The rational solutions of the mixed nonlinear Schr\"odinger equation

The mixed nonlinear Schr\"odinger (MNLS) equation is a model for the propagation of the Alfv\'en wave in plasmas and the ultrashort light pulse in optical fibers with two nonlinear effects of self-steepening and self phase-modulation(SPM), which is also the first non-trivial flow of the integrable Wadati-Konno-Ichikawa(WKI) system. The determinant representation $T_n$ of a n-fold Darboux transformation(DT) for the MNLS equation is presented. The smoothness of the solution $q^{[2k]}$ generated by $T_{2k}$ is proved for the two cases ( non-degeneration and double-degeneration ) through the iteration and determinant representation. Starting from a periodic seed(plane wave), rational solutions with two parameters $a$ and $b$ of the MNLS equation are constructed by the DT and the Taylor expansion. Two parameters denote the contributions of two nonlinear effects in solutions. We show an unusual result: for a given value of $a$, the increasing value of $b$ can damage gradually the localization of the rational solution, by analytical forms and figures. A novel two-peak rational solution with variable height and a non-vanishing boundary is also obtained.

One of widely accepted prototypes of rogue wave in one dimensional space and time is considered as Peregrine soliton [12][13][14] of the nonlinear Schrödinger equation (NLS), which usually takes the form of a single dominant peak accompanied by one deep cave at each side in a plane with a nonzero boundary. In other words, the characteristic property of the RW is localization in both space and time directions in a nonzero plane. The existence of this solution is due to modulation instability of the NLS equation [12,[15][16][17][18]. In consequence, different patterns of the RW will occur when two or more breathers with different relative phase shifts collide with each other [19][20][21][22][23][24][25][26]. One of the possible generating mechanisms for rogue waves is through the creation of breathers possessing a particular frequency, which is realized theoretically by choosing a special eigenvalue in breathers [26]. Recently,by applying Darboux transformation(DT) [27][28][29][30], the rogue waves [31][32][33] of derivative nonlinear Schrödinger equation (DNLS) are also given in the form of "Peregrine soliton".
In the field of optics, the nonlinear terms in the NLS and DNLS denote the effects of phasemodulation (SPM) and self-steepening, respectively. So it is natural and worthwhile to look for an integrable equation with these two terms from the points of view of mathematics and physics. There is indeed such an integrable equation-a mixed NLS (MNLS) equation [34,35] q t − iq xx + a(q * q 2 ) x + ibq * q 2 = 0, (1) in physics. Here q represents a complex field envelope and asterisk denotes complex conjugation, a and b are two non-negative constants, and subscript x (or t) denotes partial derivative with respect to x (or t). The MNLS equation is used to model the propagations of the Alfvén waves in plasmas [34] and the ultrashort light pulse in optical fibersc [35]. Moreover, the MNLS equation can also be given by following coupled system under a condition r = −q * . This coupled system is nothing but the first non-trivial flow of the Wadati-Konno-Ichikawa(WKI) system [36], and the corresponding Lax pair is given by the WKI spectral problem and a time flow [36] Here λ ∈ C, is called the eigenvalue(or spectral parameter), and ψ is called the eigenfunction associated with λ of the WKI system. Equations (2) and (3) are equivalent to the integrability condition U t − V x + [U, V ] = 0 of (4) and (5). In addition, the MNLS equation can be generated from other systems (or equations ) in the literatures [37][38][39][40]. The MNLS equation is exactly solved by the inverse scattering method under the non-vanishing boundary condition [41].
Later, Periodic solutions of the MNLS equation are be analyzed in terms of the Riemann's θ functions [42]. Some special solutions, such as breather solutions, of this equation are also discussed in Ref [43,44]. At the same time, the solutions of the MNLS equation have been constructed via Backlund or Darboux transformation [45][46][47][48] and the Hirota method [49,50]. What is more, using the matrix Riemann-Hilbert factorization approach, asymptotic analysis of the MNLS equation is discussed in Ref. [51,52]. Considering many wave propagation phenomena described by integrable equations in some ideal conditions, the effects of small perturbations on the MNLS equation are study by the direct soliton perturbation theory [53] and the perturbation theory based on the inverse scattering transform [54,55]. Recently, the semiclassical analysis of the MNLS has been studied in Ref. [56][57][58].
In light of the above results, two questions arise naturally. First, is there a rational solution of the MNLS equation which can be generated from a periodic seed by DT and Taylor expansion? Second, how the localization of this rational solution is affected by the two nonlinear effects through the a and b? For the first question, the first order and the second order rational solutions are given explicitly by the determinant representation of the DT and Taylor expansion with respect to the degenerate eigenvalues. To answer the second question, according to a common understanding of the role for the nonlinear effects in wave propagation, one reasonable conjecture is that the localization of this solution will be enhanced because of the appearance of the two nonlinear effects. However we shall show an unusual result: for a given value of a, the increasing value of b can damage gradually the localization of the rational solution.
The organization of this paper is as follows. In section 2, we provide a relatively simple approach to DT for the WKI system, and then the expressions of the q [n] , r [n] and ψ [n] j of the WKI system are generated by n-fold Darboux transformation. The reduction of DT for the WKI system to the MNLS equation is also discussed by choosing paired eigenvalues and eigenfunctions. In section 3, the smoothness of the solutions q [2k] generated by T 2k is proved for the two cases (non-degeneration and double-degeneration). In section 4, we present the rational solutions of the MNLS and discuss its localized properties for a given value of a. Finally, we summary our results in section 5.

Darboux transformation
Inspired by the results of the DT for the NLS [27,28] and the DNLS [29][30][31][32], the main task of this section is to present a detailed derivation of the Darboux transforation of the MNLS and the determinant representation of the n-fold transformation. It is easy to see that the spectral problem (4) and (5) are transformed to ψ [1] x = U [1] ψ [1] , U [1] under a gauge transformation By cross differentiating (6) and (7), we obtain This implies that, in order to make eqs.(2) and eq.(3) invariant under the transformation (8), it is crucial to search a matrix T so that U [1] , V [1] have the same forms as U , V . At the same time the old potential(or seed solution)(q, r) in spectral matrixes U , V are mapped into new potentials (or new solution)(q [1] , r [1] ) in transformed spectral matrixes U [1] , V [1] . Next, it is necessary to parameterize the matrix T by the eigenfunctions associated with the seed solution.
2.1 One-fold Darboux transformation of the WKI system Without losing any generality, let Darboux matrix T be the form of Here a 1 , d 1 , b 0 and c 0 are undetermined function of (x, t), which will be parameterized by the eigenfunction associated with λ 1 and seed (q, r) in the WKI spectral problem. Refer to Appendix I for detail derivation.
First of all, we introduce n eigenfunctions ψ j as Theorem 1.The elements of one-fold DT are parameterized by the eigenfunction ψ 1 associated with λ 1 as then T 1 implies following new solutions and corresponding new eigenfunction Proof. See appendix II. It is straightforward to verify that T 1 annihilate its generating function, i.e., ψ

n-fold Darboux transformation for WKI system
The main result in this subsection is the determinant representation of the n-fold DT for WKI system. To this purpose, set [29] D = a 0 0 d a, d are complex functions of x and t , According to the form of T 1 in eq.(10), the n-fold DT should be the form of In order to get diagonal matrix and anti-matrix coefficients in T n , we introduce λ and λ j by a shift. Here P 0 is a constant matrix, P i (1 ≤ i ≤ n) is the function of x and t. In particular, P 0 ∈ D if n is even and P 0 ∈ A if n is odd, which leads to a separate discussion on the determinant representation of T n in the following by means of its kernel. Specifically, from algebraic equations, coefficients P i are solved by Cramer's rule. Thus we get determinant representation of the T n . Theorem2.
Next, we consider the transformed new solutions (q [n] , r [n] ) of WKI system corresponding to the n-fold DT. Under covariant requirement of spectral problem of the WKI system, the transformed form should be with and then Substituting T n given by eq.(16) into eq. (22), and then comparing the coefficients of λ n+1 , it yields Furthermore, substitute a n , d n , b n−1 , c n−1 from eq.(18) for n = 2k and from eq.(19) for n = 2k + 1, into (23), we get new solutions (q [n] , r [n] ) of couple system in eq.(2) and eq.(3): Theorem 3. Starting from a seed q, the n-fold DT T n in theorem 2 generates new solutions Here, (1)for n = 2k, (2) for n = 2k + 1, ,

Reduction of the Darboux transformation for WKI system.
The solutions q [n] and r [n] in theorem 3 generated by the n-fold DT T n of WKI system are solutions of the coupled system in eq.(2) and eq.(3). If it keeps the reduction condition, i.e.,q [n] = −(r [n] ) * , DT of the WKI system in theorem 2 reduces to the DT of the MNLS equation, and then q [n] in theorem 3 implies automatically a new solution of the MNLS equation. In this subsection, we shall show how to choose the eigenvalues and eigenfunctions in the determinant representations of the T n in order to realize the reduction.
Under the reduction condition q = −r * , the eigenfunction ψ k = φ k ϕ k associated with eigenvalue λ k has following properties, It is trivial to verify above properties in Lax pair, eq.(4) and eq. (5), of the WKI system by a straightforward calculation. These properties of eigenfunctions for the WKI system provides one kind of possibility for choosing suitable eigenvalues and eigenfunctions such that the reduction condition q [n] = −(r [n] ) * holds in n-fold DT.
Thus q [2k] is called k-order solution of the MNLS. Similar to the reduction of the T 2k for the WKI system, T 2k+1 in eq. (19) can also be reduced to the (k+1)-fold DT of the MNLS by choosing one pure imaginary λ 2k+1 = iβ 2k+1 and k paired-eigenvalues λ 2l = −λ * 2l−1 (l = 1, 2, · · · , k) with corresponding eigenfunctions according to properties (i) and (ii). Of course, there are many other ways to select eigenvalues and eigenfunctions in order to do reduction of n-fold DT for the WKI system.

Smoothness of the solutions q [2k]
The smoothness of the q [n] generated by DT is an important property of the solution of the MNLS equation. In this section,we shall study this property for the solution q [2k] through the T 2k with the paired eigenfunctions and eigenvalues.
It is easy to check that the k-soliton solution is generated by T 2k in Theorem 9 from a trivial seed solution-zero solution. It is an interesting problem to study degenerate cases and to apply it to get smooth solutions of the MNLS equations.
To deal with the degeneration of T 2k , we begin with the T 2 under the reduction condition in lemma 6. Let ψ 1 (λ 0 ) be a smooth eigenfunction of Lax pair of the MNLS associated with nonzero eigenvalue λ 0 , which has also continuous dependence on the λ 0 . Under a small shift of eigenvalue, it can be expanded as , |a l | and |b l |(l = 1, 2, · · · , k) are not both zero . If λ 0 is the only one zero point of ψ 1 ,i.e., ψ 1 (λ 0 ) = 0, then it is also a zero point of W 2 in T 2 . Thus λ 0 is a singularity of T 2 and the q [2] . However this singularity is removable. Lemma 10 Let λ 0 be the only one zero point of eigenfunction ψ 1 (λ 1 ), λ 0 = α 0 + iβ 0 . If α 0 = 0, then q [2] in eq.(34) is a smooth solution, and the singular T 2 infers a smooth two-fold DT T 2 .
On the one side, by (k-1) times iteration of T 2 with a fixed eigenvalue λ 0 but different eigenfunctions, a non-degenerate (2k-2) fold DT T 2k−2 is obtained because each step of iteration is non-degenerate. On the other hand, setting double degeneration, i.e.,λ i = λ 0 in ψ 2k−1 and ψ i (λ 0 ) = 0(i = 1, 3, 5, · · · , 2k − 3) in eq. (27), and using reduction conditions eq.(33), then this T 2k−2 also can be presented after performing Taylor expansion with respect to . The smoothness of q [2k −2] is provided by the smoothness of each step of iteration as we shown in eq. (42). By the determinant representation of ψ Here |b 2k−1 | λ 2k−1 =λ 0 + and q [2k −2] in the formula of two-fold DT in eq.(34), respectively, then which is a smooth solution because α 0 = 0, |b Under double degeneration mentioned above, T 2k is degenerate. Thus solution in eq.(24) is singular at λ 0 . This lemma shows that its singularity is removable. We shall give a explicit representation of q smooth by Taylor expansion as the same manner of q [4] smooth . In other words, T 2k gives a non-degenerate 2k-fold DT T 2k by Taylor expansion, and q smooth is generated from q by this non-degenerate DT. Theorem 14. Let ψ 1 be an smooth eigenfunction of Lax pair of the MNLS, has also continuous dependence on the λ 0 . Here λ 0 is only one zero point of ψ 1 . Setting λ 2l−1 → λ 1 = λ 0 (l = 1, 2, 3, · · · , k) in q [2k] of eq.(24), then higher order Taylor expansion in it with respect to leads to a smooth k-order solution q [2k] smooth of the MNLS possessing following formula.
Here  , m = 0, 1, · · · , 2k, l = 1, 2, · · · k. Note again as theorem 7 that k-order just denotes the k-fold of DT for the MNLS,which is not related to the smoothness of the solution.

Rational solutions generated by 2k-fold degenerate Darboux transformation
According to theorem 14, it is a crucial step to find a suitable zero point λ 0 of eigenfunction such that h l m1 and h l m2 are both polynomials in x and t, which will be given from an explicit formula of ψ 1 in this section. We shall find explicit forms of the eigenfunction ψ j associated with a periodic seed solution, and then present smooth rational k-order solutions according to theorem 14. Next, the explicit representations of the first order and the second order rational solutions of the MNLS equation are constructed. The explicit forms of the first four order rogue waves are also provided. Furthermore, localization of the first order rational solution is analyzed and several new patterns of the higher order rogue waves are presented. We shall show an unusual result: for a given value of a, the increasing value of b can damage gradually the localization of the rational solution.
Let a 1 and c 1 be two complex constants, then q = c 1 exp (i(a 1 x + (−bc 1 2 − a 1 2 − aa 1 c 1 2 )t)) is a periodic solution of the MNLS equation, which will be used as a seed solution of the DT. Substituting q = c 1 exp (i(a 1 x + (−bc 1 2 − a 1 2 − aa 1 c 1 2 )t)) into the spectral problem eq.(4) and eq.(5), and using the method of separation of variables and the superposition principle, the eigenfunction ψ 2k−1 associated with λ 2k−1 is given by . (49) Here and a, b, a 1 , c 1 , x, t ∈ R, C 1 , C 2 , C 3 , C 4 ∈ C. Note that 1 (x, t, λ 2k−1 ) and 2 (x, t, λ 2k−1 ) are two linear independent solutions of the spectral problem eq.(4) and eq.(5). We can only get the trivial solutions through DT of the MNLS equation by setting eigenfunction ψ 2k−1 be one of them. This is the reason of setting ψ 2k−1 as the linear superposition in eq. (49).
In order to make higher order rational solution of the MNLS, a crucial step is to find the zero point of S and the eigenfunctions ψ l such that exponential functions vanish and the indeterminate form 0 0 appear in the q [2k] . By tedious calculation, we have following lemma concerning of this fact.

Lemma 15 Let
is only one zero point of S and eigenfunction ψ 2k−1 in eq. (49). Here K 0 , S j , L j ∈ C.
Theorem 16 For the eigenfunction ψ 1 defined by eq.(49) and eq.(50), h l m1 and h l m2 are both polynomials in variables x and t. Furthermore, q Obviously, this is a case of double degeneration,and then we can apply Theorem 14 here. Because λ 0 is a zero point of S and ψ 1 in eq.(49), exp(S) will disappear when λ 1 = λ 0 , h l m1 and h l m2 are both polynomials in x and t. Therefore, take the eigenfunction ψ 1 defined by eq.(49) and eq.(50) into theorem 14, q [2k] smooth are smooth rational solutions of the MNLS. Note again as theorem 7 that k-order just denotes the k-fold of DT for the MNLS,which is not related to the smoothness of the solution or the order of polynomials.
In order to show the novel properties, we shall discuss the asymptotic behaviors of the rational 1-order solution. Case a). Let a < b, there is only one saddle point of the profile for the q [1] rational at point (0, 0) . It is a nonlocal solution with two peaks. Case b). Let a = b, the maximum amplitude of |q [1] rational | 2 is equal to 9 and the trajectory is defined by x = (−2 + 3b)t. It is a line soliton with only one peak. Of course it is nonlocal.
, there is only one maximum at (0, 0) in the profile of |q [1] rational | 2 , which is 9. This solution has one peak and one vale, and then nonlocal.
Case d). Let b < a − 3 8 a 2 , there are one maximum at (0,0) and and two minima at ( ) in the profile of the |q [1] rational | 2 . The maximum is equal to 9 and the minimum is 0. This solution has one dominant peak and two hollows. It is localized in both x and t directions, and then is called the first order rogue wave solution. The dynamical evolution of the |q [1] rational | 2 is in accord with the classical "Peregrine soliton" in the NLS equation [12] and DNLS equation [31,32].
This discussion implies an unusual result: for a given value of a, the increasing value of b can damage gradually the localization of the rational solution. This is in contrast to the usual conjecture that that the localization of this solution will be enhanced because of the appearance of the two nonlinear effects represented by a and b, according to a common understanding of the role for the nonlinear effects in wave propagation.
For a given value a = 1, the profiles of |q [1] rational | 2 in case d),case c) and case a) are given in Figure 1 with b = 1 3 , 3 4 , 3, respectively. This figure shows visually the lost of the localization of the solution due to the increasing value of b. We omit the picture of case b) because it is a standard soliton. The density plots of the solutions of case d), i.e.,rogue wave solutions, of the MNLS equation are given in figure 2 with b = 0, 1 3 , 7 15 . It is clear to see the diffusion of the peak and hollows of the first order rogue wave when the value of b is increasing in Figure 2. The explicit form of the rogue wave in Figure 2 which is obtained from q [1] rational by setting a = 1 and b = 1/3. The orders of polynomial in numerator and denominator of q [1] rw are both 4. Figure 3(a) is plotted for the contour line at height 5 of the rogue wave |q [1] rational | 2 with different values of b in Figure 2. Note that height 5 is half value of the peak over the asymptotic plane. Let a = 1, then d = 1 2 the distance from the minimum point of the |q [1] rational | 2 to the coordinate origin, which is plotted in Figure 3(b). Figure 3 shows the remarkable decrease in localization of the first order rogue wave. In particular, the distance d goes to infinity when b → 5 8 in Figure 3(b), then |q [1] rational | 2 loses completely the localization in x and t. This fact is consistent with the limit value of b in case d).
To see the novelty of the two-peak solution for case a), we would like to present its explicit form as |q [1] which is obtained by setting a = 1 from eq. (51) and is plotted in Figure 1(c). The approximate trajectory of two peaks in the profile of |q [1] twopeak | 2 are two curves defined by 20t 2 + 1 − 4x 2 = 0 if x > 1 2 . It is interesting to note that the height of two peaks is gradually increasing (decreasing). The maximum height of peak along the trajectory is 21 + 4 √ 5 and the minimum height is 21 − 4 √ 5. This two-peak solution with a variable height and a non-vanishing boundary of soliton equation has never been discovered, to the best of our knowledge. Besides, two peak soliton in present paper can not be a usual double-soliton because the rational soliton just has one eigenvalue λ 0 . However a double-soliton has two eigenvalues in general. 4.2 Analytical forms and localization of the higher order rogue wave solutions Let k = 2, a 1 = −1, c 1 = 1, K 0 = 1, S 0 = L 0 = S 1 = L 1 = 0 in theorem 16, the rational 2-order solution q [4] smooth becomes in above q [k] rw (k = 2, 3, 4) are 12,24,40. This fact supports following conjecture: In general, the degree of the polynomial of the denominator for the rational k-order solution in theorem 16 is 2k(k + 1). It is a double of the corresponding degree [19] of the rogue wave for the NLS equation due to the contribution of the square of Ω n3 in theorem 3. Figures 4, 5 and 6 are plotted for the second order, third order and fourth order rogue waves from q smooth (k = 2, 3, 4). These figures show that they are localized in both x and t direction and peaks are diffused dramatically when the value of b is increased. This observation is a strong support to show that the localization of the rogue wave for the MNLS equation is decreased remarkably by increasing the value of b. This is a unique phenomenon in a rogue wave solution of the MNLS because of the appearance of the two nonlinear terms. However, it can not happen in the rogue wave of the NLS equation.
Our method can also be applied to get other patterns of the rogue wave by selecting different values of the parameters. For example, the fundamental patterns (a simple central highest peak surrounded by several gradually decreasing peaks in two sides) of the second order, the third order and the fourth order rogue wave are plotted in Figure 7 with the help of |q smooth | 2 . Similarly, a triangle pattern, a ring-decomposition pattern ( a second order rogue wave surrounded by seven first rogue wave) and a pentagon pattern of the fourth order rogue wave are plotted in Figure 8

Conclusions and Discussions
In this paper, the determinant representation of the n-fold DT for the WKI system is given in theorem 2. By choosing paired eigenvalues and paired eigenfunctions in the form λ l ↔ ψ l = φ l ϕ l , and λ 2l = −λ * 2l−1 , ↔ ψ 2l = smooth is given in theorem 16. By a detailed analysis of the localization of the rational solutions and the rogue waves, we get an unusual result: for a given value of a, the increasing value of b can damage gradually the localization of the rational solution, and a novel two-peak rational solution with a variable height and a non-vanish boundary in section 4. Note that this two peak rational soliton just has one eigenvalue λ 0 , which can not be a usual double-soliton. We have verified the validity of theses exact and analytical solutions by symbolic computation with a computer.
Finally we would like to stress that there is no doubt of the novelty of the rational solutions presented in this paper although there exists a simple gauge transformation, for example, see refs. [43,59], between the MNLS equation and the DNLS equation. To illustrate this statement clearly, we shall use following form of the DNLS equation ofq =q(X, T ): iq T −q XX + i(q 2q * ) X = 0.
The explicit form of this gauge transformation [43,59] from the MNLS to the DNLS is with t = − T a 2 5 , x = −a 6 5 X + 2bT a 7 5 . The inverse transformation maps the DNLS to the MNLS, which is given by q = −q(X, T )a − 2 5 exp(−i −a 6 5 bX + b 2 T a 12 5 ), and T = −a between (|q| x , |q| t ) and (|q| X , |q| T ), which shows that invertible transformations in eq.(57) and eq.(58) preserve the numbers of extreme value points for a given t (or T ) and stationary points in profiles of |q| andq. In other words, this simple gauge transformation can not change the numbers of peaks or valleys in |q| and |q|. Therefore, the first-order RW and the first-order rational soliton of the DNLS [31,60,61] can not be mapped to the two-peak rational solution of the MNLS by this transformation. By the gauge transformation eq.(58), two solutions of the MNLS with a = 1 and b = 3 are generated from corresponding solutions of the DNLS in eq.
(56) with α 1 = β 1 = 1 2 and in eq. (46) with β 1 = 1 2 of Ref. [31], which are plotted in Figure  9. Similarly, two peak rational solution of the DNLS can be generated from a corresponding solution in eq. (53) of the MNLS, which is plotted in Figure 10. The two peak rational solution of the DNLS has only one eigenvalue and a non-vanishing boundary, which has never been reported in literatures. are given as eq. (14). Further, by using explicit matrix representation eq.(13) of T 1 , the new eigenfunction ψ [1] j = T 1 (λ; λ 1 )| λ=λ j ψ j for j 2 becomes Last, a tedious calculation verifies that T 1 in eq.(13) and new solutions indeed satisfy eq.(A.3) and eq.(A.5) in appendix I. In the process of verification, it is crucial to use the fact that ψ 1 satisfies eq.(4) and eq.(5) of the Lax pair associated with a seed solution q and eigenvalue λ 1 . So WKI spectral problem is covariant under transformation T 1 in eq.(13) and eq. (14). Therefore T 1 is the DT of eq.(2) and eq.(3).