Open access peer-reviewed chapter

Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1

By Zhou Guo-Quan

Submitted: February 11th 2020Reviewed: July 21st 2020Published: September 5th 2020

DOI: 10.5772/intechopen.93438

Downloaded: 112

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 λ2in the first operator of its Lax pair, while the famous NLS equation, one typical example in AKNS hierarchy, has a spectral parameter of λ. 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 Nsoliton solution by means of Darboux transformation [15]. Steudel has derived a formula for Nsoliton solution in terms of Vandermonde-like determinants by means of Bäcklund transformation [13]; but just as Chen points out in Ref. [16], 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 λwith λ, 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 Exλ, so that it tends to zero as λ, 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 [11]. We thus consider proposing a new revised IST to avoid the excessive complexity. Our N-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 detI+Q1Q2, for two N × N matrices Q1and Q2, where Iis 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 [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 uxtis usually expressed as

iut+uxx+iu2ux=0E1

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

L=iλ2σ3+λU,U=0uu¯0E2

and

M=i2λ4σ3+2λ3Uiλ2U2σ3λU3+iUxσ3E3

where λis a spectral parameter, and σ3is the third one of Pauli matrices σ1, σ2, σ3, and a bar over a letter, (e.g., u¯in (2)), represents complex conjugate. The first Lax equation is

xfxλ=LxλfxλE4

In the limit of x, u0, and

LL0=iλ2σ3; MM0=i2λ4σ3E5

The free Jost solution is a 2×2matrix.

Exλ=eiλ2xσ3;E1xλ=10eiλ2x,E2xλ=01eiλ2xE6

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

Ψxλ=ψ˜xλψxλExλ,asxE7
Φxλ=ϕxλϕ˜xλExλ,asxE8

where ψxλ=ψ1xλψ2xλT, ψ˜xλ=ψ˜1xλψ˜2xλT, etc., and superscript “T” 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 Tλis defined as

Φxλ=ΨxλTλ,E9

where

Tλ=atλb˜tλbtλa˜tλE10

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

σ2Lλ¯¯σ2=Lλ,σ2Tλ¯¯σ2=TλE11
σ2Ψxλ¯¯σ2=Ψxλ,σ2Φxλ¯¯σ2=ΦxλE12

and

σ3Ψxλσ3=Ψxλ,σ3Φxλσ3=ΦxλE13
σ3Lλσ3=Lλ,σ3Tλσ3=TλE14

Then we can get the following reduction relation and symmetry properties

iσ2ψxλ¯¯=ψ˜xλE15
iσ2φ¯xλ¯=φ˜xλE16
a˜¯λ¯=aλ;b˜¯λ¯=bλE17

and

ψxλ=σ3ψxλE18
ψ˜xλ=σ3ψ˜xλE19
aλ=aλ; bλ=bλa˜λ=a˜λ; b˜λ=b˜λE20

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 λcan be obtained by simple derivation. Let υ=υ1υ2Tψ˜xλ; Eq. (4) can be rewritten as

υ1x+iλ2υ1=λuυ2,υ2xiλ2υ2=λu¯υ1E21

Then we have

υ1xxuxυ1x+iλ2υ1/u+λ4υ1+λ2u2υ1=0E22

In the limit λ, we assume ψ˜1xλ=eiλ2x+g, substituting it into Eq. (22), then we have

iλ2+gx2+gxxuxgx/u+λ4+λ2u2=0E23

In the limit λ, gxcan be expanded as series of λ2j, j=1,2,.

igxμ=μ0+μ22λ21+E24

and

μ0=u2/2,μ2=iu¯xu/2u4/4,E25

Eq. (21) leads to gxυ1=λuυ2. Considering (25), in the limit of λ, we find a useful formula

u¯=i2limλλψ˜2xλ/ψ˜1xλE26

which expresses the conjugate of solution uin terms of the Jost solutions as λ.

On the other hand, the zeros of aλappear in pairs and can be designed by λn, n=1,2,,Nin the I quadrant, and λn+N=λnin the III quadrant. The discrete part of aλis [21, 22, 23].

aλ=n=1Nλ2λn2λ2λn2¯λn2¯λn2E27

where a0=1. It comes from our consideration of the fact that, from the sum of two Cauchy integrals

lnaλλ+0=12πiΓdλlnaλa˜λλλλ,Γ=0ii00ii0,

in order to maintain that lnaλ0,asλ0,andlnaλis finiteasλ, we then have to introduce a factor λ¯n2/λn2in (27). At the zeros of aλ, we have

ϕxλn=bnψxλn,ȧλn=ȧλn,bn+N=bnE28

Due to μ00in (24) and (25), the Jost solutions do not tend to free Jost solutions Exλin the limit of λ. 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 Θxλcan be introduced as usual

Θxλ=ϕxλ/aλ,asλinI,IIIquadrants.ψ˜xλ,asλinII,IVquadrants.E29

An alternative form of IST equation is proposed as

1λ2Θ1xλE11(xλ)eiλ2x=12πiΓdλ1λλ1λ2Θ1xλE11(xλ)eiλ2xE30

Because in the limit of λ, limλeiλ2x=0,asx>0,Imλ2>0,λin theIIIIquadrants,x<0,Imλ2<0,λin theIIIVquadrants,then the integral path Γshould 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 λ2is introduced to ensure the contribution of the integral along the big arc is vanishing. Meanwhile, our modification produces no new poles since Lax operator L0, as λ0. In the reflectionless case, the revised IST equation gives

Figure 1.

The integral path for IST of the DNLS.

ψ˜1xλ=eiλ2x+n=12N1λn2λ2λλnbnȧλnψ1xλneiλn2xeiλ2xE31

where ȧλn=daλ/dλλ=λn. Similarly, an alternative form of IST equation is proposed as follows:

1λΘ2xλeiλ2x=12πiΓdλ1λλ1λΘ2xλeiλ2xE32

where a factor λ1is introduced for the same reason as λ2in Eq. (30). Then in the reflectionless case, we can attain

ψ˜2xλ=n=12N1λnλλλnbnȧλnψ2xλneiλn2xeiλ2xE33

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,

ψ˜1xλ=eiλ2x+n=1N2λ2λnλ2λn2bnȧλnψ1x1λneiλn2xeiλ2xE34
ψ˜2xλ=n=1N2λλ2λn2bnȧλnψ2xsλneiλn2xeiλ2xE35

2.2 The raw expression of N-soliton solution

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

u¯N=i2UN/VNE36

where

UN=n=1N2bnȧλnψ2xλneiλn2xE37
VN=1+n=1N2bnλnȧλnψ1xλneiλn2xE38

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

ψ2¯xλm=eiλm2¯x+n=1N2λm2¯λnλm2¯λn2cnψ1xλneiλn2xeiλm2¯xE39
ψ¯1xλm=n=1N2λm2¯λm2¯λn2cnψ2xλneiλn2xeiλm2¯x; m=1,2,,N.E40

where cn=bn/ȧλn. We also define

fn=2cneiλn2x,wjn=2cnψjλnj=1,2;andn=1,2,N.E41
B1mn=fm¯λm2¯λm2¯λn2λnfn,B2mn=f¯mλm¯λ¯m2λn2fn;m,n=1,2,,NE42
W1=w11w12w1NT,W2=w21w22w2NT,F=f1f2fNT,G=f1/λ1f2/λ2fN/λNTE43

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

w2¯m=f¯m+n=1NB1mnw1nE44
w1¯m=n=1NB2mnw2nE45

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

W¯2=F¯+B1·W1E46
W¯1=B2·W2E47

Then

W2=I+B¯1B21FE48
W1=B¯2I+B1B¯21F¯E49

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

UN=n=1Nfnw2n=FTW2E50
VN=1+n=1Nfn/λnw1n=1+GTW1E51

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

u¯N=i2FTW21+GTW1=i2FTI+B¯1B21F1GTB¯2I+B1B¯21F¯=i2detI+B¯1B2+FFTdetI+B¯1B2detI+B1F¯GTB¯2detI+B1B¯2detI+B¯1B22iA·DD¯2E52

where

AdetI+B¯1B2+FFTdetI+B¯1B2E53
DdetI+B1B¯2E54

In the subsequent chapter, we will prove that

detI+B1F¯GTB¯2=detI+B¯1B2E55

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 P1, P2, Q1, Q2, respectively, as

P1nmB1F¯GTnm=f¯nλmλn2¯λm2fm;P2mnB¯2mn=fmλmλm2λn2¯f¯nE56
Q1nmB¯1nm=fnλn2λn2λm2¯f¯mλ¯m;Q2mnB2mn=f¯mλ¯mλm2¯λn2fnE57

Then

D¯=det(I+Q1Q2)=1+r=1N1n1<n2<<nrND¯r(n1,n2,nr)=1+r=1N1n1<<nrN1m1<<mrNQ1(n1,n2,nr;m1,m2,,mr)Q2(m1,m2,mr;n1,n2,nr)E58

where Q1n1n2nrm1m2mrdenotes 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

Q1n1n2nrm1m2mrQ2m1m2mrn1n2nr=n,mfnf¯mλn2λ¯m2λn2λ¯mn<n',m<m'λn2λn'2λm'2λm2m,nf¯mfnλ¯m2λn2λ¯mn<n',m<m'λ¯m2λm'2λn'2λn2
=1rm,nλn2fn2fm2¯λn2λm2¯2n<n',m<m'λn2λn'22λm2¯λm'2¯2E59

where

n,nn1n2nr,m,mm1m2mrE60

Similarly,

P1(n1,n2,,nr;m1,m2,,mr)P2(m1,m2,,mr;n1,n2,,nr)=(1)rn,mf¯n2fm2λm2(λ¯n2λm2)2n<n',m<m'(λm2λm'2)2(λn2¯λn'2¯)2E61

where

n,nn1n2nr;m,mm1m2mr,anddetI+B1F¯GTB¯2=detI+P1P2=1+r=1N1n1<<nrN1m1<<mrNP1n1nrm1mrP2m1mrn1nrE62

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

P1n1nrm1mrP2m1mrn1nr=Q1m1mrn1nrQ2n1nrm1mrE63

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 Mi2λ4σ3in the limit of x, it is easy to derive the time dependence of scattering date.

dλn/dt=0,daλn/dt=0;cnt=cn0ei4λn4t,cn0=bn0/ȧλn,bnt=bn0ei4λn4tE64

Then the typical soliton arguments θnand φncan be defined according to

fn2=2cn0ei2λn2xei4λn4t2cn0eθneiφnE65

where λμn+iνn, and θn=4μnνnx+4μn2νn2t=4κnxVnt; φn=2μn2νn2x+4μn2νn2216μn2νn2·t

Vn=4μn2νn2,κn=4μnνnE66

2.3.3 Calculation of determinant of D¯and A

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

Q1n1n2nrm1m2mrQ2m1m2mrn1n2nr=1rn,m2cn2c¯meθneiφneθmeiφmλn2λn2λm2¯2n<n',m<m'λn2λn'22λm2¯λm'2¯2E67

with n,nn1n2nrand m,mm1m2mr. Where use is made of Binet-Cauchy formula which is numerated in Appendix A. 3–4 in Part 2. Substituting expression (67) into formula (58) thus complete the calculation of determinant D¯.

About the calculation of the most complicate determinant Ain (52), we introduce a N×(N + 1) matrix Ω1and a (N + 1) × N matrix Ω2defined as

Ω1nm=B¯1nm=Q1nm,Ω1n0=fn,Ω2mn=B2mn=Q2mn,Ω20n=fnE68

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

det(I+B¯1B2+FFT)=det(I+Ω1Ω2)=1+r=1N1n1<<nrN0m1<<mrNΩ1(n1,n2,,nr;m1,m2,,mr)Ω2(m1,m2,,mr;n1,n2,,nr)E69

The above summation obviously can be decomposed into two parts: one is extended to m1 = 0, the other is extended to m11. Subtracted from (69), the part that is extended to m11, the remaining parts of (69) is just Ain (52) (with m1=0and m21). Due to (68), we thus have

A=det(I+Ω1Ω2)det(I+Q1Q2)=r=1N1n1<<nrN1m2<<mrNAr(n1,n2,,nr;0,m2,,mr)=r=1N1n1<n2<nrNN1m2<m3<<mrNΩ1(n1,n2,,nr;0,m2,,mr)Ω2(0,m2,,mr;n1,n2,,nr)E70

with

Ω1n1n2nr0m2mrΩ20m2mrn1n2nr=1r1n,mfn2fm2¯λm2¯λn2λm2¯2n<n',m<m'λn2λn'22λm2¯λm'2¯2=1r1n,m2cn2cm¯eθneiφneθmeiφmλm2¯λn2λm2¯2n<n',m<m'λn2λn'22λm2¯λm'2¯2E71

Here n,nn1n2nrand especially m,mm2mr, which completes the calculation of determinant Ain formula (52). Substituting the explicit expressions of D, D¯, and A into (52), we finally attain the explicit expression of N-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 eθneiφn, (n = 1,2,…, N). It can be absorbed into eθneiφnby 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, λ2=λ1, λ1=ρ1eiβ1=μ1+iν1, and

A1=Ω1n1=1m1=0Ω2m1=0n1=1=f12E72
D¯1=Q1n1=1m1=1Q2m1=1n2=1=1f14λ12/λ12λ12¯2E73
f12=2c10ei2λ12xei4λ14t;c10=b10/ȧλ1;b10=e4μ1ν1x10eiα10;b10ei2λ12xei4λ14teθ1eiφ1;θ1=4μ1ν1xx10+4μ12ν12t;φ1=2μ12ν12x+4μ12ν12216μ1ν12t+α10E74

It is different slightly from the definition in (66) for that here b10has been absorbed into the soliton center and initial phase. Then

A1=λ1λ12λ12¯eθ1eiφ1/λ¯12=i2ρ1sin2β1ei3β1eθ1eiφ1
D¯1=1λ12λ12¯2λ12λ12λ12λ12¯2e2θ1=1+ei2β1e2θ1

and

u¯1xt=i2A1D1/D¯12=4ρ1sin2β1ei3β11+ei2β1e2θ11+ei2β1e2θ12eθ1eiφ1E75

The complex conjugate of one-soliton solution u¯1xtin (75) is u1xt, which is just in conformity with that gotten from pure Marchenko formalism [24] (see the next section), up to a permitted global constant phase factor. In the case of two-soliton solution, N = 2, λ3=λ1, λ4=λ2and

λ1=ρ1eiβ1=μ1+iν1;λ2=ρ2eiβ2=μ2+iν2E76
c10=b10ȧλ1=b10λ12λ12¯2λ1λ12λ22¯λ12λ22λ12λ12¯λ22λ22¯c20=b20ȧλ2=b20λ22λ2¯2λ2λ22λ12¯λ22λ12λ12λ12¯λ22λ22¯E77
fj2=2cj0ei2λj2xei4λj4t,j=1,2C.f.1.62,bj0ei2λj2x+i4λj4teθjeiφj,j=1,2E78

where θj=4μjνjxxj0+4μj2νj2t

φj=2μj2νj2x+4μj2νj2216μj2νj2t+αj0E79

and bj0is absorbed into the soliton center and the initial phase by

bj0=e4μjνjxj0eiαj0;j=1,2E80

And we get

A2=n1=1,2m1=0Ω1n10Ω20n1+n1=1,n2=2m1=0,m2=1,2Ω1n1n20m2Ω20m2n1n2=Ω110Ω201+Ω120Ω202+Ω11201Ω20112+Ω11202Ω20212=f12+f22f14f22λ12λ222λ12¯λ12λ12¯2λ12¯λ222f24f12λ12λ222λ22¯λ22¯λ122λ22λ22¯2=λ11ei4β1λ12λ22¯λ12λ22ei4β1+β2eθ1eiφ1+λ21ei4β2λ12¯λ22λ12λ22ei4β1+β2eθ2eiφ2+λ11ei4β1ei2β2λ12¯λ22λ12¯λ22¯eθ2iφ2+λ21ei4β2ei2β1λ12λ22¯λ12¯λ22¯eθ1iφ1eθ1+θ2eiφ1+φ2ei4β1+β2=i2λ12λ22¯λ12λ22[ρ1sin2β1eiφαei3β1+4β2eθ1+iφ1+ρ2sin2β2eiφ+αei4β1+3β2eθ2+iϕ2
+ρ1sin2β1eiφαei3β1+2β2e2θ2θ1eiϕ1+ρ2sin2β2eiφ+αei2β1+3β2e2θ1θ2eiϕ2]E81

where

φ=argλ12λ22¯=arctanρ12sin2β1+ρ22sin2β2/ρ12cos2β1ρ22cos2β2α=argλ12λ22=arctanρ12sin2β1ρ22sin2β2/ρ12cos2β1ρ22cos2β2E82

and

D¯2=1+r=121n1<n221m1<m22Q1n1nrm1mrQ2m1mrn1nr=1+Q1n1=1m1=1Q2m1=1n1=1+Q1n1=1m1=2Q2m1=2n1=1+Q1n1=2m1=1Q2m1=1n1=2+Q1n1=2m1=2Q2m1=2n1=2+Q1n1=1n1=2m1=1m2=2Q2m1=1m2=2n1=1n2=2=1f14λ12λ12λ12¯2f24λ22λ22λ22¯2f12f22¯λ12λ12λ22¯2f12¯f22λ22λ12¯λ222+f1f24λ12λ22λ12¯λ22¯2λ12λ222λ12λ12¯2λ12λ22¯2λ22λ12¯2λ22λ22¯2E83
D2=1+λ12λ22¯λ12λ222ei2β1e2θ1+ei2β2e2θ2+1λ12λ22¯λ12λ222eθ1+θ2eiβ1+β2ρ1ρ2eiφ2φ1+ρ2ρ1eiφ1φ2+ei2β1+β2e2θ1+θ2E84

where

λ12λ22¯λ12λ222=ρ1/ρ2ρ2/ρ12+4sin2β1+β2ρ1/ρ2ρ2/ρ12+4sin2β1β2E85

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

u¯2=i2A2D2/D¯22E86

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 N-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 [25].

2.5 The asymptotic behaviors of N-soliton solution

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

uN=i2A¯ND¯N/DN2E87

Without the loss of generality, for λn=μn+ivn,Vn=4μn2vn2,n=1,2,,N,we assume V1<V2<<Vn<VNand define the n’th vicinity area as Γn:xxnoVnt0,n=1.2N.

As t, Nvicinity areas Γn,n=1,2,N, queue up in a descending series

ΓN,ΓN1,,Γ1E88

and in the vicinity of Γn, we have (note that κj>0)

θj=4κjxxj0Vjt,forj<n+,forj>nE89

Here the complex constant 2cn0in expression (65) has been absorbed into eθneiφnby redefinition of the soliton center xn0and the initial phase αn0.

Introducing a typical factor Fn=e2θn/λn2λn2¯2>0, n=1,2,,N; then

Dn12n=j=1nλj2¯Fjl<mλl2λm2λl2λ¯m24E90

wherel,m12n. Thus

DDn112n1+Dn12n=1+λn2¯Fnj=1n1λj2λn2λj2λ¯n24Dn112n1E91

and

A¯A¯n12n012n1=Dn1eθneiφnj=1n1λj2¯λn2¯2λj2λn2¯2ei4βjE92

In the vicinity of Γn,

uxt=i2A¯D¯/D2u1θn+Δθnφn+ΔφnE93

Here

Δθn=2j=1n1lnλj2λn2¯λj2λn22E94
Δφn=j=1n1argλj2¯λn2¯2λj2λn2¯2+4βj=2j=1n1argλj2λn2¯argλj2¯λn2¯2βjE95

then

uNn=1Nu1θn+Δθnφn+ΔφnE96

Each u1θnφn, (1,2,,n) is a one-soliton solution characterized by one parameter λn, moving along the positive direction of the x-axis, queuing up in a series with descending order number nas in series (88). As t, in the vicinity of Γn, we have (note that κj>0)

θj=4κjxxj0Vjt+,forj<n,forj>nDDNnn+1n+2N+DNn+1nn+1N
=1+λn2¯Fnj=n+1Nλj2λn2λj2λ¯n24DNnn+1n+2NE97
A¯A¯Nn+1(n,n+1,,N;0,n+1,n+2,,N)=DNn(n+1,n+2,,N)eθneiφnj=n+1N(λj2¯λn2¯)2(λj2λn2¯)2λj2λ¯j2,E98

So as t, in the vicinity of Γn,

u=i2A¯D¯/D2u1θn+Δθn+φn+Δφn+E99
Δθn+=2j=n+1Nlnλj2λn2¯λj2λn22E100
Δφn+=j=n+1Nargλj2¯λn2¯2λj2λn2¯2+4βj=2j=n+1Nargλj2λn2¯argλj2¯λn2¯2βj,E101

then as t,

uNn=1Nu1θn+Δθn+φn+Δφn+E102

That is to say, the N-soliton solution can be viewed as Nwell-separated exact one- solitons, queuing up in a series with ascending order number n: Γ1,Γ2,,ΓN.In the course going from tto t, the n’th one-soliton overtakes the solitons from the first to n1’th and is overtaken by the solitons from n+1’th to N’th. In the meantime, due to collisions, the n’th soliton got a total forward shift Δθn/κnfrom exceeding those slower soliton from the first to n1’th, and got a total backward shift Δθn+/κnfrom being exceeded by those faster solitons from n+1’th to N’th, and just equals to the summation of shifts due to each collision between two solitons, together with a total phase shift Δφn, that is,

Δxn=Δθn+Δθn/κnE103
Δφn=Δφn+ΔφnE104

2.6 N-soliton solution to MNLS equation

Finally, we indicate that the exact N-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

itυ+xxυ+xυ2υ+2βυ2υ=0E105

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

υxt=uXTei2ρX+i4ρ2TE106

with x=α1X+4ρT, t=α2T; X=αx4βt, T=α2t; ρ=βα2. Using a method that is analogous to reference [16], and applying above gauge-like transformation to Eq. (105), the MNLS equation with VBC can be transformed into DNLS equation with VBC.

iTu+XXu+iXu2u=0E107

with u=uXT. So according to (106), the N-soliton solution to MNLS equation can also be attained by a gauge-like transformation from that of DNLS equation.

The N-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 N-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 [25].

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 Nxyof GLM by applying the Lax operator xLupon 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 Fx+yand Gx+y, 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

iut+uxx+iu2ux=0E108

with vanishing boundary, x, u0. Here the subscript denotes partial derivative. Its Lax pair is given by

L=iλ2σ3+λU,U=0uu¯0E109
M=i2λ4σ3+2λ3Uiλ2U2σ3λU3+iUxσ3E110

The first Lax equation is

xfxλ=LxλfxλE111

In the case of x, u0, LL0=iλ2σ3, the free Jost solution is

Exλ=eiλ2xσ3,E1xλ=10eiλ2x;E2xλ=01eiλ2xE112

where λ2is a real squared parameter, Exλexpresses two independent solutions with two components. The Jost solutions of (4) are defined by their asymptotic properties at x±,

Ψxλ=ψ˜xλψxλExλ,asxE113
Φxλ=φxλφ˜xλExλ,asxE114

3.2 The integral representation of Jost function

As usual, we introduce the integral representation,

Ψxλ=Exλ+xdyλ2Ndxy+λNo(xy)EyλE115

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

Ndxy=Nxy1100Nxy22, Noxy=0Nxy12Nxy210

Due to the symmetry of the first Lax operator λ2iσ311=λ2iσ3¯22and λU21=λU¯12, the kernel matrix Nxyof the integral representation of Jost function should have the same symmetry as follows:

λ2Ndxy11=λ2Ndxy22¯;λNoxy21=λNoxy12¯E116

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

xLExλ=LL0Exλ=λUxExλE117
xxdyλ2Ndxy+λNo(xy)Eyλ=λ2Ndxx+λNo(xx)Exλ+xdyλ2Nxdxy+λNxo(xy)EyλE118
iλ2σ3xdyλ2Ndxy+λN(xy)Eyλ=xdyσ3λ2Ndxy+λNo(xy)σ3Eyλ=σ3λ2Ndxx+λNo(xx)σ3Exλ+xdyσ3λ2Nydxy+λNyo(xy)σ3EyλE119

and

λUxxdyλ2NdxyEyλ=xdyλUxNdxyiσ3Eyλ=iλUxNdxxσ3Exλ+ixdyλUxNydxyσ3EyλE120

Use is made of that iλ2σ3Eyλ=Eyλ, then

λUxxdyNoxyxdyλUxNoxyE121

According to equation xLΨxλ=0, adding up the l.h.s. and r.h.s., respectively, of Eq. (117)(120), (121). We obtain two equations involving with terms λ2and λoutside of the integral dyas follows:

λ2:Ndxx+σ3Ndxxσ3=0E122
λ1:UxNoxx+σ3Noxxσ3+iUxNdxxσ3=0E123

Or

U12=ux=2N12xx1+iN¯11xx,E124

and the equations in the integral dyS=0, where Sis equal to

[λ2Nxd(x,y)+λNxo(x,y)]+σ3[λ2Nyd(x,y)+λNyo(x,y)]σ3λU(x)[iNyd(x,y)σ3+λNo(x,y)]=0E125

Therefore, Eq. (125) gives two conditions to be satisfied by the kernel matrix Nxyin the integral representation of Jost solution

λ2terms:AxyNxdxy+σ3Nydxyσ3UxNoxy=0E126
λ1terms:BxyNxoxy+σ3Nyoxyσ3+iUxNydxyσ3=0E127

Since (122) is an identity, Eq. (123) or (124) gives the solution Uxor uxin 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 uxin (124) is suppressed temporarily.

3.3 Marchenko equation for DNLSE and its demonstration

In Eq. (115), the Ndxyand Noxyappear in different manner, we assume the form of Marchenko equation for DNLSE with VBC is

Ndxy+xdzNoxzFz+y=0E128
Noxy+Fx+y+xdzNdxzGz+y=0E129

where Fx+yis only with off-diagonal terms. Gx+yis considered as another function with only off-diagonal terms. We notice that the Marchenko equation needn’t involve obviously the function of spectral parameter λ.

We now show the kernel Nxydetermined by (128) and (129) indeed satisfy the conditions (126) and (127) as long as we choose a suitable form of expression for Gx+y.

Making partial derivation in (128) with respect to x and y, respectively, we obtain

NxdxyNoxxFx+y+xdzNxoxzFz+y=0E130
Nydxy+xdzNoxzFz+y=0E131

By partial integrating, Eq. (131) becomes

NydxyNoxxFx+yxdzNzoxzFz+y=0E132

Use is made of the fact that Fz+y= Fyz+y=Fzz+yin (131). Making a weighing summation as follows:

l.h.s.of20+σ3l.h.s.of22σ3Uxl.h.s.of19=0

We find

Nxd(x,y)+σ3Nyd(x,y)σ3U(x)No(x,y)No(x,x)F(x+y)σ3No(x,x)F(x+y)σ3U(x)F(x+y)+xdz{Nxo(x,z)F(z+y)σ3Nzo(x,,z)F(z+y)σ3U(x)Nd(x,,z)G(z+y)}=0E133

Since Fxis off-diagonal, Fxσ3=σ3Fx. Thus the terms involving with Fx+youtside of integral are equal to iUxNdxxσ3Fx+yby use of Eq. (123). Then (133) can be rewritten as

AxyiUxNdxxσ3Fx+y+xdzBxzFz+yiUxNzdxzσ3Fz+yUxNdxzGz+y=0E134

If we choose

Gz+y=iσ3Fz+y,E135

then

iUxNdxxσ3Fx+y+zdziUxNzd(xz)σ3Fz+y+UxNd(xz)iσ3Fz+y=0E136

Thus, Eq. (134) becomes

Axy+xdzBxzFz+y=0E137

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

Noxy+Fx+y+xdzNdxziσ3Fz+y=0E138

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

Nxoxy+Fx+yNdxxiσ3Fx+y+xdzNxdxziσ3Fz+y=0E139
Nyoxy+Fx+y+xdzNdxziσ3Fz+y=0E140

or

Nyoxy+Fx+yNdxxiσ3Fx+yxdzNzdxxiσ3Fz+y=0E141

Now we make a weighing summation as

l.h.s.of29+σ3l.h.s.of31σ3+iUxl.h.s.of21σ3=0

Hence, we have

Nxoxy+σ3Nyoxyσ3+iUxNydxyσ3+Fx+y+σ3Fx+yσ3Ndxxiσ3Fx+yσ3Ndxxiσ3Fx+yσ3+xdzNxdxziσ3Fz+yσ3Nzd(xz)iσ3Fz+yσ3+iUxNo(xz)Fz+yσ3=0E142

Noticing Fxσ3=σ3Fx, Eq. (142) becomes

Bxy+xdzAxziσ3Fz+y=0E143

We find that, as long as we choose a suitable form for Gx+yas well as Fx+yaccording 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 Noxyand Ndxy, the function fx+yin (128) and (129) can only has off-diagonal elements, we write

Fx+y=0fx+y¯fx+y0;Gz+y=0hz+y¯hz+y0=iσ3Fz+yE144

Considering the dependence of the Jost solutions on the squared spectral parameter λ2, in the reflectionless case, we choose

fx+y=n=1NCnteiλn2x+yE145

where Cntcontains a time-dependent factor ei4λn4t, which can be introduced by a standard procedure [29], due to a fact of the Lax operator Mi2λ4σ3as x±.

As is well known, Lax equations are linear equation so that a constant factor can be introduced in its solution, that is, Cn=eβn+iαnei4λn4t. It means that βnis related to the center of soliton and αnexpresses 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 limxeiλn2x=0, so we only consider the Nzero points of aλin the first quadrant of complex plane of λ(also in the upper half part of the complex plane of λ2), that is, the discrete spectrum for λ1,λ2,λN, although λn,n=12Nin the third quadrant of the complex plane of λare also the zero points of aλdue 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 λ1,λ2,,λNin the first quadrant of the complex plane of λ, the Marchenko equation will give a N-solition solution to the DNLS equation with VBC in the reflectionless case. We can assume that

fx+y=F21x+y=n=1NgnxthnyGxtHyTE146

where gnxtCnteiλn2x,hnyeiλn2y,n=1,2,,N, and

Gxtg1xtg2xtgNxt;HyTh1yh2yhNyTE147

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

N11xy=N11xHyT,N12xy=N12xH¯yTE148

Then

F12x+y=F¯12x+y=G¯xH¯yT;F12x+y=iλn2¯C¯neiλn2¯x+y=Gx¯H¯yTE149

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

N11xHyT+N12xxdzH¯zTGzHyT=0N12xH¯yTG¯xH¯yTiN11xxdzHzTG¯zH¯yT=0E150

or

N11x+N12xΔ1x=0,E151
N12xiN11xΔ2x=G¯xE152

here

Δ1x=xH¯zTGzdz,Δ2x=xHzTG¯zdzE153

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

Δ1xmn=xeiλm2¯zCneiλn2zdz=h¯mxiλm2¯λn2gnxE154
Δ2xmn=xeiλm2zC¯niλn2¯eiλn2¯zdz=hmxλn2¯λm2λn2¯g¯nxE155

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

N11x=G¯x1+iΔ1xΔ2x1Δ1xE156
N12x=G¯x1+iΔ1xΔ2x1E157

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

N11xy=G¯xI+iΔ1xΔ2x1Δ1xHyTE158
N12xy=G¯xI+iΔ1xΔ2x1H¯yTE159

then

N11(x,x)=iTr{iΔ1(x)H(x)TG¯(x)[I+iΔ1(x)Δ2(x)]1}=i{det[I+iΔ1(x)Δ2(x)+iΔ1(x)H(x)TG¯(x)]det[I+iΔ1(x)Δ2(x)]1}E160

and

N12xx=TrH¯xTG¯xI+iΔ1xΔ2x1=detI+1xΔ2x+H¯TxG¯xdetI+1xΔ2x1E161

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

uNxt=2detI+iΔiΔ2+H¯TG¯detI+iΔ1Δ2detIiΔ¯1Δ¯2iΔ¯1H¯TG·detIiΔ¯1Δ¯2detI+iΔ1Δ2=2CD¯/D2E162

where

DdetI+iΔ1Δ2,CdetI+iΔ1Δ2+H¯TG¯detI+iΔ1Δ2,E163

and we will prove that in (136)

detIiΔ¯1Δ¯2iΔ¯1H¯TG=detI+iΔ1Δ2DE164

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.

detIiΔ¯1Δ¯2iΔ¯1H¯TG=detI+iΔ¯1Δ¯2+H¯TGdetI+M1M2E165

where

Μ1nmiΔ¯1mn=hn1λn2λm2¯g¯m,M2mnΔ¯2+H¯TGmn=h¯mλm2¯λm2¯λn2gnE166

The complex constant factor cn0can be absorbed into the soliton center and initial phase by redefining

gnxthnx=Cntei2λn2x=cnoei4λn4tei2λn2xeθneiφnE167

here λn=μn+ivn, and

θn4μnvn[xxn0+4(μn2vn2)t]=4κn(xxn0υnt);κn=4μnνn;υn=4(μn2vn2);φn2(μn2vn2)x+[4(μn2vn2)216μn2vn2]t+αno;cn0e4κnxnoeiαno;n=1,2,,NE168
detIiΔ¯1Δ¯2iΔ¯1H¯TG=1+r=1N1n1<n2<<nrN1m1<m2<<mrNM1n1n2nrm1m2mrM2m1m2mrn1n2nrE169

where M1n1n2nrm1m2mrdenotes a minor, which is the determinant of a submatrix of M1, consisting of elements belonging to not only (n1, n2,…, nr) rows but also columns (m1, m2,…, mr).

M1n1n2nrm1m2mr=n,mhng¯mλn2λm2¯n<n,m<mλn2λn2λm2¯λm2¯E170
M2m1m2mrn1n2nr=n,mh¯mgnλm2¯λm2¯λn2n<n,m<mλm2¯λm2¯λn2λn2E171

where n,nn1n2nr, m,mm1m2mr, then M1n1n2nrm1m2mrM2m1m2mrn1n2nr

=1rn,meθneiφneθmeiφmλm2¯λn2λm2¯2n<n,m<mλm2¯λm2¯2λn2¯λn2¯2E172

If we define matrices Q1=iΔ1and Q2=Δ2, then we can similarly attain

D=detI+iΔ1Δ2=detI+Q1Q2=1+r=1N1n1<n2<<nrN1m1<m2<<mrNQ1n1n2nrm1m2mrQ2m1m2mrn1n2nrE173

and

Q1n1n2nrm1m2mrQ2m1m2mrn1n2nr=1rn,meθmeiφmeθneiφnλn2¯λm2λn2¯2n<n,m<mλm2λm22λn2¯λn2¯2E174

where n,nn1n2nr,m,mm1m2mr. Comparing (172) and (174), we find the following permutation symmetry between them

M1n1n2nrm1m2mrM2m1m2mrn1n2nr
=Q1m1m2mrn1n2nrQ2n1n2nrm1m2mr

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 Cin (163). In order to calculate detI+iΔ1Δ2+H¯TG¯, we introduce an N×N+1matrix Ω1and an N+1×Nmatrix Ω2

Ω1nm=iΔ1nm,Ω1n0=h¯n=h¯nλn2¯λn2¯02;Ω2mn=Δ2mn,Ω20n=g¯n=λn2¯g¯n02λn2¯E175

with n, m = 1, 2,…, N. We thus have

detI+Ω1Ω2=1+r=1N1n1n2<<nrN0m1<m2<<mrNΩ1n1n2nrm1m2mrΩ2m1m2mrn1n2nrE176

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 m1=0, m21). Due to (175), we have

C=det(I+Ω1Ω2)det(I+iΔ1Δ2)=Nr=11n1<n2<<nrN1m2<m3<<mrNΩ1(n1,n2,,nr;0,m2,,mr)Ω2(0,m2,,mr;n1,n2,,nr)E177
Ω1n1n2nr0m2mr=nh¯nmgmn<n,m<mλn2¯λn2¯λm2λm2n,m1λn2¯λm2E178
Ω2(0,m2,,mr;n1,n2,nr)=ng¯nmhmn<n,m<m(λn2¯λn2¯)(λm2λm2)n,m1λm2λn2¯(1)r+1mλm2,E179

which leads to

Ω1n1n2nr0m2mrΩ20m2mrn1n2nr=1r+1neθneiφnmeθmeiφmn<n,m<mλn2¯λn2¯2λm2λm22n,m1λm2λn2¯2mλm2,E180

here n,nn1n2nr, m,mm2mrin (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 aλ. Obviously, the N-soliton solution permits uncertain complex constants cn0n=12Nas well as an arbitrary global constant phase factor.

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

In the case of one simple pole and one-soliton solution as N=1, according to (173), (177), (174), and (180), we have

C1=Ω1n1=1m1=0Ω2m1=0n1=1=g¯1h¯111+1=g¯1h¯1E181
D1=Q1n1=1m1=1Q2m1=1n2=1=1g1h1g¯1h¯1λ12¯λ12λ12¯2=1g1h12λ12¯λ12λ12¯2E182

From (167) and (168), we have (suppose λ1=ρ1eiβ1=μ1+iv1and c10=e4κ1x10eiα10)

g1h1=C1tei2λ12x=c10ei4λ14tei2λ12x=eθ1eiφ1E183
θ1=4μ1v1xx10+4μ12v12tφ1=2μ12v12x+4μ12v12216μ12v12t+α10E184

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

u1xt=2C1D¯1D12=21λ12λ12λ12¯2e2θ1eθ1eiφ1/1λ12¯λ12λ12¯2e2θ12E185

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 N=2in the case of two-soliton solution corresponding to double simple poles, we have

u2xt=2C2D¯2/D22E186
C2=n1=1,2m1=0Ω1n10Ω20n1+n1=1,n2=2m1=0,m2=1,2Ω1n1n20m2Ω20m2n1n2=Ω1n1=1m1=0Ω2m1=0n1=1+Ω1n1=2m1=0Ω2m1=0n1=2+Ω1n1=1n2=2m1=0m2=1Ω2m1=0m2=1n2=1n2=2+Ω1n1=1n2=2m1=0m2=2Ω2m1=0m2=2n1=1n2=2=g¯1h¯1+g¯2h¯2g¯1h¯12g¯2h¯2λ12¯λ22¯2λ12λ12λ12¯2λ12λ22¯2g¯2h¯22g¯1h¯1λ12¯λ22¯2λ22λ22λ12¯2λ22λ22¯2=eθeiφ1+eθ2eiφ2λ12¯λ22¯2λ12λ12λ12¯2λ12λ22¯2e2θ1θ2eiφ2λ12¯λ22¯2λ22λ22λ12¯2λ22λ22¯2e2θ2θ1eiφ1E187
D2=1+r=121n1<n221m1<m22Q1n1n2nrm1m2mrQ2m1m2mrn1n2nr=1+Q1n1=1m1=1Q2m1=1n1=1+Q1n1=1m1=2Q2m1=2n1=1+Q1n1=2m1=1Q2m1=1n1=2+Q1n1=2m1=2Q2m1=2n1=2+Q1n1=1n2=2m1=1m2=2Q2m1=1m2=2n1=1n2=2=1g1h12λ12¯λ12λ12¯2g2h22λ22¯λ22λ22¯2g¯1h¯1g2h2λ12¯λ12¯λ222g¯2h¯2g1h1λ22¯λ12λ22¯2+g1h12g2h22λ12¯λ22¯λ12λ222λ12¯λ22¯2λ12λ12¯2λ12λ22¯2λ22λ12¯2λ22λ22¯2=1λ12¯λ12λ12¯2e2θ1λ22¯λ22λ22¯2e2θ2λ12¯λ12¯λ222eθ1θ2eiφ2φ1λ22¯λ12λ22¯2eθ1θ2eiφ1φ2+λ12¯λ22¯λ12λ222λ12¯λ22¯2λ12λ12¯2λ12λ22¯2λ22λ12¯2λ22λ22¯2e2θ1+θ2E188

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,

iut+uxx+iu2ux=0,E189

is one of the typical integrable nonlinear models, which is of a different form from the following equation:

iut+uxx+i2u2ux=0,E190

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

u=g/f,E191

where f,gare 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

u=gf¯/f2E192

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 Axt,Bxtof two variables xand t, Hirota’s bilinear derivative operator, D, is defined as

DtnDxmA·B=ttnxxmA(xt)B(xt)t=t,x=xE193

which is different from the usual derivative, for example,

DxA·B=AxBABxDx2A·B=AxxB2AxBx+ABxxDx3A·B=AxxxB3AxxBx+3AxBxxABxxxE194

where Axt,Bxtare 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:

DtnDxmA·B=1n+mDtnDxmB·AE195

for example, DxA·B=DxB·A; DxA·A=0; Dx2A·B=Dx2B·A; DxnA·1=xnA; Dxn1·A=1nxnA

DxnA·B=DxnmDxmA·B,m<nE196

➂ Suppose ηi=Ωit+Λix+η0i, i=1,2, Ωi, Λi, η0iare complex constants, then

DtnDxmexpη1·expη2=Ω1Ω2nΛ1Λ2mexpη1+η2E197

Especially, we have DtnDxmexpη1·expη2=0as Ω1=Ω2or Λ1=Λ2. 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].

F1DtDxg1·f1+F2DtDxg2·f2=0E198

where FiDtDx, i=1,2are the polynomial functions of Dt, Dx; and gi,fi, i=1,2, are the differentiable functions of two variables xand t. Using formulae in the Appendix and properties ①–③ of bilinear derivative transform numerated in the last chapter, with respect to (192), we have

ut=Dtgf¯·f2/f4=ff¯Dtg·fgfDtf·f¯/f4E199
ux=Dxgf¯·f2/f4=ff¯Dxg·fgfDxf·f¯/f4E200
uxx=ff¯Dx2g·f2Dxg·fDxf·f¯+gfDx2f·f¯/f42gf¯Dx2f·f/f4E201
u2ux=2gg¯Dxg·fg2g¯fx/f4E202

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

ff¯iDt+Dx2g·fgfiDt+Dx2f·f¯+f2Dxf3·g2Dxf·f¯igg¯=0E203

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

iDt+Dx2g·f=0E204
iDt+Dx2f·f¯=0E205
Dxf·f¯=igg¯/2E206

Functions gxt, fxtcan be expanded, respectively, as series of a small parameter ε

g=iεigiE207
f=1+iεifiE208

Substituting (207) and (208) into (204)(206) and equating the sum of the terms with the same orders of εat two sides of (204)(206), we attain

it+x2g1=0E209
iDt+Dx2f1·1+1·f¯1=0E210
Dxf1·1+1·f¯1=0E211
iDt+Dx2g2·1+g1·f1=0E212
iDt+Dx2f2·1+1·f¯2+f1·f¯1=0E213
Dxf2·1+1·f¯2+f1·f¯1=ig1·g¯1/2E214
iDt+Dx2g3·1+g2·f1+g1·f2=0E215
iDt+Dx2f3·1+1·f¯3+f2·f¯1+f1·f¯2=0E216
Dxf3·1+1·f¯3+f2·f¯1+f1·f¯2=ig2g¯1+g1g¯2/2E217
iDt+Dx2g4·1+g3·f1+g2·f2+g1·f3=0E218
iDt+Dx2f4·1+1·f¯4+f3·f¯1+f1·f¯3+f2·f¯2=0E219
Dxf4·1+1·f¯4+f3·f¯1+f1·f¯3+f2·f¯2=ig3g¯1+g2g¯2+g1g¯3/2E220
iDt+Dx2g5·1+g4·f1+g3·f2+g2·f3=0E221
iDt+Dx2f5·1+1·f¯5+f4·f¯1+f1·f¯4+f3·f¯2+f2·f¯3=0E222
Dx(f(5)·1+1·f¯(5)+f(4)·f¯(1)+f(1)·f¯(4)+f(3)·f¯(2)+f(2)·f¯(3))=i(g(4)g¯(1)+g(3)g¯(2)+g(2)g¯(3)+g(1)g¯(4))/2E223

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 g1and f1respectively as

g1=eη1,η1=Ω1t+Λ1x+η10,Ω1=iΛ12,E224
f1=0E225

From (212), one can select g2=0. From (214), we can attain

f2f¯2=i21Λ1+Λ¯1eη1+η¯1E226

where the vanishing boundary condition, u0as x, is used. Then

tf2f¯2=12Λ12Λ¯12Λ1+Λ¯1eη1+η¯1E227

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

f2+f¯2=i2Λ1Λ¯1Λ1+Λ¯12eη1+η¯1E228

From (226) and (228), we can get an expression of f2

f2=i2Λ1Λ1+Λ¯12eη1+η¯1E229

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

iDt+Dx2g1·f2=0E230

which immediately leads to

it+x2g3=0E231

in Eq. (215). Then from (215), we can select g3=0. For the same reason, from (216)(223), we can select f3,g4, g5, …; f4, f5, …all to be zero. Thus the series (207) and (208) have been successfully cut off to have limited terms as follows:

g1=eη1E232
f1=1+i2Λ1Λ1+Λ¯12eη1+η¯1E233

where εihas been absorbed into the constant eη10by redefiniing η10. In the end, we attain the one-soliton solution to the DNLS equation with VBC

u1xt=g1f¯1/f12E234

which is characterized with two complex parameters Λ1and η10and shown in Figure 1. If we redefine the parameter Λ1as Λ1i2λ¯12and λ1μ1+iv1, then

ei2λ12x+i4λ14t+η¯10/2eΘ1e1;eη¯10/2e4μ1ν1x10eiα10;Θ14μ1v1[xx10+4(μ12v12)t];Φ12(μ12v12)x+[4(μ12v12)216μ12v12]t+α10E235

Then

g1=eη1ei2λ¯2xi4λ¯4t+η10=2eΘ1eiΦ1f1=1+i2Λ1Λ1+Λ¯12eη1+η¯1=1λ¯2λ2λ¯22e2ΘE236
u1xt=g1f¯1f12=21λ12λ12λ12¯2e2Θ1eΘ1eiΦ1/1λ12¯λ12λ12¯2e2Θ12E237

It is easy to find, up to a permitted constant global phase factor e=1, the one-soliton solution (234) or (237) gotten in this paper is in perfect agreement with that gotten from other approaches [16]. By further redefining its soliton center, initial phase and λ1=ρ1eiβ1, the one-soliton solution can be changed into the usual typical form [16, 23, 26, 27].

u1xt=4λ1sin2β11+ei2β1e2Θ1eΘ1eiΦ1/1+ei2β1e2Θ12E238

On the other hand, just like in Ref. [13], we can rewrite g1and f1in a more appropriate or “standard” form

g1=eη1+φ1E239
f1=1+eη1+φ1+η¯1+φ1+θ11E240

Here

eφ1=1,eφ1=i/Λ¯1,eθ11=iΛ1iΛ¯1/2Λ1+Λ¯12,E241

which makes us easily extend the solution form to the case of n-soliton solution.

4.3.2 The two-soliton solution

For the two-soliton case, again from (209), we can select g21as

g21=eη1+eη2,ηi=Ωit+Λix+ηi0,Ωi=iΛi2,i=1,2.E242

The similar procedures to that used in the one-soliton case can be used to deduce g2and f2. From (210) and (211), we can select f21=0, then from (212), we has to select g22=0. From (213) and (214), we can get the expressions of f22f¯22and f22+f¯22, then attain f22to be

f22=iΛ12Λ1+Λ¯12eη1+η¯1+iΛ22Λ2+Λ¯22eη2+η¯2+iΛ12Λ1+Λ¯22eη1+η¯2+iΛ22Λ2+Λ¯12eη2+η¯1E243

Substituting (242) and (243) into (215), one can attain g23to be

g23=iΛ1Λ22eη1+η22Λ¯1eη¯1Λ1+Λ¯12Λ2+Λ¯12+Λ¯2eη¯2Λ2+Λ¯22Λ1+Λ¯22E244

Substituting the expressions of g21,g22,g23,f21,f22into (216) and (217), we can select that f23=0. Then from the expressions of g21,g22,g23,f21,f22,f23and (218), we can select g24=0. From (219) and (220), we can get the expressions of f24f¯24and f24+f¯24, then get f24to be

f24=Λ1Λ2Λ1Λ244Λ1+Λ¯12Λ2+Λ¯22Λ1+Λ¯22Λ2+Λ¯12eη1+η2+η¯1+η¯2E245

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

iDt+Dx2g23·f22=0E246

Then from (244), (245), (246), and (221), we can select g25=0. From (222)(223) and so on, we find that the series of (207) and (208) can be cut off by selecting g25,f25;g26,f26, all to be zero. We thus attain the last result of g2,f2to be

g2=eη1+eη2iΛ1Λ222eη1+η2Λ¯1Λ1+Λ¯12Λ2+Λ¯12eη¯1+Λ¯2Λ2+Λ¯22Λ1+Λ¯22eη¯2E247
f2=1+iΛ12Λ1+Λ¯12eη1+η¯1+iΛ22Λ2+Λ¯22eη2+η¯2+iΛ12Λ1+Λ¯22eη1+η¯2+iΛ22Λ2+Λ¯12eη2+η¯1Λ1Λ2Λ1Λ244Λ1+Λ¯12Λ2+Λ¯22Λ1+Λ¯22Λ2+Λ¯12eη1+η2+η¯1+η¯2E248

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

g2=eη1+φ1+eη2+φ2+eη1+φ1+η2+φ2+η¯1+φ1+θ12+θ11+θ21+eη1+φ1+η2+φ2+η¯2+φ2+θ12+θ12+θ22E249
f2=1+e(η1+φ1)+(η¯1+φ1)+θ11+e(η1+φ1)+(η¯2+φ2)+θ12+e(η2+φ2)+(η¯1+φ1)+θ21+e(η2+φ2)+(η¯2+φ2)+θ22+e(η1+φ1)+(η2+φ2)+(η¯1+φ1)+(η¯2+φ2)+θ12+θ11+θ12+θ21+θ22+θ12E250

where φi,θijin (249) and (250) are defined afterwards in (256). We then attain the two-soliton solution as

u2xt=g2 f¯2/f22E251

which is characterized with four complex parameters Λ1, Λ2, η10, and η20and shown in Figure 3. By redefining parameters ηi0and

Λk=i2λ¯k2,k=1,2,E252

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 g1in (209) to be

gN1=eη1+eη2++eηN,ηi=Ωit+Λix+η0i,Ωi=iΛi2,i=1,2,,NE253

then using an induction method, we can write the N-soliton solution as

gN=κj=0,11expj=12Nκjηj+φj+1j<k2NκjκkθjkE254
fN=κj=0,10expj=12Nκjηj+φj+1j<k2NκjκkθjkE255

where ηN+j=η¯jjN

eϕj=1,eφj+N=eφj=iΛ¯j1eθjk=2ΛjΛk2/iΛj·iΛk,eθj,k+N=eθj,k=iΛjiΛ¯k/2Λj+Λ¯k2eθj+N,k+N=eθjk=2Λ¯jΛ¯k2/iΛ¯jiΛ¯k,1jN1kN,E256

therein κj=0,1lrepresents a summation over κj=0,1under the condition j=1Nκj=l+j=1Nκj+N.

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(ad) 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.

Figure 2.

The evolution of one-soliton solution with time and space under parameter Λ1=−1+0.2i,η10=1 in (234).

Figure 3.

The evolution of two-soliton solution with time and space under parameter Λ1=1+0.3i,Λ2=1−0.3i,η10=η20=1 in (251).

Figure 4.

The elastic collision between two solitons at 4 typical moments: (a) t = −10(normalized time); (b) t = −1; (c) t = 1; (d) t = 10, from −10 before collision to 10 after collision.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Zhou Guo-Quan (September 5th 2020). Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1, Nonlinear Optics - From Solitons to Similaritons, İlkay Bakırtaş and Nalan Antar, IntechOpen, DOI: 10.5772/intechopen.93438. Available from:

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