Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 2

A revised and rigorously proved inverse scattering transform (IST for brevity) for DNLS + equation, with a constant nonvanishing boundary condition (NVBC) and normal group velocity dispersion, is proposed by introducing a suitable affine parameter in the Zakharov-Shabat IST integral; the explicit breather-type and pure N -soliton solutions had been derived by some algebra techniques. On the other hand, DNLS equation with a non-vanishing background of harmonic plane wave is also solved by means of Hirota ’ s bilinear formalism. Its space periodic solutions are determined, and its rogue wave solution is derived as a long-wave limit of this space periodic solution.


Breather-type and pure N-soliton solution to DNLS + equation with NVBC based on revised IST
DNLS + equation with NVBC, the concerned theme of this section, is only a transformed version of modified nonlinear Schrödinger equation with normal group velocity dispersion and a nonlinear self-steepen term and can be expressed as here the subscripts represent partial derivatives. Some progress have been made by several researchers to solve the DNLS equation for DNLS equation with NVBC, many heuristic and interesting results have been attained [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. An early proposed IST worked on the Riemann sheets can only determine the modulus of the one-soliton solution [3,15]. References [4,5,16] had attained a pure single dark/bright soliton solution. Reference [6] had derived a formula for NÀ soliton solution in terms of Vandermonde-like determinants by means of Bäcklund transformation; but just as reference [7,9] pointed out, this multi-soliton solution is still difficult to exhibit the internal elastic collisions among solitons and the typical asymptotic behaviors of multi-soliton of DNLS equation. By introducing an affine parameter in the integral of Zakharov-Shabat IST, reference [7] had found their pure NÀ soliton solution for a special case that all the simple poles (zeros of a λ ð Þ) were located on a circle of radius ρ centered at the origin, while reference [8] also found its multi-soliton solution for some extended case with N poles on a circle and M poles out of the circle, and further developed its perturbation theory based on IST. Reference [7] constructed their theory by introducing an damping factor in the integral of Zakharov-Shabat IST, to make it convergent, and further adopted a good idea of introducing an affine parameter to avoid the trouble of multi-value problem in Riemann sheets, but both of their results are assumed NÀ soliton solutions and the soliton solution gotten from their IST had a self-dependent and complicated phase factor [7][8][9], hence reference [8] had to verify an identity demanded by the standard form of a soliton solution (see expression (52) in Ref. [8]). Such kind of an identity is rather difficult to prove for N ≥ 2 case even by the use of computer techniques and Mathematica. On the other hand, author of reference [8] also admits his soliton solution is short of a rigorous verification of standard form. Then questions naturally generateswhether the traditional IST for DNLS equation with NVBC can be avoided and further improved? And whether a rigorous manifestation of soliton standard form can be given and a more reasonable and natural IST can be constructed?
A newly revised IST is thus proposed in this section to avoid the dual difficulty and the excessive complexity. An additional affine factor 1=λ, λ ¼ z þ ρ 2 z À1 ð Þ =2, is introduced in the Z-S IST integral to make the contour integral convergent in the big circle [7,[10][11][12][13]. Meanwhile, the additional two poles on the imaginary axis caused by λ ¼ 0 are removable poles due to the fact that the first Lax operator L λ ð Þ ! 0, as λ ! 0. What is more different from reference [7] is that we locate the N simple poles off the circle of radius ρ centered at origin O, which corresponds to the general case of N breather-type solitons. When part of the poles approach the circle, the corresponding part of the breathers must tend to the pure solitons, which is the case described in Ref. [8]. The resulted one soliton solution can naturally tend to the well-established conclusion of VBC case as ρ ! 0 [17][18][19][20] and the pure one soliton solution in the degenerate case. The result of this section appears to be strict and reliable.

The fundamental concepts for the IST theory of DNLS equation
Under a Galileo transformation x, t ð Þ ! x þ ρ 2 t, t ð Þ , DNLS þ Eq. (1) can be changed into with nonvanishing boundary condition: According to references [7][8][9], the phase difference between the two infinite ends should be zero. The Lax pair of DNLS þ Eq. (3) is where σ 3 , σ þ , and σ À involve in standard Pauli's matrices and their linear combination. Here and hereafter a bar over a variable represents complex conjugate. An affine parameter z and two aided parameters η, λ are introduced to avoid the trouble of dealing with double-valued functions on Riemann sheets λ z þ ρ 2 z À1 À Á =2, η z À ρ 2 z À1 À Á =2 (6) The Jost functions satisfy first Lax equation here Jost functions F x, z ð Þ∈ Ψ x, z ð Þ, Φ x, zÞ ð g f .
The free Jost function E x, z ð Þ can be easily attained as follows: which can be verified satisfying Eq. (7). The monodramy matrix is which is defined by Some useful and important symmetry properties can be found Symmetry relations in (13) lead tõ The above symmetry relations further result iñ Other important symmetry properties called reduction relations can also be easily found The above symmetry properties lead to following reduction relations among Jost functions The important symmetries among the transition coefficients further resulted from (12), (21), and (22): On the other hand, the simple poles, or zeros of a z ð Þ, appear in quadruplet, and can be designated by z n , (n ¼ 1, 2, ⋯, 2N), in the I quadrant, and z nþ2N ¼ Àz n in the III quadrant. Due to symmetry (17), (18) and (23), the n' th subset of zero points is And we arrange the 2N zeros in the first quadrant in following sequence z 1 , z 2 ; z 3 , z 4 ; ⋯; z 2NÀ1 , z 2N (25) According to the standard procedure [21,22], the discrete part of a z ð Þ can be deduced At the zeros of a z ð Þ, or poles z n , Using symmetry relation in (14), (17), (21)- (24), (27), we can prove that

Relation between the solution and Jost functions of DNLS + equation
The asymptotic behaviors of the Jost solutions in the limit of λ j j ! ∞ can be obtained by simple derivation. Let F Á1 ¼ψ x, λ ð Þ, then Eq. (7) can be rewritten as then we haveψ We assume a function g to satisfy the following equatioñ Substituting (31)-(32) into Eq. (30), we have In the limit of |z| ! ∞, g x can be expanded as sum of series of z À2 ð Þ j , j ¼ 0, 1, 2, ⋯ where Inserting formula (34) and (35) in Eq. (33) at |z| ! ∞ leads to Then we find a useful formula which expresses the conjugate of the solution u in terms of Jost functions as |z| ! ∞.

Introduction of time evolution factor
In order to make the Jost functions satisfy the second Lax equation, a time evolution factor h t, z ð Þ should be introduced by a standard procedure [21,22] in the Jost functions and the scattering data. Considering the asymptotic behavior of the second Lax operator we let h t, k ð Þ to satisfy then Due to from (39) and (40), we have Therefore, the complete Jost functions should depend on time as follows Nevertheless, hereafter the time variable in Jost functions will be suppressed because it has no influence on the treatment of Z-S equation. By a similar procedure [9,15], the scattering data has following time dependences

Zakharov-Shabat equations and breather-type N-soliton solution
here and hereafter note "" represents definition. There is an abrupt jump for Π x, z ð Þ across both real and imaginary axes where is called the reflection coefficient. Due to μ 0 6 ¼ 0 in (34), Jost solutions do not tend to the free Jost solutions E x, z ð Þin the limit of |z| ! ∞. This is their most typical property which means that the usual procedure of constructing the equation of IST by a Cauchy contour integral must be invalid. In view of these abortive experiences, we proposed a revised method to derive a suitable IST and the corresponding Z-S equation by multiplying an inverse spectral parameter 1=λ, λ ¼ z þ ρ 2 z À1 ð Þ =2, before the Z-S integrand. Meanwhile, our modification produces no new poles since the Lax operator L λ ð Þ ! 0, as λ ! 0. In another word, the both additional poles z 0 ¼ AEiρ generated by λ ¼ 0 are removable. Under reflectionless case, that is, r z ð Þ ¼ 0, the Cauchy integral along with contour Γ shown in Figure 1 gives where Note that (27), (45), and (46) have been used in (48). The minus sign before the sum of residue number in (48) comes from the clock-wise contour integrals around the 4 N simple poles when the residue theorem is used, shown in Figure 1. By a standard procedure, the time dependences of b n and c n similar to (43) can be derived In the reflectionless case, the Zakharov-Shabat equations for DNLS + equation can be derived immediately from (48) as follows here Λ λη, Λ n λ z n ð Þη z n ð Þ ¼ λ n η n , and in Eqs. (52) and (53), the terms corresponding to poles z n , (n ¼ 1, 2, ⋯, 2N), have been combined with the terms corresponding to poles z nþ2N ¼ Àz n . Substituting Eqs. (52) and (53) into formula (36) and letting z ! ∞, we attain the conjugate of the raw N-soliton solution (the time dependence is suppressed).
, respectively, in Eqs. (52) and (53), by use of reduction relations (19), (21), and (22), we can further change Eqs. (52) and (53) into the following form m ¼ 1, 2, ⋯, 2N. An implicit time dependence of the complete Jost functions ψ 1 and ψ 2 besides c n should be understood. To solve Eq. (58), we define that f n ffiffiffiffiffiffi ffi 2c n p e iΛ n x , g n i ffiffiffiffi c n 2 r z n ρλ n e iΛ n x ¼ iz n 2ρλ n f n , n ¼ 1, 2, ⋯, 2N (60) Then Eq. (58) can be rewritten as or in a more compact form The above equation gives Note that the choice of poles, z n , n ¼ 1, 2, ⋯, N ð Þ , should make det I þ B ð Þ nonzero and I þ B ð Þan invertible matrix. On the other hand, Eq. (55) can be rewritten as hereafter a superscript "T" represents transposing of a matrix. Substituting Eq. (65) into (66) leads to where with To solve Eq. (57), we define that or in a more compact form The above equation givess Note that the choice of poles, z n , n On the other hand, Eq. (56) can be rewritten as Substituting Eq. (76) into (77), we thus attain where use is made of Appendix A.1 and In the end, by substituting (67) and (78) here The solution has a standard form as (80), that is which can be proved by direct calculation for the N ¼ 1 case and by some special algebra techniques for the N > 1 case.

Verification of det
In order to prove the first identity in (82), we firstly calculate D N ¼ det I þ B ð Þ. By use of (60)-(62), Binet-Cauchy formula, (Appendix (A.2)) and an important determinant formula, (Appendix (A.3)), we have here B n 1 , n 2 , ⋯, n r ð Þis a r , th-order principal minor of B consisting of elements belonging to not only rows n 1 , n 2 , ⋯, n r ð Þbut also columns n 1 , n 2 , ⋯, n r ð Þ . Due to (62), form a typical factor as a whole, (with just totally r r À 1 ð Þ=2 pairs). Note that if we define z n ρe δ n þiβ n , with δ n > 0, β n ∈ 0, π=2 ð Þ, and further define that then the typical factor is and here n, m ∈ n 1 , n 2 , ⋯, n r ð Þ , and a typical function F n is defined as where use is made of formula (60) and (62), the time dependence of the solution naturally emerged in c n t ð Þ. Substituting Eq. (88) into (83) thus completes the computation of D N .
Secondly, let us calculate det I þ D 0 ð Þ. By use of (72) and (73), Binet-Cauchy formula, (Appendix A.2) and an important matrix formula, (Appendix A.3), we have where D 0 n 1 , n 2 , ⋯, n r ð Þis the principal minor of a r , th-order submatrix of D 0 consisting of elements belonging to not only rows n 1 , n 2 , ⋯, n r ð Þbut also columns n 1 , n 2 , ⋯, n r ð Þ , and n, m ∈ n 1 , n 2 , ⋯, n r ð Þ . Using the same tricks as used in dealing with (84) leads to According to (79) and Binet-Cauchy formula (Appendix (A.2)), similarly we have n, m ∈ n 1 , n 2 , ⋯, n r ð Þ . Using the same tricks as that used in treating (84) leads to Due to (95), comparing (83) and (93) results in the expected identity and completes the verification of the first identity in (82).

Verification of det
Our most difficult and challenging task is to prove the second identity in (82). For convenience of discussion, we define that Then the sequence of poles (25) is just in the same order as follows On the other hand, due to (28), (62), and (73), we have Then Substituting z n ¼ ρ 2 z À1 n , z m ¼ ρ 2 z À1 m into above formula and using following relation We can get an important relation between D 0 nm and Bmn On the other hand, an unobvious symmetry between matrices B nm ð Þ 2NÂ2N and Bnm ð Þ 2NÂ2N is found It can be rewritten in a more explicit form The last equation in (105) is due to (97) and (99), thus from (103) and (104), we have The determinants of matrices at both sides of (106) are equal to each other The left hand of (107) is just det I þ B ð Þ, and this completes verification of identity (82). From the verified (82), we know multi-soliton solution (80) is surely of a typical form as expected.

The explicit N-soliton solution to the DNLS + equation with NVBC
In order to get an explicit N-soliton solution to the DNLS + Eq. (1) with NVBC, firstly we need to make an inverse Galileo transformation of (2) by Þin F n x, t ð Þ in (89). Due to (51) and (85)-(87), the typical soliton kernel function F n can be rewritten as where in (114) the n'th pole-dependent constant factor has been absorbed by redefinition of the n'th soliton center and initial phase in (110)-(111).
According to the definition of A in (68)-(69), using Binet-Cauchy formula, (Appendix (A.2)), leads to where A n 1 , n 2 , ⋯, n r ð Þis the determinant of a r , th-order minor of A consisting of elements belonging to not only rows n 1 , n 2 , ⋯, n r ð Þbut also columns n 1 , n 2 , ⋯, n r ð Þ .
n, m ∈ n 1 , n 2 , ⋯, n r ð Þ . Using the same tricks as used in dealing with (84) leads to

The one and two-soliton solutions to DNLS + equation with NVBC
We give two concrete examplesthe one and two breather-type soliton solutions in illustration of the general explicit NÀ soliton formula.
which is generally called a breather solution and shown as Figure 2.

Explicit pure N-soliton solution to the DNLS + equation with NVBC
When all the simple poles are on the circle O, ρ ð Þcentered at the origin O, just as shown in Figure 4, our revised IST for DNLS + equation with NVBC will give a typical pure N-soliton solution. The discrete part of a z ð Þ is of a slightly different form from that of the case for breather-type solution, and it can be expressed as Evolution of the square amplitude of a breather-type two-soliton with respect to time and space ρ ¼ 2; δ 1 ¼ 0:4; At the zeros of a z ð Þ, we have ϕ x, z n ð Þ¼b n ψ x, z n ð Þ, _ a Àz n ð Þ ¼ À_ a z n ð Þ, b n ¼ Àb n (144) On the other hand, the zeros of a z ð Þ appear in pairs and can be designed by z n , (n ¼ 1, 2, ⋯, N), in the I quadrant, and z nþN ¼ Àz n in the III quadrant. The Zakharov-Shabat equation for pure soliton case of DNLS+ equation under reflectionless case can be derived immediatelỹ Here Different from that in breather-type case, we define z n ρe δ n e iβ n ¼ ρe iβ n , with Ác no e Ài4λ 3 n κ n t e i2Λ n x Due to b n0 ¼ Àb n0 , À ib n0 ∈ , following equations hold: Àib n0 e i2Λ n xÀ 2λ 2 n þρ 2 ð Þt ½ ε n E n e iβ n exp Àθ n þ iφ n ð Þ where Àib n0 e i2Λ n xÀ 2λ 2 ε n sgn Àib n0 ð Þ; Àib n0 j j e ν n x n0 (155) where E n is also a real constant which is only dependent upon the order number n. The constant and positive real number Àib n0 j jhas been absorbed by redefinition of the n' th soliton center x n0 in (155). Thus the determinants in formula (83) for pure soliton solution can be calculated as follows The N ¼ 2 case, that is, the pure two-soliton is also a typical illustration of the general explicit N-soliton formula. According to (158)-(162), it can be calculated as follows ð Þ (163) The evolution of pure two-soliton solution with respect to time and space is given in Figure 5. It clearly demonstrates the whole process of the elastic collision between pure two solitons. If 0 < β 2 < β 1 < π=2, then ε 1 ¼ 1, sgn E 1 ¼ 1 and ε 2 ¼ À1, sgn E 2 ¼ À1 correspond to double-dark pure 2-soliton solution as in Figure 5a; ε 1 ¼ À1, sgn E 1 ¼ 1 and ε 2 ¼ 1, sgn E 2 ¼ À1 correspond to a double-bright pure 2-soliton solution in Figure 5c; ε 1 ¼ 1, sgn E 1 ¼ 1 and ε 2 ¼ 1, sgn E 2 ¼ À1 correspond to a dark-bright-mixed pure 2-soliton solution in Figure 5b. In the limit of infinite time t ! AE∞, the pure 2-soliton solution is asymptotically decomposed into two pure 1-solitons.
By the way, it should be point out, although our method and solution have different forms from that of Refs. [7,16], they are actually equivalent to each other. In fact if the constant E n , (n ¼ 1, 2, ⋯, N), is also absorbed into the n'th soliton center x n0 just like Àib n0 does in (152)-(154), and replace β n with β 0 n ¼ π=2 À β n ∈ 0, π=2 ð Þ, the result for the pure soliton case in this section will reproduce the solution gotten in Refs. [7,9,16].
On the other hand, letting only part of the poles converge in pairs on the circle in Figure 1 and rewriting the expression of a n z ð Þ as in Ref. [7,8,12], our result can naturally generate the mixed case with both pure and breather-type multi-soliton solution.

The asymptotic behaviors of the N-soliton solution
Without loss of generality, we assume β 1 > β 2 > ⋯ > β n > ⋯ > β N ; (156), and define the n'th neighboring area as ϒ n : x À x no À υ n t $ 0, n ¼ 1:2, ⋯, N ð Þ . In the neighboring area of ϒ n , where B 1, 2, …, n ð Þ¼ε n E n e Àθ n Àiβ n Y nÀ1 j¼1 sin 2 β j À β n In the neighboring area of ϒ n , we have As t ! À∞, the N neighboring areas queue up in a descending series ϒ N , ϒ NÀ1 , ⋯, ϒ 1 , then the N-soliton solution can be viewed as N well-separated exact pure one solitons, each u 1 θ n þ Δθ À ð Þ n À Á , (1, 2, ⋯, n) is a single pure soliton characterized by one parameter β n , moving to the positive direction of the x-axis, queuing up in a series with descending order number n.
As t ! ∞, in the neighboring area of ϒ n we have where B n, n þ 1, …, N ð Þ¼ε n E n e Àθ n Àiβ n Y N j¼nþ1 sin 2 β j À β n That is, the N-soliton solution can be viewed as N well-separated exact pure one solitons, queuing up in a series with ascending order number n such as ϒ 1 , ϒ 2 , ⋯, ϒ N :.
In the process of going from t ! À∞ to t ! ∞, the n'th pure single soliton overtakes the solitons from the 1'th to n À 1'th and is overtaken by the solitons from n þ 1'th to N'th. In the meantime, due to collisions, the n'th soliton got a total forward shift Δθ À ð Þ n =ν n from exceeding those slower soliton from the 1'th to n À 1'th, got a total backward shift Δθ þ ð Þ n =ν n from being exceeded by those faster solitons from n þ 1'th to N'th, and just equals to the summation of shifts due to each collision between two solitons, that is, By introducing an suitable affine parameter in the IST and based upon a newly revised and improved inverse scattering transform and the Z-S equation for the DNLS þ equation with NVBC and normal dispersion, the rigorously proved breather-type N-soliton solution to the DNLS þ equation with NVBC has been derived by use of some special linear algebra techniques. The one-and two-soliton solutions have been given as two typical examples in illustration of the unified formula of the N-soliton solution and the general computation procedures. It can perfectly reproduce the well-established conclusions for the special limit case. On the other hand, letting part/all of the poles converge in pairs on the circle in Figure 4 and rewriting the expression of a n z ð Þ as in [7,12,13], can naturally generate the partly/wholly pure multi-soliton solution. Moreover, the exact breather-type multi-soliton solution to the DNLS þ equation can be converted to that of the MNLS equation by a gauge-like transformation [17].
Finally, the elastic collision among the breathers of the above multi-soliton solution has been demonstrated by the case of a breather-type 2-soliton solution. The newly revised IST for DNLS þ equation with NVBC and normal dispersion makes corresponding Jost functions be of regular properties and asymptotic behaviors, and thus supplies substantial foundation for its direct perturbation theory.

Space periodic solutions and rogue wave solution of DNLS equation
DNLS equation is one of the most important nonlinear integrable equations in mathematical physics, which can describe many physical phenomena in different application fields, especially in space plasma physics and nonlinear optics [1,2,16,[24][25][26][27][28][29]. We have found that DNLS equation can generate not only some usual soliton solutions such as dark/bright solitons and pure/breather-type solitons, but also some special solutionsspace periodic solutions and rogue wave solution [14].
There are two celebrated models of the DNLS equations. One equation is called Kaup-Newell (KN) equation [15]: and the other is called Chen-Lee-Liu (CLL) equation [30]: Actually, there is a gauge transformation between these two Eqs. (183) and (184) [14,30,31]. Supposing u is one of the solutions of the KN Eq. (183), then will be the solution of the CLL equation. This section focuses on the KN Eq. (183) with NVBCperiodic plane-wave background. The first soliton solution of (183) was derived by Kaup and Newell via inverse scattering transformation (IST) [3,15,32]. Whereafter, the multi-soliton solution was gotten by Nakamura and Chen by virtue of the Hirota method [30,31]. The determinant expression of the N-soliton solution was found by Huang and Chen on the basis of the Darboux transformation (DT for brevity) [33], and by Zhou et al., by use of a newly revised IST [7,[11][12][13]17].
Recently, rogue waves which seem to appear from nowhere and disappear without a trace have drawn much attention [34,35]. The most significant feature of rogue wave is its extremely large wave amplitude and space-time locality [35]. The simplest way to derive the lowest order of rogue wave, that is, the Peregrine solution [35,36], is to take the long-wave limit of an Akhmediev breather [37] or a Ma breather [38], both of which are special cases of the periodic solution. Thus, the key procedure of generating a rogue wave is to obtain an Akhmediev breather or a Ma breather. As far as we know, DT plays an irreplaceable role in deriving the rogue wave solution [39][40][41]. Because both Akhmediev breather and Ma breather can exist only on a plane-wave background; Darboux transformation has the special privilege that a specific background or, in other words, a specific boundary condition can be chosen as the seed solution used in DT. For instance, if we choose q 0 ¼ 0 as the seed solution of the DT of the KN Eq. (183), then after 2-fold DT, a new solution will be gotten under VBC: (where all the parameters are defined in Ref. [38]). Similarly, setting a seed solution q 0 ¼ c exp i ax þ Àc 2 þ a ð Þ at ½ , a plane-wave solution to Eq. (183), will generate a new solution after 2-fold DT under a plane-wave background. Therefore, there is no need to discuss the boundary conditions or background when applying DT to solve those nonlinear integrable equations. This makes DT the most effective and prevailing method in obtaining a rogue wave solution.
Compared with DT, the IST has its fatal flaw that the difficulty of dealing with the boundary condition is unavoidable, which limits the possible application of the IST. Although the KN equation has been solved theoretically by means of an improved IST for both VBC and the NVBC [7][8][9][17][18][19][20], there is no report that the KN equation could be solved under a plane-wave background by means of IST. And consequently, it appears that rogue wave solutions cannot be obtained through the IST method. This major problem is caused by the difficulty of finding appropriate Jost solutions under the plane-wave background.
On the other hand, the Hirota's bilinear-derivative transform (HBDT for brevity) [42][43][44][45][46], though not as a prevalent method as DT, has its particular advantages. The core of this method is a bilinear operator D which is defined by: where, at the left side of the above formula, a dot • between two functions A x, t ð Þand B x 0 , t 0 ð Þrepresents an ordered product. The HBDT method is very useful in dealing with periodic solutions for its convenience in computing the bilinear derivatives of an exponential function [44]: Here, F represents general function expressed by the finite or infinite power series expansion of the Hirota's bilinear differential operators. Formula (188) is the generalization of Appendix 5.5. Thus using HBDT method to find space periodic solutions of KN equation is practicable. The space periodic solutions possess the characters that they approach the plane-wave solution when |t| ! ∞ and are periodic in space. The first space periodic solution was found by Akhmediev with one parameter [45]. Actually, we can regard the space periodic solution as a special Akhmediev breather with a pure complex-valued wave number. Further, through a space periodic solution, a rogue wave solution can be constructed. This means besides DT, HBDT method is also an alternative and effective way to find rogue wave solution of KN equation.

Bilinear derivative transformation of DNLS equation
The Hirota bilinear transformation is an effective method which could help to solve the KN equation. Due to the similarity of the first equation of Lax pairs between that of DNLS equation and AKNS system, there is a direct inference and manifestation that u x, t ð Þ has a typical standard form [6,7]: where f and g are complex auxiliary functions needed to be determined. Applying the bilinear derivative transform to (189), we can rewrite the derivatives of u x, t ð Þ in the bilinear form [19,20,[42][43][44][45][46]: Directly substituting the above Eqs. (190)-(192) into (183) gives: Then the above transformed KN equation can be decomposed into the following bilinear equations: where λ is a constant which needs to be determined. Notice that if λ ¼ 0 then the above bilinear equations are overdetermined because we have only two variables but three equations. Actually, setting λ ¼ 0 is the approach to search for the soliton solution of the DNLS equation under vanishing boundary condition [19,20]. Here, we set λ as a nonzero constant to find solutions under a different boundary conditiona plane-wave background.

First order space periodic solution and rogue wave solution
Let us assume that the series expansion of the complex functions f and g in (189) are cut off, up to the 2'th power order of ϵ, and have the following formal form: Substituting f and g into Eqs. (194)-(196) yields a system of equations at the ascending power orders of ϵ, which allows for determination of its coefficients [14,19,20]. We have 15 equations [14,19,20] corresponding to the different orders of ϵ. After solving all the equations, then we can obtain the solution of the DNLS equation: (199) where Notice that ρ and M are real; b 1 and φ 0 are complex constants, so there are two restrictions for a valid calculation: (1) the wave number p must be a pure imaginary number; (2) the angular frequency Ω must not be purely imaginary number and must furthermore satisfy the quadratic dispersion relation: According to the test rule for a one-variable quadratic, there is a threshold condition under which Ω will not be a pure imaginary number: The asymptotic behavior of this breather is apparent. Because the wave number p is a pure imaginary number, the breather is a periodic function of x. The quadratic dispersion relation (204) which has the same parameters as u RW . And this solution υ c, RW has exactly the same form as the result given by ref. [46].

Second-order periodic solution
Taking the similar procedures described previously could help us to derive the 2nd-order space periodic solution. Assume the auxiliary functions f and g to have higher order expansions in terms of ϵ: Similarly, substituting f and g into the bilinear Eqs. (194)-(196) leads to the 27 equations [14,19,20] corresponding to different orders of ϵ. Solving these equations is tedious and troublesome but worthy and fruitful. The results are expressed in the following form: . The max amplitude is equal to 3 at the point In this section, the 1st order and the 2nd order space periodic solutions of KN equation have been derived by means of HBDT. And after an integral transformation, these two breathers can be transferred into the solutions of CLL equation. Meanwhile, based on the long-wave limit, the simplest rogue wave model has been obtained according to the 1st order space periodic solution. Furthermore, the asymptotic behaviors of these breathers have been discussed in detail. As |t| ! ∞, both breathers will regress into the plane wave with a phase shift.
In addition, the generalized form of these two breathers is obtained, which gives us an instinctive speculation that higher order space periodic solutions may hold this generalized form, but a precise demonstration is needed. Moreover, higher order rogue wave models cannot be constructed directly by the long-wave limit of a higher order space periodic solution because the higher order space periodic solution has multiple wave numbers p i , we are also interested in seeking an alternative method besides DT that could help us to determine the higher order rogue wave solutions.

Concluding remarks
In the end, as the author of the above two parts, part 1 and 2, I want to give some concluding remarks. As a whole, the two parts had taken the DNLS equation as a reference, systematically introduced several principal methods, such as IST, GLM (Marchenko) method, HBDT, to solve an integrable nonlinear equation under VBC and NVBC. We had gotten different kinds of soliton solutions, such as the light/ dark soliton, the breather-type soliton, the pure soliton, the mixed breather-type and pure soliton, and especially the rogue-wave solution. We had also gotten soliton solutions in a different numbers, such as the one-soliton solution, the two-soliton solution, and the N-soliton solution. Nevertheless, I regret most that I had not introduced the Bäcklund transform or Darboux transform to search for a rogue wave solution or a soliton solution to the DNLS equation, just like professor Huang N.N., one of my guiders in my academic research career, had done in his paper [33]. Another regretful thing is that, limited to the size of this chapter, I had not introduced an important part of soliton studies, the perturbation theory for the nearlyintegrable perturbed DNLS equation. Meanwhile, this chapter have not yet involved in the cutting-edge research of the higher-order soliton and rogue wave solution to the DNLS equation, which remain to be studied and concluded in the future.
The above formula also holds for the case of det I þ Ω 1 Ω 2 ð ÞWith Ω 1 to be a N Â N þ 1 ð Þmatrix and Ω 2 a N þ 1 ð ÞÂN matrix.
A4, For a squared matrix C with elements C jkÀ ¼ f j g k x j À y k À Á À1 , A5, Some useful blinear derivative formulae.