## Abstract

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.

### Keywords

- soliton
- nonlinear equation
- derivative nonlinear Schrödinger equation
- inverse scattering transform
- Zakharov-Shabat equation
- Marchenko formalism
- Hirota’s bilinear derivative transform
- rogue wave

## 1. Introduction

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 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 *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 [11]. 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 *N* × *N* matrices *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 [21].

### 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

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

and

where

In the limit of

The free Jost solution is a

The Jost solutions of (4) are defined by their asymptotic behaviors as

where

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

where

It is easy to find from (2) and (9) that

and

Then we can get the following reduction relation and symmetry properties

and

#### 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

Then we have

In the limit

In the limit

and

Eq. (21) leads to

which expresses the conjugate of solution

On the other hand, the zeros of

where

in order to maintain that

Due to

#### 2.1.3 The revised IST and Zakharov-Shabat equation for DNLS equation with VBC

The 2 × 1 column function

An alternative form of IST equation is proposed as

Because in the limit of

where

where a factor

Taking the symmetry and reduction relation (18) and (28) into consideration, from (31) and (33), we can obtain the revised Zakharov-Shabat equation for DNLS equation with VBC, that is,

### 2.2 The raw expression of N -soliton solution

Substituting Eqs. (34) and (35) into formula (26), we thus attain the

where

Let *m* = 1, 2,…, *N*, respectively, in Eqs. (34) and (35), and make use of the symmetry and reduction relation (15), we can attain

where

where superscript “T” represents transposition of a matrix. Then Eqs. (39) and (40) can be rewritten as

where *m* = 1, 2,…, *N*. They can be rewritten in a more compact matrix form.

Then

where *N* × *N* identity matrix. On the other hand, from (37) and (38), we know

Substituting Eqs. (48), (49) into (50) and (51) and then substituting (50) and (51) into formula (36), we thus attain

where

In the subsequent chapter, we will prove that

It is obvious that formula (52) has the usual standard form of soliton solution. Here in formula (52), some algebra techniques have been used and can be found in Appendix A.1 in Part 2.

### 2.3 Explicit expression of N -soliton solution

#### 2.3.1 Verification of standard form for the N -soliton solution

We only need to prove that Eq. (55) holds. Firstly, we define *N* × *N* matrices *P*_{1}, *P*_{2}, *Q*_{1}, *Q*_{2}, respectively, as

Then

where *Q*_{1} consisting of elements belonging to not only rows (*n*_{1}, *n*_{2},…, *nr*) but also columns (*m*_{1},*m*_{2},…, *mr*). Here use is made of Binet-Cauchy formula in the Appendices A.2–4 in Part 2. Then

where

Similarly,

where

It is easy to find a kind of permutation symmetry existed between expressions (59) and (61), that is,

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 [21]. Due to the fact that the second Lax operator

Then the typical soliton arguments

where

#### 2.3.3 Calculation of determinant of D ¯ and A

Substituting expression (64) and (65) into formula (59) and then into (58), we have

with

About the calculation of the most complicate determinant *N*×(*N* + 1) matrix *N* + 1) × *N* matrix

with

The above summation obviously can be decomposed into two parts: one is extended to *m*_{1} = 0, the other is extended to

with

Here *A* into (52), we finally attain the explicit expression of

An interesting conclusion is found that, besides a permitted well-known constant global phase factor, there is also an undetermined constant complex parameter *b*_{n0} before each of the typical soliton factor *n* = 1,2,…, *N*). It can be absorbed into

### 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,

It is different slightly from the definition in (66) for that here

and

The complex conjugate of one-soliton solution *N* = 2,

where

and

And we get

where

and

where

Substituting (81) and (84) into formula (52), we thus get the two-soliton solution to the DNLS equation with VBC

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

### 2.5 The asymptotic behaviors of N -soliton solution

The complex conjugate of expression (52) gives the explicit expression of *N*-soliton solution as

Without the loss of generality, for *n*’th vicinity area as

As

and in the vicinity of

Here the complex constant

Introducing a typical factor

and

In the vicinity of

Here

then

Each

So as

then as

That is to say, the *N*’th. In the meantime, due to collisions, the

### 2.6 N -soliton solution to MNLS equation

Finally, we indicate that the exact

is also integrable [23] 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 [23] formulated as

with

with

The

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

Gel’fand-Levitan-Marchenko (GLM for brevity) equations can be viewed as an integral-transformed version of IST for those integrable nonlinear equations [21, 24, 28].

In this section, a simple method is used to derive and solve Marchenko equation (or GLM equation) for DNLS E with VBC [28]. Firstly, starting from the first Lax equation, we derive two conditions to be satisfied by the kernel matrix

### 3.1 The lax pair and its Jost functions of DNLS equation

DNLS equation is usual expressed as

with vanishing boundary,

The first Lax equation is

In the case of

where

### 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

Substitute Eq. (115) into the first Lax Eq. (111). By simply partial integration, we have the following terms:

and

Use is made of that

According to equation

Or

and the equations in the integral

Therefore, Eq. (125) gives two conditions to be satisfied by the kernel matrix

Since (122) is an identity, Eq. (123) or (124) gives the solution *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

### 3.3 Marchenko equation for DNLSE and its demonstration

In Eq. (115), the

where

We now show the kernel

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

We find

Since

If we choose

then

Thus, Eq. (134) becomes

Now substituting (135) into (129), we find

Making partial derivation with respect to *x* and *y*, respectively, on the l.h.s. of Eq. (138), we have

or

Now we make a weighing summation as

Hence, we have

Noticing

We find that, as long as we choose a suitable form for

Considering the dependence of the Jost solutions on the squared spectral parameter

where

As is well known, Lax equations are linear equation so that a constant factor can be introduced in its solution, that is, *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 *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 *N*-solition solution to the DNLS equation with VBC in the reflectionless case. We can assume that

where

Here and hereafter the superscript T represents transposing of a matrix. On the other hand, we assume that

Then

Substituting (146)–(149) into the Marchenko equation (128) and (129), we have

or

here

Both of them are *N* × *N* matrices and their matrix element are, respectively, expressed as

From (151) and (152), we immediately get

from (148), (156), and (157), we have

then

and

Substituting (160) and (161) into Eq. (124), we thus attain the *N*-soliton solution as follows in a pure Marchenko formalism.

where

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.

where

The complex constant factor

here

where *n*_{1}, *n*_{2},…, *nr*) rows but also columns (*m*_{1}, *m*_{2},…, *mr*).

where

If we define matrices

and

where

Using above identity, comparing (169), (172), (173), and (174), we find that identity (164) holds and complete the computation of *D.*

Secondly, we compute the most complicate determinant

with *n*, *m* = 1, 2,…, *N*. We thus have

The above summation obviously can be decomposed into two parts: one is extended to *m*_{1} = 0 and the other extended to *m*_{1} ≥ 1. Subtracted from (176), the part that is extended to *m*_{1} ≥ 1, the remaining parts of (176) is just *C* in Eq. (163) (with

which leads to

here *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 *N*-soliton solution permits uncertain complex constants

### 3.5 The special examples for one- and two-soliton solutions

In the case of one simple pole and one-soliton solution as

From (167) and (168), we have (suppose

Then from (181) and (182), we attain the one-soliton solution

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

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:

which had been solved in Ref. [14] by using HBDT. We have paid special attention to the following solution form in it [14]:

where

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 *D*, is defined as

which is different from the usual derivative, for example,

where

for example,

➂ Suppose

Especially, we have

### 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

Substituting the above expressions (199)–(202) into Eq. (189), the latter can be reduced to [26, 27].

We can extract the needed bilinear derivative equations from Eq. (203) as follows:

Functions

Substituting (207) and (208) into (204)–(206) and equating the sum of the terms with the same orders of

The above equations, (209)–(223), contain the whole information needed to search for a soliton solution of the DNLS equation with VBC.

### 4.3 Soliton solution of the DNLS equation with VBC based on HBDT

#### 4.3.1 One-soliton solution

For the one-soliton case, due to (209)–(211) and considering the transform property ③, we can select

From (212), one can select

where the vanishing boundary condition,

Substituting (226) and (227) into Eq. (213), we can attain

From (226) and (228), we can get an expression of

Due to (224) and (229), we can also easily verify that

which immediately leads to

in Eq. (215). Then from (215), we can select

where

which is characterized with two complex parameters

Then

It is easy to find, up to a permitted constant global phase factor

On the other hand, just like in Ref. [13], we can rewrite

Here

which makes us easily extend the solution form to the case of

#### 4.3.2 The two-soliton solution

For the two-soliton case, again from (209), we can select

The similar procedures to that used in the one-soliton case can be used to deduce

Substituting (242) and (243) into (215), one can attain

Substituting the expressions of

Due to (243) and (244), we can also easily verify that

Then from (244), (245), (246), and (221), we can select

It can also be rewritten in a standard form as follows:

where

which is characterized with four complex parameters

we can easily transform it to a two-soliton form given in Ref. [23], 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

then using an induction method, we can write the

where

therein

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 [23] 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.