Asymptotic behaviors of

## Abstract

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.

### Keywords

- soliton
- nonlinear equation
- derivative nonlinear Schrödinger equation
- inverse scattering transform
- Zakharov-Shabat equation
- Hirota’s bilinear derivative method
- DNLS equation
- space periodic solution
- rogue wave

## 1. 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

A newly revised IST is thus proposed in this section to avoid the dual difficulty and the excessive complexity. An additional affine factor *N* simple poles off the circle of radius *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

### 1.1 The fundamental concepts for the IST theory of DNLS equation

Under a Galileo transformation

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

where

The Jost functions satisfy first Lax equation

here Jost functions

The free Jost function

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 to

The above symmetry relations further result in

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 *n*’ th subset of zero points is

And we arrange the

According to the standard procedure [21, 22], the discrete part of

At the zeros of

Using symmetry relation in (14), (17), (21)–(24), (27), we can prove that

## 2. Relation between the solution and Jost functions of DNLS^{+} equation

The asymptotic behaviors of the Jost solutions in the limit of

then we have

We assume a function

Substituting (31)–(32) into Eq. (30), we have

In the limit of

where

Inserting formula (34) and (35) in Eq. (33) at

Then we find a useful formula

which expresses the conjugate of the solution

### 2.1 Introduction of time evolution factor

In order to make the Jost functions satisfy the second Lax equation, a time evolution factor

we let

then

Due to

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

### 2.2 Zakharov-Shabat equations and breather-type N -soliton solution

A 2 × 1 column function

here and hereafter note “

where

is called the reflection coefficient. Due to

or

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

In the reflectionless case, the Zakharov-Shabat equations for DNLS^{+} equation can be derived immediately from (48) as follows

here

Letting

Then Eq. (58) can be rewritten as

or in a more compact form

The above equation gives

Note that the choice of poles,

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

with

or in a more compact form

The above equation givess

Note that the choice of poles,

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) into (54), we attain the ^{+}Eq. (3) under NVBC and reflectionless case (note that the time dependence of soliton solution naturally emerges in

here

The solution has a standard form as (80), that is

which can be proved by direct calculation for the

## 3. Verification of standard form and the explicit breather-type multi-soliton solution

### 3.1 Verification of det I + B ′ = det I + B

In order to prove the first identity in (82), we firstly calculate

here

in (84),

and further define that

then the typical factor is

and

here

where use is made of formula (60) and (62), the time dependence of the solution naturally emerged in

Secondly, let us calculate

where

Thirdly, let us calculate

According to (79) and Binet-Cauchy formula (Appendix (A.2)), similarly we have

Due to (95), comparing (83) and (93) results in the expected identity and completes the verification of the first identity in (82).

### 3.2 Verification of det I + D ′ ¯ = det I + B

Our most difficult and challenging task is to prove the second identity in (82). For convenience of discussion, we define that

then

or

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

We can get an important relation between

On the other hand, an unobvious symmetry between matrices

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

### 3.3 The explicit N -soliton solution to the DNLS^{+} equation with NVBC

In order to get an explicit **-**soliton solution to the DNLS^{+}Eq. (1) with NVBC, firstly we need to make an inverse Galileo transformation of (2) by

where in (114) the

Secondly, we need to calculate determinant

where

^{+}Eq. (1) with NVBC under reflectionless case, based upon a revised and improved inverse scattering transform. Due to the limitation of space, the asymptotic behaviors of the

## 4. The one and two-soliton solutions to DNLS^{+} equation with NVBC

We give two concrete examples – the one and two breather-type soliton solutions in illustration of the general explicit

In the case of one-soliton solution,

we have

where not as that in (114), we define

It is different slightly from the definition in Eq. (114) for the reason that an additional minus sign “^{+}Eq. (1) with NVBC.

which is generally called a breather solution and shown as Figure 2.

Formula (130) includes the one-soliton solution of the DNLS equation with VBC as its limit case. In the limit of

Substituting (131) and (132) into (130), we can attain

If we redefine

The degenerate case for

with

where

In the case of breather-type two-soliton solution,

which is just the same as that defined in (108)–(115), the pole

Similarly we can attain

Substituting (141)–(142) into (140) completes the calculation of breather-type two-soliton solution. The evolution of breather-type two-soliton solution with respect to time and space is given in Figure 3. It clearly display the whole process of the elastic collision between two breather solitons, and in the limit of infinite time

### 4.1 Explicit pure *N*-soliton solution to the DNLS^{+} equation with NVBC

When all the simple poles are on the circle ^{+} equation with NVBC will give a typical pure *N*-soliton solution. The discrete part of

At the zeros of

On the other hand, the zeros of

Here

Different from that in breather-type case, we define

An inverse Galileo transformation

Due to

where

where *n*. The constant and positive real number *n*’ th soliton center

Substituting (149)–(157) into (158)–(161), and substituting (158)–(161) into the following formula, we attain the explicit pure *N*-soliton solution

The *N*-soliton formula. According to (158)–(162), it can be calculated as follows

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

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 *En*, (*n*’th soliton center

On the other hand, letting only part of the poles converge in pairs on the circle in Figure 1 and rewriting the expression of

### 4.2 The asymptotic behaviors of the *N*-soliton solution

Without loss of generality, we assume *n*’th neighboring area as

where

In the neighboring area of

With

As

the *n*.

As

where

That is, the *n* such as

In the process of going from

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

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

## 5. 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 solutions – space 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 NVBC – periodic 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

(where all the parameters are defined in Ref. [38]). Similarly, setting a seed 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

where, at the left side of the above formula, a dot

Here,

### 5.1 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

where

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

### 5.2 Solution of bilinear equations

#### 5.2.1 First order space periodic solution and rogue wave solution

Let us assume that the series expansion of the complex functions

Substituting

with

where

Notice that

According to the test rule for a one-variable quadratic, there is a threshold condition under which

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

If we set

And

where

and due to

Hereto, we have completed the computation of the 1st-order space periodic solution, the space-time evolution of its module is depicted in Figure 6. In what follows, we will take the long-wave limit, that is, p → 0, to construct a rogue wave solution. Supposing

where

Consequently, the rogue wave solution can be derived according to Eq. (198):

where

Here

As we discussed in the Introduction section, there is a gauge transformation between KN Eq. (183) and CLL Eq. (184). Thus, it is instructive to use the integral transformation Eq. (185) to construct a solution of Eq. (184). Substituting the solution (198) into (185), further algebra computation will lead to a space periodic solution of the CLL equation:

where,

which has the same parameters as

#### 5.2.2 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

Similarly, substituting

with

where

Of course, for a valid and complete calculation, we are faced with the same situation as the 1st-order breather:

And the threshold conditions for each complex-valued

The space-time evolution of the module of the 2nd order space periodic solution (223) is shown in Figure 8. Paying attention to the form of this breather and the previous one, we will notice that this breather can exactly degenerate into the 1st-order breather if we take

where

where the coefficient *M* is defined by:

On the other hand, this breather possesses the same feature as the former one that it is periodic with respect to variable *x* due to the pure imaginary numbers

Thus, we will have four combinations of

Choice of | ||||
---|---|---|---|---|

From Table 1, we could draw the conclusion that this breather will also degenerate into the background plane wave as

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

## 6. 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 nearly-integrable 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.

Some useful formulae.

**A1**, If

**A2**, Binet-Cauchy formula: For a squared

*r*’th-order principal minor of

**A3**, For a *N* × *N* matrix *N* × *N* matrix

where *Q*_{1} consisting of elements belonging to not only rows (

The above formula also holds for the case of

**A4**, For a squared matrix

**A5,** Some useful blinear derivative formulae.

where