Rogue waves on the periodic background in the extended mKdV equation

We construct new exact solutions of the extended mKdV (emKdV) equation. The exact solutions are obtained by nonlinearization of spectral problem associated with the travelling periodic waves and using the one-fold, two-fold Darboux transformations. We consider the dnoidal and cnoidal travelling periodic waves of the emKdV equation. Since the dnoidal travelling periodic wave is modulationally stable, the algebraic solitons propagating on dnoidal wave background. However, since the cnoidal travelling periodic wave is modulationally unstable, the rogue waves generated on the cnoidal wave background.


Introduction
For the Ablowitz-Kaup-Newell-Segur (AKNS) spectral problems, there are two local group constraints that generate local integrable mKdV equations [1]. There exist some effective methods to compute analytical solutions to mKdV equation. The author of Ref. [2] use the binary Darboux transformations to obtain the soliton solutions for matrix mKdV equations from the zero seed solution and Wang et al. used the finite-gap integration approach and Whitham modulation theory to give the exact solutions for the defocusing complex mKdV equation with step-like initial condition [3]. The algebraic method based on the nonlinearization of Lax pair [4][5][6] can be used to get explicit solutions for the AKNS spectral problems. The authors of Refs. [7,8] employ the nonlinear steepest descent method to study the long-time asymptotic behavior of the solutions of the mKdV equation and the focusing Kundu-Eckhaus equation. N-soliton solutions in both (1+1)-dimensions and (2+1)-dimensions was obtained through the Hirota direct method [9] and for nonlocal PT-symmetric mKdV equation have been discussed by the Riemann-Hilbert problems [10].
Rogue waves also called killer waves are short-lived, large amplitude waves that occur locally. Rogue waves were discovered in the deep ocean, the field of optics [11] and the capillary waves [12]. The formation of rogue waves may be connected with the modulation instability of the background wave. Peregrine was the first to show that modulation instability can be lead to a rapid increase in the wave amplitude [13]. In order to construct rogue waves on the periodic background, Chen and Pelinovsky [14] first combine the nonlinearization a e-mail: ypzhen@seu.edu.cn (corresponding author) of spectral problem with the Darboux transformation method, and then by using these two approaches, rogue waves on the periodic background have obtained for the NLS equation [15,16], modified Korteweg-de Vries equation [14,17], Hirota equation [18,19], derivative NLS equation [20,21], sine-Gordon equation [22,23] and some other equations [24][25][26].
Two types of traveling periodic waves of mKdV equation are expressed by the Jacobian elliptic functions which are dnoidal and cnoidal periodic waves [14] and stability of periodic wave was studied in [27,28], where it was concluded that the dnoidal periodic wave is modulationally stable and the cnoidal periodic wave is modulationally unstable, rogue waves only generated on the background of cnoidal periodic wave and steady propagation of an algebraic solitons propagating on the background of dnoidal periodic wave. Recently, they generalized this results to the discrete mKdV and investigated modulational stability of the traveling periodic waves and obtained similar results [29]. This conclusion also can be extended to higher-order mKdV [30,31] and they have been studied fifth-order Ito equation and seventhorder mKdV equation separately.
The most general traveling periodic waves of mKdV equation written as a rational function of Jacobian elliptic functions [17], which first appeared in the Ref. [32]. It shows that Darboux transformations with the periodic eigenfunctions, the new solutions remain in the class of the same travelling periodic waves and Darboux transformations with the non-periodic eigenfunctions produce rogue waves generated on the background. The authors of Ref. [33,34] discussed fifth-order Ito equation and seventh-order mKdV equation separately.
In this paper, we consider the emKdV equation in the following form where α, β are arbitrary constants. Let q per =q(x − ct) be a travelling periodic wave of the emKdV Eq. (1) with the period L. We say that q(x, t) is a rogue wave on the background of the periodic The spectral instability of periodic waves represented by the Jacobi elliptic functions were investigated in the focused NLS [35]. For travelling periodic waves in the emKdV equation, from Ref. [36,37] it is clear that that the dnoidal travelling periodic wave is modulationally stable. Therefore, it is not a rogue wave in the sense of the definition (2). However, since the cnoidal travelling periodic wave is modulationally unstable, the rogue wave generated on the cnoidal wave background.
The article is organized as follows. In Sect. 2, we derive the traveling periodic wave solutions given by the Jacobian dnoidal and cnoidal elliptic functions of the emKdV equation, and then we use the nonlinearization of the Lax pair and get the periodic eigenfunctions of emKdV equation spectral problem related to the Jacobian elliptic functions. In Sect. 3, we compute the second, linearly independent solution of the Lax equations. We construct respectively the algebraic soliton propagating on the dnoidal wave background and the rogue waves generated on the cnoidal wave background using the one-fold and two-fold Darboux transformations in Sects. 4 and 5. Section 6 gives the conclusion.

The nonlinearization method
The emKdV Eq. (1) can be represented as the compatibility condition for the following Lax pair of linear equations and where where ψ = (ψ 1 , ψ 2 ) T and λ ∈ C is the spectral parameter. Then, we consider two families of travelling periodic waves of emKdV Eq. (1) are expressed by the Jacobian elliptic function and where k ∈ (0, 1) is the elliptic modulus.
Using the nonlinearization method, the following proposition give the precise expressions for eigenvalues λ 1 and periodic eigenfunctions (ψ 1 , ψ 2 ) T of the Lax pair (3) and (4) related to the travelling periodic wave solutions (5) and (6) of the emKdV Eq. (1).

Proposition 1
The travelling periodic wave solutions (5) and (6) satisfying where a 0 and a 1 are real constants given by Further information, for dnoidal elliptic function solution (5), we have and for cnoidal elliptic function solution (6), we have Then, the periodic eigenfunctions (ψ 1 , ψ 2 ) T related to the travelling periodic wave solutions (5) and (6) satisfying For dnoidal elliptic function solution (5), it follow from (8) and (9) that therefore, we get two particular real eigenvalues For cnoidal elliptic function solution (6), it follow from (8) and (10) that therefore, we obtain a pair of conjugate complex roots

Proposition 2 Let ψ = (ψ 1 , ψ 2 ) T be a solution to the Lax Eqs. (3) and
for dnoidal elliptic function solution (5), we have and for cnoidal elliptic function solution (6), we have Proof Substituting (16) into (3) and using (3), we have By using the relation (11), we can rewrite it as we rewrite (19) in the equivalent form Integrating it with the boundary condition φ(0) = 0, we have where α(t) is a constant of integration in x and may be depend on t. Substituting (16) into (4) and using (4), we have with the help of (5), (6), (11), (13) and (15), after the complex calculation. For dnoidal elliptic function solution (5), we have which can be yield the representation (17). For cnoidal elliptic function solution (6), we have which can be yield the representation (18).

New solution obtained from the dnoidal travelling periodic wave
The following lemma gives the one-fold Darboux transformation of the emKdV equation (1) Lemma 1 Let q is a solution of the emKdV Eq. (1), and (f 1 , g 1 ) T be a nonzero solution of the Lax pairs (3) and (5) with the eigenvalue λ 1 , then is a new solution of the emKdV equation (1).
with the help of (11), we can rewrite (24) as Figure 1 shows that when we choose α = β = 1, the solution surface of the algebraic soliton (25) propagating on the background of the dnoidal periodic wave (5) for the elliptic modulus k = 0.5 or k = 0.99.

New solution obtained from the cnoidal travelling periodic wave
The following lemma gives the two-fold Darboux transformation of the emKdV Eq. (1) Lemma 2 Let q is a solution of the emKdV Eq. (1), and (f i , g i ) T , i = 1, 2, be a nonzero solutions of the Lax pairs (3) and (5) with the eigenvalues λ i , i = 1, 2,, then is a new solution of the emKdV equation (1).
We use the two-fold Darboux transformation (26) to the cnoidal elliptic function solution (6) to obtain a new solution to the emKdV equation (1). Since the cnoidal elliptic function solution (6) is modulationally unstable, the rogue waves generated on the cnoidal periodic background.
(30) Figure 2 shows that when we choose α = β = 1, the solution surface of the rogue waves generated on Fig. 2 The solution surface for the rogue waves generated on the background of the cnoidal periodic wave with α = β = 1 and k = 0.5 or k = 0.99 the background of the cnoidal periodic wave (6) for the eigenvalue λ = λ 1 and λ =λ 1 given by (15).

Conclusion
In this paper, we construct the exact solutions for the emKdV equation. Since the dnoidal travelling periodic wave is modulationally stable, we use the one-fold Darboux transformation to construct the algebraic soliton propagating on the dnoidal wave background. On the other hand, since the cnoidal travelling periodic wave is modulationally unstable, we use the two-fold Darboux transformation to construct the rogue waves generated on the cnoidal wave background.