eigenvalues in the periodic spectral problem away from the imaginary axis. We show that Darboux transformations with the periodic eigenfunctions remain in the class of the same periodic travelling waves of the mKdV equation. In a general setting, there are exactly three symmetric pairs of eigenvalues away from the imaginary axis, and we give a new representation of the second non-periodic solution to the Lax equations for the same eigenvalues. We show that Darboux transformations with the non-periodic solutions to the Lax equations produce rogue waves on the periodic background, which are either brought from infinity by propagating algebraic solitons or formed in a finite region of the time-space plane.


Introduction
We address periodic travelling waves of the modified Korteweg-de Vries (mKdV) equation, which we take in the normalized form: (1.1) As is well-known since the pioneer paper [1], the mKdV equation (1.1) is a compatibility condition of the following pair of two linear equations written for the vector ϕ = (ϕ 1 , ϕ 2 ) t : and Assuming ϕ(x, t) ∈ C 2,2 (R × R) and u(x, t) ∈ C 3,1 (R × R), the compatibility condition ϕ xt = ϕ tx is equivalent to the mKdV equation (1.1) satisfied in the classical sense.
Among the periodic travelling wave solutions, the mKdV equation (1.1) admits the normalized constant wave u(x, t) = 1 and two families of the normalized periodic waves given by u(x, t) = dn(x − ct; k), c = 2 − k 2 (1.4) and u(x, t) = kcn(x − ct; k), c = 2k 2 − 1, (1.5) where dn and cn are Jacobian elliptic functions and k ∈ (0, 1) is the elliptic modulus (see Chapter 8.1 in [21] for review of elliptic functions and integrals). The normalized constant wave u(x, t) = 1 is linearly and nonlinearly stable in the time evolution of the mKdV equation (1.1) in the sense that any small perturbation to the constant wave in the energy space H 1 (R) remains small in the H 1 (R) norm globally in time, see, e.g., [28,31]. Among the exact solutions to the mKdV equation (1.1) on the normalized constant wave, we note the following algebraic soliton where x 0 ∈ R is arbitrary. The algebraic soliton propagates on the normalized constant background with the speed c 0 = 6. In our previous work [15], we have constructed new solutions on the periodic background given by the normalized periodic waves (1.4) and (1.5). In doing so, we have adopted the formal algebraic method from [13,14,18] and elaborated the following algorithm for constructing new solutions to the mKdV equation (1.1): 1. Impose a constraint between a solution u to the mKdV equation (1.1) and a solution ϕ = (p 1 , q 1 ) t to the Lax system (1.2)-(1.3) with λ = λ 1 and deduce closed differential equations on u. These equations are satisfied if u is a periodic travelling wave.
2. Characterize the set of admissible values for λ 1 and the relations between u and the squared components p 2 1 + q 2 1 , p 2 1 − q 2 1 , and p 1 q 1 , The solution ϕ = (p 1 , q 1 ) t is periodic in x and is travelling in t.
3. Obtain the second solution ϕ = (p 1 ,q 1 ) t to the Lax system (1.2)-(1.3) for the same values of λ 1 . The second solution is non-periodic, it grows linearly in x and t almost everywhere as |x| + |t| → ∞.
4. Apply Darboux transformation with the second solution ϕ = (p 1 ,q 1 ) t and obtain new solutions to the mKdV equation (1.1) on the periodic background u.
As the main outcome of step 2 in [15], we obtained two pairs of admissible values for λ 1 with Re(λ 1 ) = 0. For the dn-periodic wave (1.4), the two pairs are real ±λ + and ±λ − with For the cn-periodic wave (1.5), the two pairs are complex-conjugate ±λ + and ±λ − with λ ± = 1 2 (k ± i 1 − k 2 ). (1.8) As the main outcome of step 4 in [15], we constructed an algebraic soliton propagating on the background of the dn-periodic wave (1.4) and a fully localized rogue wave on the (x, t) plane growing and decaying on the background of the cn-periodic wave (1.5). Since the dn-periodic wave (1.4) converges to the constant wave u(x, t) = 1 as k → 0, the algebraic soliton on the dn-periodic wave background generalizes the exact solution (1.6). The rogue wave on the cn-periodic wave background satisfies the following mathematical definition of a rogue wave. (1.9) Definition 1 correspond to the physical interpretation of a rogue wave as the wave that comes from nowhere and disappears without any trace. Rogue waves in physics are associated with the gigantic waves on the ocean's surface and in optical fibers which arise due to the modulation instability of the background wave [29,30]. Several recent publications were devoted to numerical and analytical studies of rogue waves on the background of periodic [3,23], quasi-periodic [5,6,12], and multi-soliton [7,8,9] wave patterns in the framework of the nonlinear Schrödinger (NLS) equation. Since internal waves are modeled by the mKdV equation [22], formation of rogue internal waves was also studied in the framework of the mKdV equation (1.1) as a result of multi-soliton interactions [34,35,36,37].
The difference between the two outcomes of the algorithm applied in [15] to the normalized periodic waves (1.4) and (1.5) in the mKdV equation is related to the fact that the dn-periodic waves are modulationally stable with respect to perturbations of long periods, whereas the cnperiodic waves are modulationally unstable [10,11].
The purpose of this work is to consider the most general periodic travelling wave of the mKdV equation (1.1) and to characterize explicitly location of eigenvalues λ with Re(λ) = 0 in the periodic spectral problem (1.2). Although it may seem to be an incremental goal, advancement from the normalized periodic waves (1.4) and (1.5) to the most general periodic wave of the mKdV equation (1.1) require us to consider Riemann Theta functions of genus g = 2, which are expressed as rational functions of Jacobian elliptic functions. As is well known [4,19], Riemann Theta functions of genus g represent quasi-periodic solutions to many integrable evolution equations including the mKdV equation (1.1). Hence, having successfully solved the problem for 0 ≤ g ≤ 2, we can move to the next goal of solving this problem for general g.
The algebraic method developed here is different from construction of multi-soliton solutions on the background of quasi-periodic solutions developed in [20] by using commutation methods. It is also different from other analytical techniques for explicit characterization of eigenvalues related to the periodic solutions of the mKdV equation in the Whitham modulation theory [26,27] (see also [25,33] for earlier works).
Let us now present the main results of our work. The travelling wave to the mKdV equation (1.1) has the form u(x, t) = u(x − ct), where c is wave speed. The wave profile u satisfies the third-order differential equation: (1.10) Integrating it once yields the second-order differential equation: (1.11) where e is the integration constant. Integrating it once again yields the first-order invariant: where d is another integration constant. Thus, the most general periodic travelling wave in the mKdV equation (1.1) is characterized by the parameters (c, d, e). The previous case considered in [15] corresponds to e = 0.
Our first result is about classification of the most general periodic travelling wave solution to the mKdV equation (1.1). As is well-known (see, e.g., [40]), there exists two explicit families of the periodic solutions to equations (1.11) and (1.12) depending on parameters (c, d, e). When the polynomial P (u) := u 4 − cu 2 + d − 2eu, (1.13) admits four real roots ordered as u 4 ≤ u 3 ≤ u 2 ≤ u 1 , where (u 1 , u 2 , u 3 , u 4 ) are related to the parameters (c, d, e), the exact periodic solution to the system (1.11) and (1.12) is given by , (1.14) where ν > 0 and κ ∈ (0, 1) are parameters given by When the polynomial P (u) in (1.13) admits two real roots b ≤ a and two complex-conjugate roots α±iβ, where (a, b, α, β) are related to parameters (c, d, e), the exact periodic solution to the system (1.11) and (1.12) is given by where δ > 0, ν > 0, and κ ∈ (0, 1) are parameters given by (1.17) The trivial case when the polynomial P (u) in (1.13) admits no real roots does not produce any real solution to the system (1.11) and (1.12). The following theorem characterizes the periodic travelling waves to the mKdV equation. . There exist −∞ < d 1 < d 2 < ∞ such that for every d ∈ (d 1 , d 2 ), the system (1.11) and (1.12) admits the exact periodic solution in the form (1.14) with (1.15) and three other periodic solutions of the same period obtained with the following three symmetry transformations  Remark 2. The proof of Theorem 1 is elementary. It is based on the phase-plane analysis and properties of the Jacobian elliptic functions. We included Theorem 1 for clarity of our presentation.
Remark 3. The periodic travelling wave of the mKdV equation (1.1) in Theorem 1 is also the periodic travelling wave of the following Gardner equation: where a ∈ R is arbitrary. Indeed, if u(x, t) ∈ C 3,1 (R × R) satisfies the mKdV equation (1.1) and is represented by u(x, t) = a + v(x − 6a 2 t, t) with a ∈ R, then v(x, t) ∈ C 3,1 (R × R) satisfies the Gardner equation (1.20). The Gardner equation is commonly used in modelling of internal waves [22].
Next, we use the algebraic method from [13,14,18] and relate the solutions u in Theorem 1 with squared eigenfunctions of the periodic spectral problem (1.2). Compared to our previous work in [15], we have to use two squared eigenfunctions for two different eigenvalues λ in order to obtain periodic solutions of the third-order differential equation (1.10). The following theorem represents the outcome of the algebraic method.
Theorem 2. The spectral problem (1.2) with u given by the periodic waves (1.14) and (1.16) admits three pairs ±λ 1 , ±λ 2 , ±λ 3 of eigenvalues with Re(λ) = 0 that corresponds to the periodic eigenfunctions ϕ of the same period. For the periodic wave (1.14), the eigenvalues are located at For the periodic wave (1.16), the eigenvalues are located at Remark 4. The algebraic method in the proof of Theorem 2 gives also explicit characterization of the periodic eigenfunctions ϕ of the spectral problem (1.2) at the three pairs of eigenvalues λ. If u(x, t) = u(x − ct) is a travelling wave solution to the mKdV equation (1.1), the time evolution problem (1.3) is also satisfied with the solution ϕ(x, t) = ϕ(x − ct).
Remark 5. The algebraic method does not allow us to conclude that no other eigenvalues λ with Re(λ) = 0 exist in the periodic spectral problem (1.2).
Next, we proceed with the multi-fold Darboux transformations [24,32] by using the general form proven with explicit computations in Appendix A of [15]. Since we only have up to three pairs of simple eigenvalues in Theorem 2 in a generic case, we should only use one-fold, two-fold, and three-fold Darboux transformations. The corresponding Darboux transformations are given by the explicit expressions:ũ whereũ stands for a new solution to the mKdV equation (1.1) and ϕ j = (p j , q j ) t , j = 1, 2, 3 stands for a nonzero solution to the Lax system (1. Theorem 3. Assume u 4 < u 3 < u 2 < u 1 such that u 1 + u 2 + u 3 + u 4 = 0, u 1 + u 2 = 0, u 1 + u 3 = 0, and u 2 + u 3 = 0. The one-fold Darboux transformation with the periodic eigenfunctions for each eigenvalue in (1.21) transforms the periodic wave (1.14) to the periodic wave of the same period obtained after the corresponding symmetry transformation in (1.18) and the reflection u → −u. The two-fold Darboux transformation with the periodic eigenfunctions for any two eigenvalues from (1.21) transforms the periodic wave (1.14) to the periodic wave of the same period obtained after the complementary third symmetry transformation in (1.18). The three-fold Darboux transformation with the periodic eigenfunctions for all three eigenvalues in (1.21) maps the periodic wave (1.14) to itself reflected with u → −u. Remark 6. Under the term "complementary third transformation" in Theorem 3, we mean that if (λ 1 , λ 2 ) is selected in (1.21), then (S3) is selected in (1.18), and so on.
Remark 7. The three eigenvalues in (1.21) with the three periodic eigenfunctions used in the onefold transformation (1.23) recover the three symmetry transformations in (1.18). It is interesting to note that due to the constraint u 1 +u 2 +u 3 +u 4 = 0, the choice of each eigenvalue indicate explicitly the choice of the transformation between u 1 , u 2 , u 3 , and u 4 , e.g., λ 1 = (u 1 + u 2 )/2 corresponds to (S1) with u 1 ↔ u 2 and u 3 ↔ u 4 , and so on.  Finally, we use the algorithm above and construct new solutions to the mKdV equation (1.1). To do so, we obtain the closed form expression for the second solution ϕ = (p 1 ,q 1 ) t of the Lax system (1.2)-(1.3) with λ = λ 1 given by an eigenvalue in Theorem 2. The explicit representation of the second solution has been already obtained in our previous work [15], however, the expression obtained there is singular at any point of (x, t) where one of the two components of the periodic solution ϕ = (p 1 , q 1 ) t vanishes. As a result, the corresponding expression was only used for the eigenvalue λ + in (1.7) but not for the eigenvalue λ − . Here we obtain a different explicit representation of the second solution ϕ which is free of singularities for every eigenvalue λ in Theorem 2. As a result, we are able to construct new solutions to the mKdV equation by using the Darboux transformations for every second solution to the Lax system (1.2)-(1.3) corresponding to eigenvalues in Theorem 2. The following two theorems describe the new solutions as a function of (x, t).
Theorem 5. Assume u 4 < u 3 < u 2 < u 1 such that u 1 + u 2 + u 3 + u 4 = 0, u 1 + u 2 = 0, u 1 + u 3 = 0, and u 2 + u 3 = 0. Under three non-degeneracy conditions (7.5), (7.6), and (7.7) below, there exist c 1 , c 2 , c 3 = c such that the second solutions to the Lax system (1.2)-(1.3) for the eigenvalues λ 1 , λ 2 , λ 3 in (1.21) are linearly growing in x and t everywhere on the (x, t) plane except for the straight lines x − c 1,2,3 t = ξ 1,2,3 , where ξ 1 , ξ 2 , ξ 3 are phase parameters which are not uniquely defined. The one-fold Darboux transformation with the second solution for each eigenvalue λ 1,2,3 in (1.21) adds an algebraic soliton with the corresponding speed c 1,2,3 on the background of the periodic wave (1.14) transformed by the symmetry (S 1,2,3 ) and reflection u → −u. The two-fold Darboux transformation with the second solutions for any two eigenvalues from (1.21) adds two algebraic solitons with the corresponding two wave speeds on the background of the periodic wave (1.14) transformed by the complementary symmetry. The three-fold Darboux transformation with the second solutions for all three eigenvalues in (1.21) adds all three algebraic solitons with the three wave speeds on the background of the periodic wave (1.14) reflected with u → −u.  The paper is organized as follows. Section 2 describes the algebraic method which allows us to complete step 1 in the algorithm above. Section 3 characterizes the most general periodic wave of the mKdV equation (1.1) and gives the proof of Theorem 1.
Step 2 of the algorithm and the proof of Theorem 2 are given in Section 4. The proof of Theorems 3 and 4 can be found in Section 5.
Step 3 of the algorithm is developed in Section 6, where the second non-periodic solution to the Lax system (1.

The algebraic method
We have showed in [15] that imposing a constraint u = p 2 1 + q 2 1 between a solution u to the mKdV equation (1.1) and components of the solution ϕ = (p 1 , q 1 ) t to the Lax system (1.2)-(1.3) with fixed parameter λ = λ 1 results in the second-order differential equation on u. This equation is given by (1.11) for e = 0 and it does not recover the most general periodic wave of the mKdV equation (1.1). Here we extend the algebraic method of [13,14,18] by imposing the constraint between a solution u to the mKdV equation (1.1) and components of two solutions ϕ = (p 1 , q 1 ) t and ϕ = (p 2 , q 2 ) t to the Lax system (1.2)-(1.3) with two fixed parameters λ = λ 1 and λ = λ 2 . Assume λ 1 = ±λ 2 . The spectral problem (1.2) for two vectors ϕ = (p 1 , q 1 ) t and ϕ = (p 2 , q 2 ) t at λ 1 and λ 2 can be written as the Hamiltonian system of degree two generated from the Hamiltonian function Since H 0 is x-independent, we introduce the constant E 0 := 4H 0 (p 1 , q 1 , p 2 , q 2 ) in addition to parameters (λ 1 , λ 2 ). As is shown in [14], the Hamiltonian system (2.2)-(2.3) of degree two is integrable in the sense of Liouville and there exists another conserved quantity Since H 1 is x-independent, we introduce another constant E 1 := 4H 1 (p 1 , q 1 , p 2 , q 2 ). Thus, the algebraic method includes four parameters (λ 1 , λ 2 , E 0 , E 1 ). It remains to establish differential equations on the class of admissible solutions u.

Lax-Novikov equations
By differentiating (2.1) in x and using the Hamiltonian system (2.2)-(2.3), we obtain the following first-order differential equation: By differentiating (2.5) in x and using the Hamiltonian system (2.2)-(2.3) and the relation (2.6), we obtain the following second-order differential equation: where we have introduced a parameter Taking yet another derivative of (2.7) in x and using the Hamiltonian system (2.2)-(2.3) again, we obtain the following third-order differential equation: Differential equations (2.5), (2.7), and (2.9) are not closed for u. However, we show that one more differentiation gives a closed fourth-order differential equation on u. To do so, we first note that it follows from (2.1), (2.3), (2.4), (2.5), and (2.7) that (2.10) By using (2.6), this relation can be rewritten in the equivalent form By differentiating (2.9) in x and using the Hamiltonian system (2.2)-(2.3) and the relation (2.11), we obtain a closed fourth-order differential equation on u: where we have introduced another parameter The fourth-order differential equation (2.12) belongs to the class of Lax-Novikov equations (see [39] and references therein), which combine the stationary flows of the mKdV equation and higher-order members of its integrable hierarchy.

Conserved quantities for the stationary fourth-order equation
Let us introduce where the components of W (λ) are given explicitly by In order to simplify the presentation, we denote derivatives of u in x by subscripts. By using (2.1), (2.5), (2.7), and (2.9), the expression for W 12 can be rewritten in the equivalent form: . (2.17) By using (2.6), (2.11), and (2.13), the expression for W 11 can also be rewritten in the equivalent form: . Substituting the representations (2.17) and (2.18) into det[W (λ)] and removing the double poles at λ = ±λ 1 and λ = ±λ 2 yield the following two constraints: These two equations (2.21) represent lower-order invariants for the fourth-order differential equation (2.12). By performing elementary operations, the system of two constraints (2.21) can be rewritten in the equivalent form: In order to verify conservation of (2.22) and (2.23), we have checked that differentiating these constraints in x recovers the fourth-order differential equation (2.12).

Dubrovin equations
One can characterize solutions u to the fourth-order differential equation (2.12) from the algebrogeometric point of view, which is now commonly accepted in the context of quasi-periodic solutions of integrable equations [4,19]. Since the numerator of W 12 (λ) is a polynomial of degree three, it admits three roots denoted as µ 1 , µ 2 , and µ 3 . Writing the representation (2.17) in the form , (2.24) yields the following differential expressions for µ 1 , µ 2 , and µ 3 : By substituting the representations (2.17) and (2.18) into det [W (λ)], removing the double poles with the constraints (2.21), and simplifying the remaining expressions, we obtain the following representation: By substituting (2.24) and (2.28) into (2.20) and evaluating (2.20) at λ = µ 1,2,3 , we recover the Dubrovin equations [17] in the form: . (2.29) These equations characterize the algebro-geometric structure of the quasi-periodic solutions.

Degeneration procedure
For the scopes of our work, we reduce the algebraic method in order to recover the third-order differential equation (1.10) instead of the fourth-order equation (2.12). This scope is achieved by imposing the constraint µ 3 = 0 on Dubrovin variables satisfying the system of equations (2.25), (2.26), and (2.27). Indeed, if µ 3 = 0, equation (2.27) is equivalent to the third-order differential equation (1.10) rewritten again as whereas the other two equations (2.25) and (2.26) yield relations The presence of two Dubrovin variables indicates that the general solution to the third-order equation (2.30) is expressed by the Riemann Theta function of genus two [4,19]. Substituting (2.30) into (2.12) yields the following invariant for the third-order equation (2.30): .
These representations coincide with those used in [41] for integration of ultra-elliptic solutions to the cubic NLS equation.
Remark 10. It may be possible to modify the algebraic method by starting with polynomials of even degree for W 12 (λ) and odd degree for W 11 (λ) and to derive the same solutions as those obtained by the degeneration procedure here. This possible modification remains to be developed.
By substituting differential equations (2.30) and (2.33) into the constraints (2.22) and (2.23), we rewrite the constraints in the equivalent form: In order to simplify characterization of the general solution of the third-order equation (2.30), we integrate it once and obtain the differential equation (1.11) rewritten again as The system (2.38) and (2.39) is solved in the explicit form: and The relation (2.40) and the definition (2.13) imply that the parameter E 1 is uniquely expressed by Summarizing, the three parameters (c, d, e) for the differential equations (2.30), (2.36), and (2.37) are related to the parameters (λ 1 , λ 2 , E 0 ) of the algebraic method with the help of relation (2.8), (2.40), and (2.41), whereas parameter E 1 is uniquely expressed by (2.42).
Remark 11. Compared to the algebraic method with only one eigenfunction ϕ = (p 1 , q 1 ) t in [15], where solutions to the second-order equation (2.36) with e = 0 are obtained, we have flexibility to recover the same solutions with the two eigenfunctions ϕ = (p 1 , q 1 ) t and ϕ = (p 2 , q 2 ) t as in (2.1), as long as there are two nonzero eigenvalues for λ 1 and λ 2 such that λ 1 = ±λ 2 . Thus, solutions to the second-order equation (2.36) with e = 0 can be recovered by two equivalent algebraic methods.

Proof of Theorem 1
Here we develop the phase-plane analysis of the second-order differential equation (1.11). Therefore, we treat it as a dynamical system on the phase plane  Since 2Q(u) = ∂ u H(u, 0), the function H(u, 0) has three extremal points with two minima at u * and u * * * and one maximum at u * * . Let h * := H(u * , 0), h * * := H(u * * , 0), and h * * * := H(u * * * , 0). Recall that H(u, 0) = P (u)−d, where P (u) is given by (1.13). For every d ∈ (d 1 , d 2 ) with d 1 = −h * * and d 2 = − max{h * , h * * * }, the polynomial P (u) has four real roots ordered as u 4 < u 3 < u 2 < u 1 also shown on Figure 1. There exists two periodic solutions which correspond to closed orbits on the phase plane: one closed orbit surrounds the center point (u * , 0) and the other closed orbit surrounds the center point (u * * * , 0). These periodic solutions are shown on Figure 2(a,b).  In order to complete the proof of Theorem 1, it remains to verify the exact representations (1.14)-(1.15) and (1.16)-(1.17) of the periodic solutions as rational functions of the Jacobian elliptic functions. The following two lemmas state the corresponding exact representations. Lemma 1. Let P (u) in (1.13) admit four real roots ordered as u 4 ≤ u 3 ≤ u 2 ≤ u 1 . The exact periodic solution to the first-order invariant (1.12) in the interval [u 2 , u 1 ] is given by where ν > 0 and κ ∈ (0, 1) are parameters given by The roots satisfy the constraint u 1 + u 2 + u 3 + u 4 = 0 and are related to the parameters (c, d, e) by (3.4) Proof. By factorizing P (u) by its roots, we rewrite the first-order invariant (1.12) in the form: Expanding the quartic polynomial and comparing it with the coefficients in (1.12), we verify the constraint u 1 + u 2 + u 3 + u 4 = 0 and the relations (3.4). In order to prove that (3.2) with (3.3) is an exact solution of (3.5), we substitute u(x) = u 4 + (u 1 − u 4 )(u 2 − u 4 )/v(x) into (3.5) and obtain the equivalent equation: Then, we substitute v(x) = (u 2 − u 4 ) + (u 1 − u 2 )w(x) into (3.6) and obtain the equivalent equation , (3.9) which also exists in the interval [u 2 , u 1 ] with u(0) = u 2 and u(L/2) = u 1 . The symmetry transformation (S3) in (1.18) generates another solution From Table 8.152 of [21], we have the transformation formula

Lemma 2.
Let P (u) in (1.13) admit two real roots ordered as b ≤ a and two complex conjugated roots labeled as α ± iβ. The exact periodic solution to the first-order invariant (1.12) is given by where δ > 0, ν > 0, and κ ∈ (0, 1) are parameters given by .

(3.16)
The roots satisfy the constraint a + b + 2α = 0 and are related to the parameters (c, d, e) by Proof. By factorizing P (u) by its roots, we rewrite the first-order invariant (1.12) in the form: Expanding the quartic polynomial and comparing it with the coefficients in (1.12), we verify the constraint a + b + 2α = 0 and the relations (3.17). In order to prove that (3.15) with (3.16) is an exact solution of (3.18), we substitute u(x) = a + (b − a)/v(x) into (3.18) and obtain the equivalent equation: Then, we substitute v(x) = 1 + δ 1 + cn(νx; κ) 1 − cn(νx; κ) into (3.19) and obtain three equations on δ, ν, and κ which yield (3.16). Since the Jacobian elliptic function cn(νx; κ) is periodic in x with the period L := 4K(κ)/ν, where K(κ) is the complete elliptic integral, the solution u in (3.15) is L-periodic. It is unique up to the translation symmetry   Setting a = k and β = √ 1 − k 2 yields the normalized cnoidal periodic wave (1.5).
Remark 18. The explicit expressions (4.4) of the three eigenvalues for the periodic wave (3.2) can be recovered from formulas (58) and (62) in [26], where they are derived by a different technique. The explicit expressions (4.10) of the three eigenvalues for the periodic wave (3.15) are new to the best of our knowledge.

Proof of Theorems 3 and 4
Let us first establish algebraic relations on the squared eigenfunctions in the periodic spectral problem (1.2) with the periodic solution u to the mKdV equation (1.1). We rewrite equations (2.1), (2.5), (2.7), and (2.9) as a system of linear equations for squared eigenfunctions: where the subscripts denote the derivatives in x. Solving the algebraic system (5.1) with Cramer's rule yields the relations and similar relations for p 2 2 + q 2 2 and p 2 2 − q 2 2 obtained by the transformation λ 1 ↔ λ 2 . If u satisfies differential equations (2.30) and (2.36), the expressions (5.2) and (5.3) are simplified to the form .

(5.5)
We also rewrite equations (2.6) and (2.10) as another system of linear equations for squared eigenfunctions: (5.6) Solving the algebraic system (5.6) with Cramer's rule yields the relations and a similar relation for 4λ 2 p 2 q 2 obtained by the transformation λ 1 ↔ λ 2 . If u satisfies the differential equation ( .
We now proceed with the proof of Theorem 3. The following three lemmas represent the outcomes of the one-fold, two-fold, and three-fold Darboux transformations for the periodic wave (1.14) with the periodic eigenfunctions of the spectral problem (1.2).
We show with explicit computations that the transformation (5.10) applied to the periodic wave (3.2) with the eigenvalue λ j produces the same periodic wave after the corresponding symmetry transformation (Sj) and the reflection u → −u.
By using (3.2), (3.4), and (4.4), we perform lengthy but straightforward computations to obtain the following expressions: where ν and κ are given by (3.3). Substituting (5.11) and (5.12) into (5.10) for u 1 +u 2 = 0 produces a new solution in the form: The new solution is obtained from the periodic wave (3.2) by the symmetry transformation (S1) in (1.18) and the reflection u → −u, see the explicit form (3.9). By repeating the previous computations for the eigenvalue λ 2 in (4.4), we obtain the following expressions: and eu + 2λ 2 2 (c − 4λ 2 2 ) = Substituting (5.14) and (5.15) into the one-fold transformation for u 1 + u 3 = 0 produces a new solution in the form:ũ , (5.16) which is obtained from the periodic wave (3.2) by the symmetry transformation (S2) in (1.18) and the reflection u → −u, see the explicit form (3.8).
By repeating the previous computations for the eigenvalue λ 3 in (4.4), we obtain the following expressions: .
The representation (5.22) is another linear fractional transformation between two solutions of the mKdV equation (1.1). We show with explicit computations that this transformation applied to the periodic wave (3.2) with the eigenvalue pair (λ 1 , λ 2 ) produces the same periodic wave after the complementary symmetry transformation (S3). By using (3.2), (3.4), and (4.4), we obtain the following expressions: and where ν and κ are given by (3.3). Substituting these expressions into (5.22) for u 1 + u 2 = 0 and u 1 + u 3 = 0 produces a new solution in the form: The new solution is obtained from the periodic wave (3.2) by the symmetry transformation (S3) in (1.18). Repeating the same computations for the eigenvalue pair (λ 1 , λ 3 ) in (4.4) for u 1 + u 2 = 0 and u 2 + u 3 = 0 produces the periodic wave (3.2) after the symmetry transformation (S2) in (1.18).

Remark 21.
The two-fold transformation with two eigenvalues (λ 1 , λ 2 ) can be thought as a composition of two one-fold transformations with eigenvalues λ 1 and λ 2 . Indeed, by Lemma 5, the one-fold transformation with λ 1 performs symmetry transformation (S1) and reflection, whereas the onefold transformation with λ 2 performs symmetry transformation (S2) and reflection. Composition of (S1) and (S2) yields (S3), whereas two reflections annihilate each other. Proof. By using (3.15), (3.17), and (4.10), we obtain the following expressions: where δ, ν, and κ are defined by (3.16) with α = −(a + b)/2. Substituting these expressions into (5.10) with λ 1 → λ 3 for a + b = 0 produces a new solutioñ Proof. By using (3.15), (3.17), and (4.10), we obtain the following expressions: and where δ, ν, and κ are defined by (3.16) with α = −(a + b)/2. Substituting these expressions into (5.22) for a + b = 0 produces a new solutioñ Let ϕ = (p 1 , q 1 ) t be the periodic eigenfunction of the spectral problem (1.2) with λ = λ 1 and denote the second linearly independent solution the same spectral problem (1.2) with the same λ = λ 1 by ϕ = (p 1 ,q 1 ) t . Since the coefficient matrix in (1.2) has zero trace, the Wronskian determinant between the two solutions is constant in x and nonzero. To keep consistency with our previous work [15], we normalize the Wronskian by 2, hence p 1q1 −p 1 q 1 = 2. (6.1) In the previous work [15], we constructed the second solution in the explicit form: where θ 1 satisfies a certain scalar equation in x and t, which can be easily integrated. The corresponding expressions were used in [15] to construct the rogue waves on the periodic background, however, it was found that the expressions may be undefined if there exists a point of (x, t) for which either p 1 or q 1 vanishes.
Here we consider a different representation of the second solution which is free of the technical problem above. The following lemma represents the second solution in the explicit form.
Proof. By substituting the representation (6.3) to the spectral problem (1.2) with λ = λ 1 and using the same spectral problem (1.2) for ϕ = (p 1 , q 1 ) t , we obtain (6.5) Thanks to (5.4) and (5.9), this equation can be integrated to the form where ψ 1 is the constant of integration in x that may depend on t.
On the other hand, by substituting the representation (6.3) to the time evolution system (1.3) with λ 1 and using the same system (1.3) for ϕ = (p 1 , q 1 ) t , we obtain (6.7) Substituting (5.5), (5.9), and (6.5) into (6.7) yields the scalar equation from which we obtain (6.6), where ψ 1 is now constant both in x and t.
Remark 25. Without loss of generality, thanks to the translational invariance in (x, t), we can set ψ 1 = 0 so that if u is even in x, then φ 1 is odd in x at t = 0.
Remark 27. If λ 1 / ∈ R, the representation (6.3) is non-singular if e = 0 and singular if e = 0. This follows from the expression (5.4) which shows that p 2 1 +q 2 1 ∼ = e+4λ 2 1 u with e ∈ R and u ∈ R. If e = 0, the cn-periodic solution (1.5) vanish at some points of (x, t), which lead to the singular behavior of p 1 andq 1 given by (6.3). Note in this case, the previous representation (6.2) is non-singular, as it follows from our previous work [15]. where with φ 1 given by (6.4) for the choice of ψ 1 = 0. Figure 3 shows three new solutions computed from the periodic wave (1.14) with the parameter values: The three solutions are generated with three different choices for the eigenvalue λ 1 given by (1.21). Each solution displays an algebraic soliton propagating on the background of the periodic travelling wave. Substituting (6.3) into the two-fold transformation (1.24) yields the new solution to the mKdV equation (1.1) in the form: where for j = 1, 2 The expression for φ 1 is given by (6.4) with ψ 1 = 0. Note that the numerical factor (λ 2 1 − λ 2 2 ) in (6.4) cancels with the denominators in the expressions (5.4), (5.5), and (5.9) for the squared eigenfunctions. Therefore, the expression for φ 2 and φ 3 can be obtained with the transformation λ 1 → λ 2 and λ 1 → λ 3 respectively.
Three choices exist for an eigenvalue pair (λ 1 , λ 2 ) from the three real eigenvalues in (1.21). Figure 4 shows three new solutions computed from the periodic wave (1.14) with the same choice of parameters as in (7.2). The three solutions display three different choices of two algebraic solitons propagating on the background of the periodic travelling wave. Substituting (6.3) into the three-fold transformation (1.25) yields a new solution in the explicit form that is similar to (7.1) and (7.3). There is only one choice for the three real eigenvalues (λ 1 , λ 2 , λ 3 ) in (1.21). Figure 5 shows the new solution computed from the periodic wave (1.14) with the same choice of parameters as in (7.2). The new solution displays three algebraic solitons propagating on the background of the periodic travelling wave.
Remark 28. The first limit in (7.4) tells us that the new solutionû approaches the transformed periodic waveũ almost everywhere on the (x, t) plane as |x|+|t| → ∞ except for the lines x−c 1,2,3 t = ξ 1,2,3 . Algebraic solitons propagate along these lines with the wave speeds c 1,2,3 .
Remark 29. The second limit in (7.4) tells us the amplitude of the algebraic soliton on the periodic backgroundũ. The periodic wave u in (3.2) has two extremal points u(0) = u 1 and u(L/2) = u 2 and our convention is u 4 < u 3 < u 2 < u 1 . Table 1 shows the periodic backgroundũ and the new solutionû at two extremal points of u. Table 1: Characteristics of rogue waves on the periodic background (1.14).
We next represent graphically the outcomes of the one-fold, two-fold, and three-fold Darboux transformations for the periodic wave (1.  Figure 7 shows the new solution computed from the periodic wave (1.16) with the same choice of parameters as in (7.10). The new solution displays a fully localized rogue wave on the background of the periodic travelling wave. Figure 8 shows the outcome of the three-fold transformation computed from the periodic wave (1.16) with the same choice of parameters as in (7.10). All three eigenvalues (λ 1 , λ 2 , λ 3 ) are selected from the list (1.22). The new solution displays both the algebraic soliton propagating on the background of the periodic travelling wave and a localized rogue wave at the center.
The proof of Theorem 6 relies on Lemma 13 and the following lemma.
Remark 30. By Lemma 15, the new solutionû after the one-fold transformation with eigenvalue λ 3 approaches the transformed periodic waveũ everywhere on the (x, t) plane as |x| + |t| → ∞ except for the line x − c 0 t = ξ 0 , where the algebraic soliton propagates. The periodic wave u in (3.15) has two extremal points u(0) = a and u(L/2) = b and our convention is b < a. Table 2 shows the periodic backgroundũ and the new solutionû at two extremal points of u.
Remark 31. By Lemma 15, the new solutionû after the two-fold transformation with eigenvalues (λ 1 , λ 2 ) approaches the transformed periodic waveũ everywhere on the (x, t) plane as |x| + |t| → ∞. Hence, this is the proper rogue wave in the sense of Definition 1. Table 2 shows the magnification factors of the rogue wave defined as a ratio between the extremal values ofû andũ, e.g. M = max{|2a − b|/|b|, |2b − a|/|a|}.

Remark 32.
The three-fold transformation with all three eigenvalues (λ 1 , λ 2 , λ 3 ) produces both the rogue wave and the algebraic soliton with the wave speed c 0 . Table 2 shows that the new waveû has triple magnification compared to the periodic backgroundũ.

Conclusion
We have addressed the most general periodic travelling wave solutions to the mKdV equation (1.1) and obtained the periodic solutions to the Lax system (1.2)-(1.3) for three particular pairs of eigenvalues λ away from the imaginary axis. For the family of periodic waves (1.14) generalizing the dn-periodic wave (1.4), the three pairs of eigenvalues ±λ 1 , ±λ 2 , ±λ 3 are real. For the family of periodic waves (1.16) generalizing the cn-periodic wave (1.5), one pair ±λ 3 is real and two pairs ±λ 1 , ±λ 2 form a quadruplet of complex eigenvalues. By using the Darboux transformations, we have showed that transformations involving the periodic eigenfunctions remain in the class of the same periodic wave solutions, whereas transformations involving second nonperiodic solutions to the Lax system (1. Based on the summary above, it is natural to conjecture that the solution u to the mKdV equation (1.1) expressed by quasi-periodic Riemann Theta function of genus g is related to exactly g + 1 pairs of eigenvalues in the spectral problem (1.2) with the quasi-periodic eigenfunctions of the same periods. Moreover, location of these eigenvalues is related to parameters of the Riemann Theta functions. It is also natural to conjecture that no other eigenvalues λ with the quasiperiodic eigenfunctions exist away from the imaginary axis. To the best of our knowledge, these mathematical questions have not been solved in the literature, in spite of the large amount of publications on the mKdV equation. Solving these problems in future looks an interesting question of fundamental significance with many potential applications.