Based upon different methods such as a newly revised version of inverse scattering transform, Marchenko formalism, and Hirota’s bilinear derivative transform, this chapter aims to study and solve the derivative nonlinear Schrödinger (DNLS for brevity) equation under vanishing boundary condition (VBC for brevity). The explicit one-soliton and multi-soliton solutions had been derived by some algebra techniques for the VBC case. Meanwhile, the asymptotic behaviors of those multi-soliton solutions had been analyzed and discussed in detail.
- nonlinear equation
- derivative nonlinear Schrödinger equation
- inverse scattering transform
- Zakharov-Shabat equation
- Marchenko formalism
- Hirota’s bilinear derivative transform
- rogue wave
Derivative nonlinear Schrödinger (DNLS for brevity) equation is one of the several rare kinds of integrable nonlinear models. Research of DNLS equation has not only mathematic interest and significance, but also important physical application background. It was first found that the Alfven waves in space plasma [1, 2, 3] can be modeled with DNLS equation. The modified nonlinear Schrödinger (MNLS for brevity) equation, which is used to describe the sub-picosecond pulses in single mode optical fibers [4, 5, 6], is actually a transformed version of DNLS equation. The weak nonlinear electromagnetic waves in ferromagnetic, anti-ferromagnetic, or dielectric systems [5, 6, 7, 8, 9] under external magnetic fields can also be modeled by DNLS equation.
Although DNLS equation is similar to NLS equation in form, it does not belong to the famous AKNS hierarchy at all. As is well known, a nonlinear integrable equation can be transformed to a pair of Lax equation satisfied by its Jost functions, the original nonlinear equation is only the compatibility condition of the Lax pair, that is, the so-called zero-curvature condition. Another fact had been found by some scholars that those nonlinear integrable equations which have the same first operator of the Lax pair belong to the same hierarchy and can deal with the same inverse scattering transform (IST for brevity). As a matter of fact, the DNLS equation has a squared spectral parameter of in the first operator of its Lax pair, while the famous NLS equation, one typical example in AKNS hierarchy, has a spectral parameter of . Thus, the IST of the DNLS equation is greatly different from that of the NLS equation which is familiar to us. In a word, it deserves us to demonstrate several different approaches of solving it as a typical integrable nonlinear equation.
In this chapter, we will solve the DNLS equation under two kinds of boundary condition, that is, the vanishing boundary condition (VBC for brevity) and the non-vanishing boundary condition (NVBC for brevity), by means of three different methods – the revised IST method, the Marchenko formalism, and the Hirota’s bilinear derivative method. Meanwhile, we will search for different types of special soliton solution to the DNLS equation, such as the light/dark solitons, the pure solitons, the breather-type solitons, and the rogue wave solution, in one- or multi-soliton form.
2. An N-soliton solution to the DNLS equation based on a revised inverse scattering transform
For the VBC case of DNLS equation, which is just the concerned theme of the section, some attempts and progress have been made to solve the DNLS equation. Since Kaup and Newell proposed an IST with a revision in their pioneer works [10, 11], one-soliton solution was firstly attained and several versions of raw or explicit multi-soliton solutions were also obtained by means of different approaches [12, 13, 14, 15, 16, 17, 18, 19, 20]. Huang and Chen have got a soliton solution by means of Darboux transformation . Steudel has derived a formula for soliton solution in terms of Vandermonde-like determinants by means of Bäcklund transformation ; but just as Chen points out in Ref. , Steudel’s multi-soliton solution is difficult to demonstrate collisions among solitons and still has a too complicate form to be used in the soliton perturbation theory of DNLS equation, although it can easily generate compute pictures. Since the integral kernel in Zakharov-Shabat (Z-S for brevity) equation does not tend to zero in the limit of spectral parameter with , the contribution of the path integral along the big circle (the out contour) is also nonvanishing, the usual procedure to perform inverse scattering transform encounters difficulty and is invalid. Kaup thus proposed a revised IST by multiplying an additional weighing factor before the Jost solution , so that it tends to zero as , thus the modified Z-S kernel should lead to vanishing contribution of the integral along the big circle of Cauchy contour. Though the one-soliton solution has been found by the obtained Z-S equation of their IST, it is very difficult to derive directly its multi-soliton solution by their IST due to the existence of a complicated phase factor which is related to the solution itself . We thus consider proposing a new revised IST to avoid the excessive complexity. Our -soliton solution obviously has a standard multi-soliton form. It can be easily used to discuss its asymptotic behaviors and then develop its direct perturbation theory. On the other hand, in solving Z-S equation for DNLS with VBC, unavoidably we will encounter a problem of calculating determinant , for two N × N matrices and , where is a N × N identity matrix. Our work also shows Binet-Cauchy formula and some other linear algebra techniques, (Appendices A.1–4 in Part 2), play important roles in the whole process, and actually also effective for some other nonlinear integrable models .
2.1 The revised inverse scattering transform and the Zakharov-Shabat equation for DNLS equation with VBC
2.1.1 The fundamental concepts for the IST theory of DNLS equation
DNLS equation for the one-dimension wave function is usually expressed as
with VBC, where the subscripts stand for partial derivative. Eq. (1) is also called Kaup-Newell (KN for brevity) equation. Its Lax pair is given by
where is a spectral parameter, and is the third one of Pauli matrices , , , and a bar over a letter, (e.g., in (2)), represents complex conjugate. The first Lax equation is
In the limit of , , and
The free Jost solution is a matrix.
The Jost solutions of (4) are defined by their asymptotic behaviors as .
where , , etc., and superscript “” represents transposing of a matrix here and afterwards.
Since the first Lax equation of DNLS is similar to that of NLS, there are some similar properties of the Jost solutions. The monodromy matrix is defined as
Then we can get the following reduction relation and symmetry properties
2.1.2 Relation between Jost functions and the solutions to the DNLS equation
The asymptotic behaviors of the Jost solutions in the limit of can be obtained by simple derivation. Let ; Eq. (4) can be rewritten as
Then we have
In the limit , we assume , substituting it into Eq. (22), then we have
In the limit , can be expanded as series of , .
which expresses the conjugate of solution in terms of the Jost solutions as .
where . It comes from our consideration of the fact that, from the sum of two Cauchy integrals
in order to maintain that , we then have to introduce a factor in (27). At the zeros of , we have
Due to in (24) and (25), the Jost solutions do not tend to free Jost solutions in the limit of . 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 and abortive, thus a newly revised procedure to derive a suitable IST and the corresponding Z-S equation is proposed in our group.
2.1.3 The revised IST and Zakharov-Shabat equation for DNLS equation with VBC
The 2 × 1 column function can be introduced as usual
An alternative form of IST equation is proposed as
Because in the limit of , then the integral path should be chosen as shown in Figure 1, where the radius of big circle tends to infinite, while the radius of small circle tends to zero. And the factor is introduced to ensure the contribution of the integral along the big arc is vanishing. Meanwhile, our modification produces no new poles since Lax operator , as . In the reflectionless case, the revised IST equation gives
where . Similarly, an alternative form of IST equation is proposed as follows:
where a factor is introduced for the same reason as in Eq. (30). Then in the reflectionless case, we can attain
2.2 The raw expression of -soliton solution
where . We also define
where m = 1, 2,…, N. They can be rewritten in a more compact matrix form.
In the subsequent chapter, we will prove that
2.3 Explicit expression of -soliton solution
2.3.1 Verification of standard form for the -soliton solution
We only need to prove that Eq. (55) holds. Firstly, we define N × N matrices P1, P2, Q1, Q2, respectively, as
where denotes a minor, which is the determinant of a submatrix of Q1 consisting of elements belonging to not only rows (n1, n2,…, nr) but also columns (m1,m2,…, mr). Here use is made of Binet-Cauchy formula in the Appendices A.2–4 in Part 2. Then
Comparing (58) with (62) and making use of (63), we thus complete verification of Eq. (55). The soliton solution is surely of a typical form as that in NLS equation and can be expressed as formula (52).
2.3.2 Introduction of time evolution function
The time evolution factor of the scattering data can be introduced by standard procedure . Due to the fact that the second Lax operator in the limit of , it is easy to derive the time dependence of scattering date.
Then the typical soliton arguments and can be defined according to
where , and ;
2.3.3 Calculation of determinant of and
About the calculation of the most complicate determinant in (52), we introduce a N×(N + 1) matrix and a (N + 1) × N matrix defined as
with . We thus have
The above summation obviously can be decomposed into two parts: one is extended to m1 = 0, the other is extended to . Subtracted from (69), the part that is extended to , the remaining parts of (69) is just in (52) (with and ). Due to (68), we thus have
Here and especially , which completes the calculation of determinant in formula (52). Substituting the explicit expressions of , , and A into (52), we finally attain the explicit expression of -soliton solution to the DNLS equation under VBC and reflectionless case, based upon a newly revised IST technique.
An interesting conclusion is found that, besides a permitted well-known constant global phase factor, there is also an undetermined constant complex parameter bn0 before each of the typical soliton factor , (n = 1,2,…, N). It can be absorbed into by redefinition of soliton center and its initial phase factor. This kind of arbitrariness is in correspondence with the unfixed initial conditions of the DNLS equation.
2.4 The typical examples for one- and two-soliton solutions
We give two concrete examples – the one- and two-soliton solutions as illustrations of the general explicit soliton solution.
In the case of one-soliton solution, N = 1, , , and
It is different slightly from the definition in (66) for that here has been absorbed into the soliton center and initial phase. Then
The complex conjugate of one-soliton solution in (75) is , which is just in conformity with that gotten from pure Marchenko formalism  (see the next section), up to a permitted global constant phase factor. In the case of two-soliton solution, N = 2, , and
and is absorbed into the soliton center and the initial phase by
And we get
Once again we find that, up to a permitted global constant phase factor, the above two-soliton solution is equivalent to that gotten in Ref. [23, 24], verifying the validity of our formula of -soliton solution and the reliability of those linear algebra techniques. As a matter of fact, a general and strict demonstration of our revised IST for DNLS equation with VBC has been given in one paper by use of Liouville theorem .
2.5 The asymptotic behaviors of -soliton solution
The complex conjugate of expression (52) gives the explicit expression of N-soliton solution as
Without the loss of generality, for we assume and define the n’th vicinity area as .
As , vicinity areas , queue up in a descending series
and in the vicinity of , we have (note that )
Here the complex constant in expression (65) has been absorbed into by redefinition of the soliton center and the initial phase .
Introducing a typical factor ; then
In the vicinity of ,
Each , () is a one-soliton solution characterized by one parameter , moving along the positive direction of the x-axis, queuing up in a series with descending order number as in series (88). As , in the vicinity of , we have (note that )
So as , in the vicinity of ,
then as ,
That is to say, the -soliton solution can be viewed as well-separated exact one- solitons, queuing up in a series with ascending order number : In the course going from to , the ’th one-soliton overtakes the solitons from the first to ’th and is overtaken by the solitons from ’th to N’th. In the meantime, due to collisions, the ’th soliton got a total forward shift from exceeding those slower soliton from the first to ’th, and got a total backward shift from being exceeded by those faster solitons from ’th to ’th, and just equals to the summation of shifts due to each collision between two solitons, together with a total phase shift , that is,
2.6 -soliton solution to MNLS equation
Finally, we indicate that the exact -soliton solution to the DNLS equation can be converted to that of MNLS equation by a gauge-like transformation. A nonlinear Schrödinger equation including the nonlinear dispersion term expressed as
is also integrable  and called modified nonlinear Schrödinger (MNLS for brevity) equation. It is well known that MNLS equation well describes transmission of femtosecond pulses in optical fibers [4, 5, 6] and is related to DNLS equation by a gauge-like transformation  formulated as
with , ; , ; . Using a method that is analogous to reference , and applying above gauge-like transformation to Eq. (105), the MNLS equation with VBC can be transformed into DNLS equation with VBC.
with . So according to (106), the -soliton solution to MNLS equation can also be attained by a gauge-like transformation from that of DNLS equation.
The -soliton solution to the DNLS equation with VBC has been derived by means of a IST considered anew and some special linear algebra techniques. The one- and two-soliton solutions have been given as two typical examples in illustration of the general formula of the -soliton solution. It is found to be perfectly in agreement with that gotten in the following section based on a pure Marchenko formalism or Hirota’s Bilinear derivative transformation [24, 26, 27]. The demonstration of the revised IST considered anew for DNLS equation with VBC has also been given by use of Liouville theorem .
The newly revised IST technique for DNLS equation with VBC supplies substantial foundation for its direct perturbation theory.
3. A simple method to derive and solve Marchenko equation for DNLS equation
In this section, a simple method is used to derive and solve Marchenko equation (or GLM equation) for DNLS E with VBC . Firstly, starting from the first Lax equation, we derive two conditions to be satisfied by the kernel matrix of GLM by applying the Lax operator upon the integral representation of Jost function for DNLSE. Secondly, based on Lax equation, a strict demonstration has been given for the validness of Marchenko formalism. At last, the Marchenko formalism is determined by choosing a suitable and , and their relation (135) has been constructed. The one and multi-soliton solution in the reflectionless case is attained based upon a pure Marchenko formalism by avoiding direct use of inverse scattering data and verified by using direct substitution method with Mathematica.
3.1 The lax pair and its Jost functions of DNLS equation
DNLS equation is usual expressed as
with vanishing boundary, , . Here the subscript denotes partial derivative. Its Lax pair is given by
The first Lax equation is
In the case of , , , the free Jost solution is
where is a real squared parameter, expresses two independent solutions with two components. The Jost solutions of (4) are defined by their asymptotic properties at ,
3.2 The integral representation of Jost function
As usual, we introduce the integral representation,
where the superscripts d and o mean the diagonal and off-diagonal elements, respectively. According to the conventional operation in IST, the time variable is suppressed temporarily. Here
Due to the symmetry of the first Lax operator and , the kernel matrix of the integral representation of Jost function should have the same symmetry as follows:
Use is made of that , then
and the equations in the integral , where is equal to
Therefore, Eq. (125) gives two conditions to be satisfied by the kernel matrix in the integral representation of Jost solution
Since (122) is an identity, Eq. (123) or (124) gives the solution or in terms of N(x,x), thus the first Lax equation gives two conditions (126) and (127) which should be satisfied by the integral kernel N(x,y). Note that the time variable of in (124) is suppressed temporarily.
3.3 Marchenko equation for DNLSE and its demonstration
In Eq. (115), the and appear in different manner, we assume the form of Marchenko equation for DNLSE with VBC is
where is only with off-diagonal terms. is considered as another function with only off-diagonal terms. We notice that the Marchenko equation needn’t involve obviously the function of spectral parameter .
Making partial derivation in (128) with respect to x and y, respectively, we obtain
By partial integrating, Eq. (131) becomes
Use is made of the fact that = in (131). Making a weighing summation as follows:
If we choose
Thus, Eq. (134) becomes
Making partial derivation with respect to x and y, respectively, on the l.h.s. of Eq. (138), we have
Now we make a weighing summation as
Hence, we have
Noticing , Eq. (142) becomes
We find that, as long as we choose a suitable form for as well as according to Eq. (135), Eq. (128) and (129) will just satisfy the two conditions (126) and (127) derived from the first Lax Eq. (111). On the other hand, owing to the symmetry properties of and , the function in (128) and (129) can only has off-diagonal elements, we write
Considering the dependence of the Jost solutions on the squared spectral parameter , in the reflectionless case, we choose
where contains a time-dependent factor , which can be introduced by a standard procedure , due to a fact of the Lax operator as .
As is well known, Lax equations are linear equation so that a constant factor can be introduced in its solution, that is, . It means that is related to the center of soliton and expresses the initial phase up to a constant factor. Thus, the time-independent part of Cn is inessential and can be absorbed or normalized only by redefinition of the soliton center and initial phase. On the other hand, notice the terms generated by partial integral in (133)–(142), in order to ensure the convergence of the partial integral, we must let , so we only consider the zero points of in the first quadrant of complex plane of (also in the upper half part of the complex plane of ), that is, the discrete spectrum for , although in the third quadrant of the complex plane of are also the zero points of due to symmetry of Lax operator and transition matrix. Then Eq. (145) corresponds to the N-soliton solution in the reflectionless case, and we have completed the derivation and manifestation of Marchenko equation (128) and (129), (144), and (145) for DNLSE with VBC.
3.4 A multi-soliton solution of the DNLS equation based upon pure Marchenko formalism
When there are N simple poles in the first quadrant of the complex plane of , the Marchenko equation will give a N-solition solution to the DNLS equation with VBC in the reflectionless case. We can assume that
where , and
Here and hereafter the superscript T represents transposing of a matrix. On the other hand, we assume that
Both of them are N × N matrices and their matrix element are, respectively, expressed as
and we will prove that in (136)
By means of some linear algebraic techniques, especially the Binet-Cauchy formula for some special matrices (see the Appendices 2–3 in Part2), the determinant D and C can be expanded explicitly as a summation of all possible principal minors. Firstly, we can prove identity (164) by means of Binet-Cauchy formula.
The complex constant factor can be absorbed into the soliton center and initial phase by redefining
here , and
where denotes a minor, which is the determinant of a submatrix of , consisting of elements belonging to not only (n1, n2,…, nr) rows but also columns (m1, m2,…, mr).
where , , then
If we define matrices and , then we can similarly attain
Secondly, we compute the most complicate determinant in (163). In order to calculate , we introduce an matrix and an matrix
with n, m = 1, 2,…, N. We thus have
The above summation obviously can be decomposed into two parts: one is extended to m1 = 0 and the other extended to m1 ≥ 1. Subtracted from (176), the part that is extended to m1 ≥ 1, the remaining parts of (176) is just C in Eq. (163) (with , ). Due to (175), we have
which leads to
here , in (178)–(180). Finally, substituting (174) into (173), (180) into (177), and (173 and 177) into (162), we thus attain the explicit N-soliton solution to the DNLS equation with VBC under the reflectionless case, based on a pure Marchenko formalism and in no need of the concrete spectrum expression of . Obviously, the N-soliton solution permits uncertain complex constants as well as an arbitrary global constant phase factor.
3.5 The special examples for one- and two-soliton solutions
By further redefinition of its soliton center and initial phase, the single soliton solution can be further rewritten as usual standard form. It is easy to find, up to a permitted well-known constant global phase factor, the one-soliton solution to DNLS equation gotten in the pure Marchenko formalism is in perfectly agreement with that gotten from other approaches [23, 24, 26, 27].
As in the case of two-soliton solution corresponding to double simple poles, we have
Up to a permitted constant global phase factor, the two-soliton solution gotten above is actually equivalent to that gotten from both IST and Hirota’s method [23, 24, 26, 27], verifying the validity of the algebraic techniques that is used and our formula of the generalized multi-soliton solution. Because Marchenko equations (128), (129), (144), and (145) had been strictly proved, the multi-soliton solution is certainly right as long as we correctly use the algebraic techniques, especially Binet-Cauchy formula for the principal minor expansion of some special matrices.
4. Soliton solution of the DNLS equation based on Hirota’s bilinear derivative transform
Bilinear derivative operator D had been found and defined in the early 1970s by Hirota R., a Japanese mathematical scientist [30, 31, 32, 33]. Hirota’s bilinear-derivative transform (HBDT for brevity) can be used to deal with some partial differential equation and to find some special solutions, such as soliton solutions and rogue wave solutions [26, 27, 32]. In this section, we use HBDT to solve DNLS equation with VBC and search for its soliton solution. The DNLS equation with VBC, that is,
is one of the typical integrable nonlinear models, which is of a different form from the following equation:
where are usually complex functions. Solution (191) is suitable for Eq. (190) and NLS equation, and so on, but not suitable for the DNLS equation. Just due to this fact, their work cannot deal with Eq. (189) at the same time. As is well known, rightly selecting an appropriate solution form is an important and key step to apply Hirota’s bilinear derivative transform to an integrable equation like Eq. (189). Refs. [13, 16, 17, 23], etc., have proved the soliton solution of the DNLS equation must has following standard form
here and henceforth a bar over a letter represents complex conjugation.
In view of the existing experiences of dealing with the DNLS equation, in the present section, we attempt to use the solution form (192) and HBDT to solve the DNLS equation. We demonstrate our solving approach step by step, and naturally extend our conclusion to the n-soliton case in the end.
4.1 Fundamental concepts and general properties of bilinear derivative transform
For two differentiable functions of two variables and , Hirota’s bilinear derivative operator, D, is defined as
which is different from the usual derivative, for example,
where are two functions derivable for an arbitrary order, and the dot between them represents a kind of ordered product. Hirota’s bilinear derivative has many interesting properties. Some important properties to be used afterwards are listed as follows:
for example, ; ; ; ;
➂ Suppose , , , , are complex constants, then
Especially, we have as or . Some other important properties are listed in the Appendix.
4.2 Bilinear derivative transform of DNLS equation
After a suitable solution form, for example, (192) has been selected, under the Hirota’s bilinear derivative transform, a partial differential equation like (189) can be generally changed into [20, 26, 27].
where , are the polynomial functions of , ; and , , are the differentiable functions of two variables and . Using formulae in the Appendix and properties ①–③ of bilinear derivative transform numerated in the last chapter, with respect to (192), we have
We can extract the needed bilinear derivative equations from Eq. (203) as follows:
Functions , can be expanded, respectively, as series of a small parameter
4.3 Soliton solution of the DNLS equation with VBC based on HBDT
4.3.1 One-soliton solution
where the vanishing boundary condition, as , is used. Then
which immediately leads to
in Eq. (215). Then from (215), we can select . For the same reason, from (216)–(223), we can select , , …; , , …all to be zero. Thus the series (207) and (208) have been successfully cut off to have limited terms as follows:
where has been absorbed into the constant by redefiniing . In the end, we attain the one-soliton solution to the DNLS equation with VBC
which is characterized with two complex parameters and and shown in Figure 1. If we redefine the parameter as and , then
It is easy to find, up to a permitted constant global phase factor , the one-soliton solution (234) or (237) gotten in this paper is in perfect agreement with that gotten from other approaches . By further redefining its soliton center, initial phase and , the one-soliton solution can be changed into the usual typical form [16, 23, 26, 27].
On the other hand, just like in Ref. , we can rewrite and in a more appropriate or “standard” form
which makes us easily extend the solution form to the case of -soliton solution.
4.3.2 The two-soliton solution
For the two-soliton case, again from (209), we can select as
The similar procedures to that used in the one-soliton case can be used to deduce and . From (210) and (211), we can select , then from (212), we has to select . From (213) and (214), we can get the expressions of and , then attain to be
Substituting the expressions of into (216) and (217), we can select that . Then from the expressions of and (218), we can select . From (219) and (220), we can get the expressions of and , then get to be
Then from (244), (245), (246), and (221), we can select . From (222)–(223) and so on, we find that the series of (207) and (208) can be cut off by selecting , all to be zero. We thus attain the last result of to be
It can also be rewritten in a standard form as follows:
which is characterized with four complex parameters , , , and and shown in Figure 3. By redefining parameters and
we can easily transform it to a two-soliton form given in Ref. , up to a permitted constant global phase factor.
4.3.3 Extension to the N-soliton solution
Generally for the case of N-soliton solution, if we select in (209) to be
then using an induction method, we can write the -soliton solution as
therein represents a summation over under the condition .
Here, we have some discussion in order. Because what concerns us only is the soliton solutions, our soliton solution of DNLS equation with VBC is only a subset of the whole solution set. Actually in the whole process of deriving the bilinear-form equations and searching for the one and two-soliton solutions, some of the latter results are only the sufficient but not the necessary conditions of the former equations. Thereby some possible modes might have been missing. For example, the solutions of Eqs. (209)–(211) are not as unique as in (224) and (225), some other possibilities thus get lost here. This is also why we use a term “select” to determine a solution of an equation. In another word, we have selected a soliton solution. Meanwhile, we have demonstrated in Figures 2 and 3, the three-dimensional evolution of the one- and two-soliton amplitude with time and space, respectively. The elastic collision of two solitons in the two-soliton case has been demonstrated in Figure 4(a–d) too. It can be found that each soliton keeps the same form and characteristic after the collision as that before the collision. In this section, by means of introducing HBDT and employing an appropriate solution form (192), we successfully solve the derivative nonlinear Schrödinger equation with VBC. The one- and two-soliton solutions are derived and their equivalence to the existing results is manifested. The N-soliton solution has been given by an induction method. On the other hand, by using simple parameter transformations (e.g., (235) and (252)), the soliton solutions attained here can be changed into or equivalent to that gotten based on IST, up to a permitted global constant phase factor. This section impresses us so greatly for a fact that, ranked with the extensively used IST  and other methods, the HBDT is another effective and important tool to deal with a partial differential equation. It is especially suitable for some integrable nonlinear models.