Open access peer-reviewed chapter - ONLINE FIRST

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

By Zhou Guo-Quan

Submitted: July 2nd 2020Reviewed: July 22nd 2020Published: September 22nd 2020

DOI: 10.5772/intechopen.93450

Downloaded: 28

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

iutuxx+iu2ux=0,E1

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 Nsoliton solution in terms of Vandermonde-like determinants by means of Bäcklund transformation; but just as reference [7, 9] pointed out, this multi-soliton solution is still difficult to exhibit the internal elastic collisions among solitons and the typical asymptotic behaviors of multi-soliton of DNLS equation. By introducing an affine parameter in the integral of Zakharov-Shabat IST, reference [7] had found their pure Nsoliton solution for a special case that all the simple poles (zeros of aλ) were located on a circle of radius ρcentered at the origin, while reference [8] also found its multi-soliton solution for some extended case with Npoles on a circle and Mpoles out of the circle, and further developed its perturbation theory based on IST. Reference [7] constructed their theory by introducing an damping factor in the integral of Zakharov-Shabat IST, to make it convergent, and further adopted a good idea of introducing an affine parameter to avoid the trouble of multi-value problem in Riemann sheets, but both of their results are assumed Nsoliton solutions and the soliton solution gotten from their IST had a self-dependent and complicated phase factor [7, 8, 9], hence reference [8] had to verify an identity demanded by the standard form of a soliton solution (see expression (52) in Ref. [8]). Such kind of an identity is rather difficult to prove for N2case even by the use of computer techniques and Mathematica. On the other hand, author of reference [8] also admits his soliton solution is short of a rigorous verification of standard form. Then questions naturally generates – whether the traditional IST for DNLS equation with NVBC can be avoided and further improved? And whether a rigorous manifestation of soliton standard form can be given and a more reasonable and natural IST can be constructed?

A newly revised IST is thus proposed in this section to avoid the dual difficulty and the excessive complexity. An additional affine factor 1/λ, λ=z+ρ2z1/2, is introduced in the Z-S IST integral to make the contour integral convergent in the big circle [7, 10, 11, 12, 13]. Meanwhile, the additional two poles on the imaginary axis caused by λ=0are removable poles due to the fact that the first Lax operator Lλ0,as λ0. What is more different from reference [7] is that we locate the N simple poles off the circle of radius ρcentered at origin O, which corresponds to the general case of N breather-type solitons. When part of the poles approach the circle, the corresponding part of the breathers must tend to the pure solitons, which is the case described in Ref. [8]. The resulted one soliton solution can naturally tend to the well-established conclusion of VBC case as ρ0[17, 18, 19, 20] and the pure one soliton solution in the degenerate case. The result of this section appears to be strict and reliable.

1.1 The fundamental concepts for the IST theory of DNLS equation

Under a Galileo transformation xtx+ρ2tt, DNLS +Eq. (1) can be changed into

iutuxx+iu2ρ2ux=0,E2

with nonvanishing boundary condition:

uρ,asx.E3

According to references [7, 8, 9], the phase difference between the two infinite ends should be zero. The Lax pair of DNLS+Eq. (3) is

L=iλ2σ3+λUU=0uu¯0=uσ++u¯σE4
M=i2λ4σ32λ3U+iλ2U2ρ2σ3λU2ρ2U+Uxσ3E5

where σ3, σ+, and σinvolve in standard Pauli’s matrices and their linear combination. Here and hereafter a bar over a variable represents complex conjugate. An affine parameter zand two aided parameters η, λare introduced to avoid the trouble of dealing with double-valued functions on Riemann sheets

λz+ρ2z1/2,ηzρ2z1/2E6

The Jost functions satisfy first Lax equation

xFxz=LxzFxz,E7

here Jost functions FxzΨxzΦ(xz).

Ψxz=ψ˜xzψxzExz,asxE8
Φxz=ϕxzϕ˜xzExz,asxE9

The free Jost function Exzcan be easily attained as follows:

Exz=I+ρz1σ2expiληxσ3,E10

which can be verified satisfying Eq. (7). The monodramy matrix is

Tz=azb˜zbza˜z,E11

which is defined by

Φxz=ΨxzTzE12

Some useful and important symmetry properties can be found

σ1L¯zσ1=Lz¯,σ3Lzσ3=LzE13

Symmetry relations in (13) lead to

ψ˜xz=σ1ψxz¯¯,ϕ˜xz=σ1ϕxz¯¯E14
ψ˜xz=σ3ψ˜xz,ψxz=σ3ψxzE15
σ1Txz¯¯σ1=Tz,σ3Tzσ3=TzE16

The above symmetry relations further result in

a˜z¯=az¯,b˜z=bz¯E17
az=az,bz=bzE18

Other important symmetry properties called reduction relations can also be easily found

λρ2z1=λz,ηρ2z1=ηz,Lxρ2z1=LxzE19
Exρ2z1=ρ1zI+ρz1σ2eiηλxσ3σ2E20

The above symmetry properties lead to following reduction relations among Jost functions

ψ˜xρ2z1=iρ1zψxz,ψxρ2z1=iρ1zψ˜xzE21
ϕxρ2z1=iρ1zϕ˜xz,ϕ˜xρ2z1=iρ1zϕxzE22

The important symmetries among the transition coefficients further resulted from (12), (21), and (22):

σ2Tρ2z1σ2=Tz,a˜ρ2z1=az,b˜ρ2z1=bzE23

On the other hand, the simple poles, or zeros of az, appear in quadruplet, and can be designated by zn, (n=1,2,,2N), in the I quadrant, and zn+2N=znin the III quadrant. Due to symmetry (17), (18) and (23), the n’ th subset of zero points is

z2n1z2n=ρ2z¯2n11z2n1z2n=ρ2z¯2n11E24

And we arrange the 2Nzeros in the first quadrant in following sequence

z1,z2;z3,z4;;z2N1,z2NE25

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

az=n=12Nz2zn2z2z¯n2z¯nznE26

At the zeros of az, or poles zn, n=122N12N, we have

ϕxzn=bnψxzn,ȧzn=ȧznE27

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

b¯2n=b2n1,c¯2n1=ρ2z2n2c2nE28

2. Relation between the solution and Jost functions of DNLS+ equation

The asymptotic behaviors of the Jost solutions in the limit of λcan be obtained by simple derivation. Let F·1=ψ˜xλ, then Eq. (7) can be rewritten as

ψ˜1x+iλ2ψ˜1=λuψ˜2,ψ˜2xiλ2ψ˜2=λu¯ψ˜1,E29

then we have

ψ˜1xxψ˜1x+iλ2ψ˜1ux/u+λ4ψ˜1λ2u2ψ˜1=0E30

We assume a function gto satisfy the following equation

ψ˜1xλ=eiληx+gE31
ψ˜1x=iλη+gxψ˜1,ψ˜1xx=iλη+gx2+gxxψ˜1E32

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

gxx+gx22iλη+ux/ugxληux/u+λ2λ2η2λ2u2=0E33

In the limit of z, gxcan be expanded as sum of series of z2j, j=0,1,2,

gxμ=μ0+μ2z2+μ4z4+,E34

where

μ0=iρ2u2/2=O1E35

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

u=ψ˜1λψ˜2λη12iρ2u2iψ˜1ψ˜2u2z,asz

Then we find a useful formula

u¯=ilimzzψ˜2xz/ψ˜1xz,E36

which expresses the conjugate of the solution uin terms of Jost functions as z.

2.1 Introduction of time evolution factor

In order to make the Jost functions satisfy the second Lax equation, a time evolution factor htzshould be introduced by a standard procedure [21, 22] in the Jost functions and the scattering data. Considering the asymptotic behavior of the second Lax operator

Mxtzi2λ4σ32λ3ρσ1,asx,E37

we let htkto satisfy

/tMxtzh1tzψxtz=0,asxE38

then

/ti2λ4σ32λ3ρσ1h1tzE2xz=0.E39

Due to

E2xz=z1eiληxeiληxT,E40

from (39) and (40), we have

htz=ei2λ3ηt.E41

Therefore, the complete Jost functions should depend on time as follows

htzψ˜xz,h1tzψxz;htzϕxz,h1tzϕ˜xzE42

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

azt=az0,btz=b0ei4λ3ηtE43

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

A 2 × 1 column function Πxzis introduced as usual

Πxzϕxz/az,aszinI,IIIquadrants.ψ˜xz,aszinII,IVquadrants.E44

here and hereafter note “” represents definition. There is an abrupt jump for Πxzacross both real and imaginary axes

ϕxz/azψ˜xz=rzψxz,E45

where

rz=bz/azE46

is called the reflection coefficient. Due to μ00in (34), Jost solutions do not tend to the free Jost solutions Exzin the limit of z. This is their most typical property which means that the usual procedure of constructing the equation of IST by a Cauchy contour integral must be invalid. In view of these abortive experiences, we proposed a revised method to derive a suitable IST and the corresponding Z-S equation by multiplying an inverse spectral parameter 1/λ,λ=z+ρ2z1/2, before the Z-S integrand. Meanwhile, our modification produces no new poles since the Lax operator Lλ0, as λ0. In another word, the both additional poles z0=±generated by λ=0are removable. Under reflectionless case, that is, rz=0, the Cauchy integral along with contour Γshown in Figure 1 gives

Figure 1.

The integral path for IST of the DNLS+.

1λΠxzE·1(xz)eiληx=12πiΓdz1zz1λΠxzE·1(xz)eiληxE47

or

ψ˜xz=eiληx+λn=14N1λn1znzcnψxzneiλnηnxeiληxE48

where

cnbn/ȧznȧzn=daz/dzz=zn;n=1,2,,4NE49

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 bnand cnsimilar to (43) can be derived

bnt=bn0ei4λn3ηnt,cn=cn0ei4λn3ηtE50
cn0=bn0/ȧn;n=1,2,,4NE51

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

ψ˜1xz=eiΛx+λn=12N2zλn1z2zn2cnψ1xzneiΛnxeiΛxE52
ψ˜2xz=z1eiΛx+λn=12N2znλn1z2zn2cnψ2xzneiΛnxeiΛxE53

here Λλη,Λnλznηzn=λnηn, and in Eqs. (52) and (53), the terms corresponding to poles zn, (n=1,2,,2N), have been combined with the terms corresponding to poles zn+2N=zn. Substituting Eqs. (52) and (53) into formula (36) and letting z, we attain the conjugate of the raw N-soliton solution (the time dependence is suppressed).

u¯N=UN/VNE54
UNρ1n=12Niznρλncnψ2xzneiΛnxE55
VN1+n=12Ncnλnψ1xzneiΛnxE56

Letting z=ρ2zm1, (m=1,2,,2N), respectively, in Eqs. (52) and (53), by use of reduction relations (19), (21), and (22), we can further change Eqs. (52) and (53) into the following form

ψ1xzm=zm1eiΛmx+n=12Nλmcnλnzm22ρ3iρ4zm2zn2ψ1xzneiΛn+ΛmxE57
ψ2xzm=eiΛmx+n=12Nλmzncnλnzm·2ρiρ4zm2zn2ψ2xzneiΛn+ΛmxE58

m=1,2,,2N. An implicit time dependence of the complete Jost functions ψ1and ψ2besides cnshould be understood. To solve Eq. (58), we define that

φ2niznρλncn2ψ2xzn,φ2φ21φ22φ22NE59
fn2cneiΛnx,gnicn2znρλneiΛnx=izn2ρλnfn,n=1,2,,2NE60
ff1f2f2N,gg1g2g2NE61

BMatrixBnm2N×2N, with

Bnmfnρizn2ρ4zm2fm,m,n=1,2,,2N.E62

Then Eq. (58) can be rewritten as

φ2m=gmn=12Nφ2nBnm,m=1,2,,2NE63

or in a more compact form

φ2=gφ2B.E64

The above equation gives

φ2=gI+B1.E65

Note that the choice of poles, zn,n=12N, should make detI+Bnonzero and I+Ban invertible matrix. On the other hand, Eq. (55) can be rewritten as

UN=ρ1φ2fT,E66

hereafter a superscript “T” represents transposing of a matrix. Substituting Eq. (65) into (66) leads to

UN=ρ1gI+B1fT=ρdetI+BfTgdetI+B=ρdetI+AdetI+BE67

where

ABfTgE68

with

AnmBnmfngm=zmznλn/ρ2λmBnm.E69

To solve Eq. (57), we define that

φ1micm2zmρλmψ1xzm,φ1φ11φ12φ12NE70
fmi2cm·ρzmeiΛmx=iρzmfm;gm=cm2·1λmeiΛmx=izm2ρλmfmE71
f=f1f2f2N;g=g1g2g2N;E72
Dnmfnρizn2ρ4zm2fm=ρ2znzmBnmc.f.1.70and1.61E73

with n,m=1,2,,2N. Then Eq. (57) can be rewritten as

φ1m=gmn=12Nφ1nDnm,m=1,2,,2NE74

or in a more compact form

φ1=gφ1DE75

The above equation givess

φ1=gI+D1E76

Note that the choice of poles, zn,n=12N, should make detI+Dnonzero and I+Dan invertible matrix. On the other hand, Eq. (56) can be rewritten as

VN=1n=12Nφ1nfn=1φ1fTE77

Substituting Eq. (76) into (77), we thus attain

VN=1gI+D1fT=detI+DfTgdetI+DdetI+BdetI+DE78

where use is made of Appendix A.1 and

BnmDfTgnm=zmznλnρ2λmDnm=λnλmBnmE79

In the end, by substituting (67) and (78) into (54), we attain the N-soliton solution to the DNLS+Eq. (3) under NVBC and reflectionless case (note that the time dependence of soliton solution naturally emerges in cnt):

u¯xt=UNVN=ρdetI+AdetI+DdetI+BdetI+BρCNDND¯N2,E80

here

CNdetI+A,D¯NdetI+BE81

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

detI+B=detI+B=detI+D¯D¯,E82

which can be proved by direct calculation for the N=1case and by some special algebra techniques for the N>1case.

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

3.1 Verification of detI+B=detI+B

In order to prove the first identity in (82), we firstly calculate D¯N=detI+B. By use of (60)(62), Binet-Cauchy formula, (Appendix (A.2)) and an important determinant formula, (Appendix (A.3)), we have

D¯NdetI+B=1+r=12N1n1<<nr2NBn1n2nr,E83

here Bn1n2nris a r,th-order principal minor of Bconsisting of elements belonging to not only rows n1n2nrbut also columns n1n2nr. Due to (62),

Bn1n2nr=n,mfnρizn2ρ4zm2fmn<mizn2zm2·iρ4zn2ρ4zm2E84

in (84), n,mn1n2nr. The technique of calculating Bn1n2nris to couple term izn2ρ4zm21with term izm2ρ4zn2into pair, (nm), in the denominator of n,m, (with totally rr1/2pairs), and transplant them into the denominator of n<m, and combine with izn2zm2iρ4zm2ρ4zn2in n<mto form a typical factor as a whole, (with just totally rr1/2pairs). Note that if we define

znρeδn+iβn,withδn>0,βn0π/2,E85

and further define that

zn2/ρ2=e2δn+i2βntanhΘn,E86

then the typical factor is

izn2zm2iρ4zm2ρ4zn2izn2ρ4zm2izm2ρ4zn2=zn2/ρ2zm2/ρ21zn2zm2/ρ42=tanh2ΘnΘmE87

and

Bn1n2nr=nfn2ρizn2ρ4zn2n<mtanh2ΘnΘm=nFnn<mtanh2ΘnΘmE88

here n,mn1n2nr, and a typical function Fnis defined as

FnBnn=fn2ρizn2ρ4zn2=2ρizn2ρ4zn2cntei2ΛnxE89

where use is made of formula (60) and (62), the time dependence of the solution naturally emerged in cnt. Substituting Eq. (88) into (83) thus completes the computation of D¯N.

Secondly, let us calculate detI+D. By use of (72) and (73), Binet-Cauchy formula, (Appendix A.2) and an important matrix formula, (Appendix A.3), we have

detI+D=1+r=12N1n1<n2<<nr2NDn1n2nr,E90

where Dn1n2nris the principal minor of a r,th-order submatrix of Dconsisting of elements belonging to not only rows n1n2nrbut also columnsn1n2nr, and

Dn1n2nr=n,mfnzn2ρ4zm2fmn<mzn2zm2ρ4zm2ρ4zn2E91

n,mn1n2nr. Using the same tricks as used in dealing with (84) leads to

Dn1n2nr=n1rfn2ρizn2ρ4zn2n<mzn2zm2ρ4zm2ρ4zn2zn2ρ4zm2zm2ρ4zn2
=nρzn2Fnn<mtanh2ΘnΘmE92

Thirdly, let us calculate detI+DfTgdetI+Bwith BDfTg.

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

detI+DfTg=detI+B=1+r=12N1n1<n2<<nr2NBn1n2nrE93
Bn1n2nr=n,mznzmλnρ2λmfnρizn2ρ4zm2fmn<mizn2zm2iρ4zm2ρ4zn2E94

n,mn1n2nr. Using the same tricks as that used in treating (84) leads to

Bn1n2nr=nznρ2zn2·fn2ρizn2ρ4zn2n<mtanh2ΘnΘm
=nfn2ρizn2ρ4zn2n<mtanh2ΘnΘm=nFnn<mtanh2ΘnΘmBn1n2nrE95

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 detI+D¯=detI+B

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

zn̂ρ2zn1¯E96

then

z2n=z2n1=ρ2z2n11¯,z2n1=z2n̂=ρ2z2n1¯E97

or

2n̂=2n1,2n1̂=2n,n=12NE98

Then the sequence of poles (25) is just in the same order as follows

z2̂,z1̂;z4̂,z3̂;;z2N,z2N1E99

On the other hand, due to (28), (62), and (73), we have

Dnm=ρ2znzm·fnρizn2ρ4zm2fmE100

Then

Dnm¯=ρ2zn¯zm¯·4c¯nc¯m·ρiz¯n2ρ4zm2¯·eiηn¯λn¯+ηm¯λm¯xE101

Substituting z¯n=ρ2zn̂1,z¯m=ρ2zm̂1into above formula and using following relation

η¯nλ¯n=η¯n̂λ¯n̂,c¯n=ρ2zn̂2cn̂E102

We can get an important relation between Dnmand Bm̂n̂

D¯nm=fm̂ρizm̂2ρ4zn̂2fn̂=Bm̂n̂=Bn̂m̂TE103

On the other hand, an unobvious symmetry between matrices Bnm2N×2Nand Bn̂m̂2N×2Nis found

diagσ1σ12N×2NBnm2N×2Ndiagσ1σ12N×2N=Bn̂m̂2N×2NE104

It can be rewritten in a more explicit form

σ1000σ10σ1B11B12B1,2NB21B22B2,2NB2N,1B2N,2B2N,2Nσ1000σ10σ1=B22B21B2,2NB2,2N1B12B11B1,2NB1,2N1B2N,2B2N,1B2N,2NB2N,2N1B2N1,2B2N1,1B2N1,2NB2N1,2N1=B1̂1̂B1̂2̂B1̂2N^B2̂1̂B2̂2̂B2̂,2N^B2N1^,1̂B2N1^,2̂B2N1^,2N^B2N^,1̂B2N^,2̂B2N^,2N^E105

The last equation in (105) is due to (97) and (99), thus from (103) and (104), we have

σ100σ1I+B2N×2Nσ100σ1=I+D¯TE106

The determinants of matrices at both sides of (106) are equal to each other

detσ1NdetI+Bdetσ1N=detI+D¯T=detI+D¯E107

The left hand of (107) is just detI+B, and this completes verification of identity (82). From the verified (82), we know multi-soliton solution (80) is surely of a typical form as expected.

3.3 The explicit N-soliton solution to the DNLS+ equation with NVBC

In order to get an explicit N-soliton solution to the DNLS+Eq. (1) with NVBC, firstly we need to make an inverse Galileo transformation of (2) by xtxρ2ttin Fnxtin (89). Due to (51) and (85)(87), the typical soliton kernel function Fncan be rewritten as

Fn=2ρizn2ρ4zn2bn0ȧznexpi2Λnx2λn2+ρ2tE108
Fnexpθn+iφnE109
θnxt=ρ2sin2βnch2δnxxn0ρ22+cos2βnch4δnch2δntνnxυntxn0E110
φnxt=ρ2cos2βnsh2δnxρ22+cos4βnch2δncos2βnt+φn0μnxξnt+φn0E111
μn=ρ2cos2βnsh2δn,νn=ρ2sin2βnch2δnE112
υn=ρ22+ch4δncos2βnch2δn,ξn=ρ22+ch2δncos4βncos2βnE113
2ρizn2ρ4zn2bn0ȧznexpνnxn0expiφn0E114
tanhΘnΘm=shδnδmcosβnβm+ichδnδmsinβnβmshδn+δmcosβn+βm+ichδn+δmsinβn+βmE115

where in (114) the n’th pole-dependent constant factor has been absorbed by redefinition of the n’th soliton center and initial phase in (110)(111).

Secondly, we need to calculate determinant CN=detI+AdetI+BfTg. According to the definition of Ain (68)(69), using Binet-Cauchy formula, (Appendix (A.2)), leads to

CN=detI+A=detI+BfTg=1+r=12N1n1<n2<<nr2NAn1n2nrE116

where An1n2nris the determinant of a r,th-order minor of Aconsisting of elements belonging to not only rows n1n2nrbut also columnsn1n2nr.

An1n2nf=n,mznzmλnρ2λmfnρizn2ρ4zm2fmn<mizn2zm2iρ4zm2ρ4zn2E117

n,mn1n2nr. Using the same tricks as used in dealing with (84) leads to

An1n2nr=nzn2ρ2fn2ρizn2ρ4zn2n<mtanh2ΘnΘm=nFntanhΘnn<mtanh2ΘnΘmE118

n,mn1n2nr. Substituting (108)(115) into (88) and (83) gives the explicit values of DN¯detI+Band DN. Substituting (118) into (116) then completes calculation of CNin (81). In the end, by substituting (83) and (116) into (80), we thus attain an explicit breather-type N-soliton solution of the DNLS+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 N-soliton solution are just similar to that of the pure N-soliton solution in Ref. [7] and thus not discussed here, but it should be emphasized that in the limit of t±, the N-soliton solution surely can be viewed as summation of Nsingle solitons with a definite displacement and phase shift of each soliton in the whole process of elastic collisions.

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 Nsoliton formula.

In the case of one-soliton solution, N=1,z1ρeδ1eiβ1, z2=ρ2z¯11=ρeδ1eiβ1, and δ1>0,β10π/2, using formula (82), (88), (116), (108)(115), and

c10=b10ȧz1=b10z12z¯122z1z12z¯22z12z22z1z2z¯1z¯2E119
c20=b20ȧz2=b20z22z¯222z2z22z¯12z22z12z1z2z¯1z¯2E120

we have

D¯1=1+Bn1=1+Bn1=2+Bn1=1n2=2=1+F1+F2+F1F2tanh2Θ1Θ2=1+sin2β1sinh2δ1eiβ1eθ1eδ1eiφ1+eδ1eiφ1ei2β1e2θ1E121
C1=1+An1=1+An1=2+An1=1n2=2=1+F1tanhΘ1+F2tanhΘ2+F1F2tanhΘ1tanhΘ2tanh2Θ1Θ2=1+sin2β1sinh2δ1ei3β1eθ1e3δ1eiφ1+e3δ1eiφ1ei6β1e2θ1E122

where not as that in (114), we define

b10ei2λ1η1x2λ12teθ1eiφ1,b10=eν1x10eiφ10E123
b20ei2λ2η2x2λ22teθ2eiφ2E124
θ1xtν1xυ1tx10E125
φ1xtμ1xξ1t+φ10,E126
withμ1=ρ2cos2β1sh2δ1,ν1=ρ2sin2β1ch2δ1,andE127
υ1=ρ22+ch4δ1cos2β1ch2δ1,ξ1=ρ22+ch2δ1cos4β1cos2β1E128
θ2=θ1,φ2=φ1E129

It is different slightly from the definition in Eq. (114) for the reason that an additional minus sign “” before b20can support (131)(133) due to b20=b¯10. Substituting (121)(122) into the following formula gives the one-soliton solution of DNLS+Eq. (1) with NVBC.

u¯1xt=ρC1D1/D¯12,oru1xt=ρC¯1D¯1/D12,E130

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

Figure 2.

The evolution of one-breather solution in time and space.

Formula (130) includes the one-soliton solution of the DNLS equation with VBC as its limit case. In the limit of ρ0,δ1but an invariant ρeδ1, we have

ρC14λ1sin2β1ei3β1eθ1eiφ1E131
D¯11ei2β1e2θ1,andD11ei2β1e2θ1E132

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

u1xt=4λ1sin2β1ei3β1eθ1eiφ11ei2β1e2θ1/1ei2β1e2θ12E133

If we redefine z1ρeδ1eiπ/2β1,z2ρeδ1eiπ/2β1, then u¯1xt=q1xt, the complex conjugate of one-soliton solution (133), completely reproduce the one-soliton solution that gotten in [17, 18, 19, 20, 23], under the VBC limit with ρ0,δ1, but ρeδ1=2λ1invariant, up to a permitted global constant phase factor. This verifies the validity of our formula of N-Soliton solution and the reliability of the newly revised inverse scattering transform.

The degenerate case for N=1,or the so-called pure one soliton solution, is also a typical illustration of the present improved IST. It can be dealt with by letting δ10. The simple poles z1=ρeiβ1and z2=ρ2z¯11=ρeiβ1are coincident, so do z3=z1and z4=ρeiβ1. Meanwhile μ10,φ10, ν1=ρ2sin2β1, ib10R. Especially for the degenerate case, we have

az=z2z12z2z12¯z¯1z1,c10=b10ȧz1=b10z12z¯122z1z1z¯1E134
ib10ei2λ1η1x2λ12tεeθ1,θ1xtν1xυ1tx10E135

with ν1=ρ2sin2β1, υ1=ρ21+2cos2β1, ε=sgnib10. Then we have

D¯=1+εeiβ1eθ1,orD=1+εeiβ1eθ1;C=1+z12F1/ρ12=1+εei3β1eθ1E136
u¯1xt=ρC1D1D¯12=ρ1+εei3β1eθ11+εeiβ1eθ11+εeiβ1eθ12=ρ14εsin2β1eθ1eiβ1+eθ1eiβ1+2εE137

where ε=11corresponds to dark (bright) soliton. Similarly if we redefine that β1π/2β1, then solution (137) is just the same as that gotten in [4, 5, 11, 12, 16] and called one-parameter pure soliton. This further convinces us of the validity and reliability of the newly revised IST for NVBC.

In the case of breather-type two-soliton solution, N=2, we define that

z1ρeδ1eiβ1,z2=ρeδ1eiβ1,z3ρeδ3eiβ3,z4=ρeδ3eiβ3E138
Fjeθjeiφj,j=1,2,3,4E139

which is just the same as that defined in (108)(115), the pole zj-related constant complex factor is absorbed into the j’th soliton center and the initial phase. Using formula (80)(82), (88), (116), (108)(115), we have

u¯2xt=ρC2D2/D¯22,oru2xt=ρC¯2D¯2/D22E140
D¯2=detI+B=1+Bn1=1+Bn1=2+Bn1=3+Bn1=4+Bn1=1n2=2+Bn1=1n2=3+Bn1=1n2=4+Bn1=2n2=3+Bn1=2n2=4+Bn1=3n2=4+Bn1=1n2=2n3=3+Bn1=1n2=2n3=4+Bn1=1n2=3n3=4+Bn1=2n2=3n4=4+Bn1=1n2=2n3=3n4=4=1+F1+F2+F3+F4+F1F2tanh2Θ1Θ2+F1F3tanh2Θ1Θ3+F1F4tanh2Θ1Θ4+F2F3tanh2Θ2Θ3+F2F4tanh2Θ2Θ4+F3F4tanh2Θ3Θ4+F1F2F3tanh2Θ1Θ2tanh2Θ1Θ3tanh2Θ2Θ3+F1F2F4tanh2Θ1Θ2tanh2Θ1Θ4tanh2Θ2Θ4+F1F3F4tanh2Θ1Θ3tanh2Θ1Θ4tanh2Θ3Θ4+F2F3F4tanh2Θ2Θ3tanh2Θ2Θ4tanh2Θ3Θ4+F1F2F3F4tanh2Θ1Θ2tanh2Θ1Θ3tanh2Θ1Θ4·tanh2Θ2Θ3tanh2Θ2Θ4tanh2Θ3Θ4E141

Similarly we can attain C2from (116) and (118) as follows

C2=detI+A=1+F1tanΘ1+F2tanΘ2+F3tanΘ3+F4tanΘ4+F1F2tanhΘ1tanhΘ2tanh2Θ1Θ2+tanhΘ1tanhΘ3F1F3tanh2Θ1Θ3+F1F4tanhΘ1tanhΘ4tanh2Θ1Θ4+F2F3tanhΘ2tanhΘ3tanh2Θ2Θ3+F2F4tanhΘ2tanhΘ4tanh2Θ2Θ4+F3F4tanhΘ3tanhΘ4tanh2Θ3Θ4+F1F2F3tanhΘ1tanhΘ2tanhΘ3tanh2Θ1Θ2tanh2Θ1Θ3tanh2Θ2Θ3+F1F2F4tanhΘ1tanhΘ2tanhΘ4tanh2Θ1Θ2tanh2Θ1Θ4tanh2Θ2Θ4+F1F3F4tanhΘ1tanhΘ3tanhΘ3tanh2Θ1Θ3tanh2Θ1Θ4tanh2Θ3Θ4+F2F3F4tanhΘ2tanhΘ3tanhΘ4tanh2Θ2Θ3tanh2Θ2Θ4tanh2Θ3Θ4+F1F2F3F4tanhΘ1tanhΘ2tanhΘ3tanhΘ4tanh2Θ1Θ2tanh2Θ1Θ3tanh2Θ1Θ4·tanh2Θ2Θ3tanh2Θ2Θ4tanh2Θ3Θ4E142

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 t±, the breather-type two-soliton is asymptotically decomposed into two breather-type 1-solitons.

Figure 3.

Evolution of the square amplitude of a breather-type two-soliton with respect to time and space ρ=2; δ1=0.4; δ3=0.6; β1=π/5.0; β3=π/2.2; x10=0; x20=0 =0; x30=0; x40=0; φ10=0; φ20=0; φ30=0; φ40=0.

4.1 Explicit pure N-soliton solution to the DNLS+ equation with NVBC

When all the simple poles are on the circle Oρcentered at the origin O, just as shown in Figure 4, our revised IST for DNLS+ equation with NVBC will give a typical pure N-soliton solution. The discrete part of azis of a slightly different form from that of the case for breather-type solution, and it can be expressed as

Figure 4.

Integral contour as all poles are on the circle of radiusρ.

az=n=1Nz2zn2z2zn2¯zn¯zn;ȧzn=2znzn2z¯n2z¯nznm=1,mnNzn2zm2zn2z¯m2z¯mzmE143

At the zeros of az, we have

ϕxzn=bnψxzn,ȧzn=ȧzn,b¯n=bnE144

On the other hand, the zeros of azappear in pairs and can be designed by zn, (n=1,2,,N), in the I quadrant, and zn+N=znin the III quadrant. The Zakharov-Shabat equation for pure soliton case of DNLS+ equation under reflectionless case can be derived immediately

ψ˜1xz=eiΛx+λn=1N2zλn1z2zn2cnψ1xzneiΛnxeiΛxE145
ψ˜2xz=z1eiΛx+λn=1N2znλn1z2zn2cnψ2xzneiΛnxeiΛxE146

Here Λ=κλ,Λn=κnλn; Letting z=ρ2zm1, m=1,2,,N, then

ψ1xzm=zm1eiΛmx+n=1Nλmcnλnzm2·2ρ3iρ4zm2zn2ψ1xzneiΛn+ΛmxE147
ψ2xzm=eiΛmx+n=1Nλmzncnλnzm·2ρiρ4zm2zn2ψ2xzneiΛn+ΛmxE148

Different from that in breather-type case, we define znρeδneiβn=ρeiβn, with βn0π/2,δn=0, i=12N, specially we have

cn0=bn0/ȧzn=ibn0ρsin2βneiβnk=1;knNsinβn+βksinβnβkE149
tanh2ΘnΘm=sin2βnβm/sin2βn+βmE150

An inverse Galileo transformation xtxρ2ttchanges Fnxtinto

Fnfn2ρizn2ρ4zn2=2ρizn2ρ4zn2cnoei4λn3κntei2Λnx=ibn0k=1;knNsinβn+βksinβnβkei2Λnx2λ2+ρ2t+iβnE151

Due to b¯n0=bn0,ibn0R, following equations hold:

Fn=eiβnk=1;knNsinβn+βksinβnβkibn0ei2Λnx2λn2+ρ2tεnEneiβnexpθn+iφnE152

where

ibn0ei2Λnx2λn2+ρ2tεnexpθn+iφnE153
θnxtνnxυntxn0,φn=0,E154
εnsgnibn0;ibn0eνnxn0E155
νn=ρ2sin2βn,υn=ρ21+2cos2βnE156
Enk=1;knNsinβn+βksinβnβkE157

where Enis also a real constant which is only dependent upon the order number n. The constant and positive real number ibn0has been absorbed by redefinition of the n’ th soliton center xn0in (155). Thus the determinants in formula (83) for pure soliton solution can be calculated as follows

D¯NdetI+B=1+r=1N1n1<<nrNBn1n2nrE158
Bn1n2nr=nfn2ρizn2ρ4zn2n<mizn2zm2iρ4zm2ρ4zn2izn2ρ4zm2izm2ρ4zn2=nFnn<mtanh2ΘnΘm=nεnEneiβneθnn<msin2βnβmsin2βn+βm;n,mn1n2nrE159
CN=detI+A=1+r=1N1n1<n2<<nrNAn1n2nrE160
An1n2nf=nzn2ρ2Fnn<mtanh2ΘnΘm
=nεnEnei3βneθnn<msin2βnβmsin2βn+βm;n,mn1n2nrE161

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

u¯NρCNDN/D¯N2oruNρC¯ND¯N/DN2E162

The N=2case, that is, the pure two-soliton is also a typical illustration of the general explicit N-soliton formula. According to (158)(162), it can be calculated as follows

D¯2=1+B1+B2+B12=1+ε1E1eiβ1eθ1+ε2E2eiβ2eθ2+ε1ε2E1E2eiβ1+β2eθ1+θ2sin2β1β2/sin2β1+β2=1+ε1sinβ1+β2sinβ1β2eiβ1eθ1ε2sinβ1+β2sinβ1β2eiβ2eθ2ε1ε2eiβ1+β2eθ1+θ2E163
C2=1+A1+A2+A12=1+ε1sinβ1+β2sinβ1β2ei3β1eθ1ε2sinβ1+β2sinβ1β2ei3β2eθ2ε1ε2ei3β1+β2eθ1+θ2E164
u¯2xt=ρC2D2/D¯22E165

The evolution of pure two-soliton solution with respect to time and space is given in Figure 5. It clearly demonstrates the whole process of the elastic collision between pure two solitons. If 0<β2<β1<π/2, then ε1=1,sgnE1=1and ε2=1,sgnE2=1correspond to double-dark pure 2-soliton solution as in Figure 5a; ε1=1,sgnE1=1and ε2=1,sgnE2=1correspond to a double-bright pure 2-soliton solution in Figure 5c; ε1=1,sgnE1=1and ε2=1,sgnE2=1correspond to a dark-bright-mixed pure 2-soliton solution in Figure 5b. In the limit of infinite time t±, the pure 2-soliton solution is asymptotically decomposed into two pure 1-solitons.

Figure 5.

Evolution of pure two soliton solution in time and space. (a) dark-dark pure 2-soilton, (b) dark-bright pure 2-soilton, and (c) bright-bright pure 2-soilton.

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=1,2,,N), is also absorbed into the n’th soliton center xn0just like ibn0does in (152)(154), and replace βnwith βn=π/2βn0π/2, the result for the pure soliton case in this section will reproduce the solution gotten in Refs. [7, 9, 16].

On the other hand, letting only part of the poles converge in pairs on the circle in Figure 1 and rewriting the expression of anzas in Ref. [7, 8, 12], our result can naturally generate the mixed case with both pure and breather-type multi-soliton solution.

4.2 The asymptotic behaviors of the N-soliton solution

Without loss of generality, we assume β1>β2>>βn>>βN; υ1<υ2<<υn<<υNin (156), and define the n’th neighboring area as ϒn:xxnoυnt0,n=1.2N. In the neighboring area of ϒn,

θj=νjxxj0υjt{,forj<n+,forj>nE166
D¯B12n1+B12n1nE167
CA12n1+A12n1nE168

where

B12n=εnEneθniβnj=1n1sin2βjβnsin2βj+βnB12n1E169
A12n1n=εnEneθn+i3βnj=1n1sin2βjβnsin2βj+βnA12n1E170

In the neighboring area of ϒn, we have

uu1θn+ΔθnE171

With

Δθn=2j=1n1lnsinβjβnsinβj+βnE172

As t, the Nneighboring areas queue up in a descending series ϒN,ϒN1,,ϒ1, then

uNn=1Nu1θn+ΔθnE173

the N-soliton solution can be viewed as Nwell-separated exact pure one solitons, each u1θn+Δθn, (1,2,,n) is a single pure soliton characterized by one parameter βn, moving to the positive direction of the x-axis, queuing up in a series with descending order number n.

As t, in the neighboring area of ϒnwe have

θj=νjxxj0υjt{+,forj<n,forj>nE174
D¯Bnn+1N+Bn+1n+2NE175
CAnn+1N+An+1n+2NE176

where

Bnn+1N=εnEneθniβnj=n+1Nsin2βjβnsin2βj+βnBn+1n+2NE177
Ann+1N=εnEneθn+i3βnj=n+1Nsin2βjβnsin2βj+βnAn+1n+2NE178
uu1θn+Δθn+E179
Δθn+=2n+1Nlnsinβj+βnsinβjβnE180
uNn=1Nu1θn+Δθn+E181

That is, the N-soliton solution can be viewed as Nwell-separated exact pure one solitons, queuing up in a series with ascending order number n such as ϒ1,ϒ2,,ϒN..

In the process of going from tto t, the n’th pure single soliton overtakes the solitons from the 1’th 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/νnfrom exceeding those slower soliton from the 1’th to n1’th, got a total backward shift Δθn+/νnfrom 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, that is,

Δxn=Δθn+Δθn/νnE182

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+equation with NVBC and normal dispersion, the rigorously proved breather-type N-soliton solution to the DNLS+equation with NVBC has been derived by use of some special linear algebra techniques. The one- and two-soliton solutions have been given as two typical examples in illustration of the unified formula of the N-soliton solution and the general computation procedures. It can perfectly reproduce the well-established conclusions for the special limit case. On the other hand, letting part/all of the poles converge in pairs on the circle in Figure 4 and rewriting the expression of anzas in [7, 12, 13], can naturally generate the partly/wholly pure multi-soliton solution. Moreover, the exact breather-type multi-soliton solution to the DNLS+equation can be converted to that of the MNLS equation by a gauge-like transformation [17].

Finally, the elastic collision among the breathers of the above multi-soliton solution has been demonstrated by the case of a breather-type 2-soliton solution. The newly revised IST for DNLS+equation with NVBC and normal dispersion makes corresponding Jost functions be of regular properties and asymptotic behaviors, and thus supplies substantial foundation for its direct perturbation theory.

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]:

iut+uxx+iu2u¯x=0E183

and the other is called Chen-Lee-Liu (CLL) equation [30]:

ivt+vxx+ivv¯vx=0E184

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

v=uexpi2xu2dxE185

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 q0=0as the seed solution of the DT of the KN Eq. (183), then after 2-fold DT, a new solution will be gotten under VBC:

q2=4iαβiα1cosh2Γ+β1sinh2Γ3α12β12cosh2Γ2+β122E186

(where all the parameters are defined in Ref. [38]). Similarly, setting a seed solution q0=cexpiax+c2+aat, a plane-wave solution to Eq. (183), will generate a new solution after 2-fold DT under a plane-wave background. Therefore, there is no need to discuss the boundary conditions or background when applying DT to solve those nonlinear integrable equations. This makes DT the most effective and prevailing method in obtaining a rogue wave solution.

Compared with DT, the IST has its fatal flaw that the difficulty of dealing with the boundary condition is unavoidable, which limits the possible application of the IST. Although the KN equation has been solved theoretically by means of an improved IST for both VBC and the NVBC [7, 8, 9, 17, 18, 19, 20], there is no report that the KN equation could be solved under a plane-wave background by means of IST. And consequently, it appears that rogue wave solutions cannot be obtained through the IST method. This major problem is caused by the difficulty of finding appropriate Jost solutions under the plane-wave background.

On the other hand, the Hirota’s bilinear-derivative transform (HBDT for brevity) [42, 43, 44, 45, 46], though not as a prevalent method as DT, has its particular advantages. The core of this method is a bilinear operator Dwhich is defined by:

DtnDxmABttnxxmAxtBxtt=t;x=xE187

where, at the left side of the above formula, a dot between two functions Axtand Bxtrepresents an ordered product. The HBDT method is very useful in dealing with periodic solutions for its convenience in computing the bilinear derivatives of an exponential function [44]:

FDxDyDtexpkx+ly++ωtexpkx+ly++ωt=Fkkllωωexpk+kx+l+ly++ω+ωtE188

Here, Frepresents general function expressed by the finite or infinite power series expansion of the Hirota’s bilinear differential operators. Formula (188) is the generalization of Appendix 5.5. Thus using HBDT method to find space periodic solutions of KN equation is practicable. The space periodic solutions possess the characters that they approach the plane-wave solution when tand are periodic in space. The first space periodic solution was found by Akhmediev with one parameter [45]. Actually, we can regard the space periodic solution as a special Akhmediev breather with a pure complex-valued wave number. Further, through a space periodic solution, a rogue wave solution can be constructed. This means besides DT, HBDT method is also an alternative and effective way to find rogue wave solution of KN equation.

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 uxthas a typical standard form [6, 7]:

uxt=gf¯/f2E189

where fand gare complex auxiliary functions needed to be determined. Applying the bilinear derivative transform to (189), we can rewrite the derivatives of uxtin the bilinear form [19, 20, 42, 43, 44, 45, 46]:

ut=ff¯DtgfgfDtff¯/f4E190
uxx=ff¯Dx2gf2DxgfDxff¯+gfDx2ff¯2gf¯Dx2ff/f4E191
u2ux=2gg¯Dxgf+g2Dxg¯f/f4E192

Directly substituting the above Eqs. (190)(192) into (183) gives:

ff¯iDt+Dx2gfgfiDt+Dx2ff¯+f2Dxf3g2Dxff¯igg¯=0E193

Then the above transformed KN equation can be decomposed into the following bilinear equations:

iDt+Dx2λgf=0E194
iDt+Dx2λff¯=0E195
Dxff¯=igg¯/2E196

where λis a constant which needs to be determined. Notice that if λ=0then the above bilinear equations are overdetermined because we have only two variables but three equations. Actually, setting λ=0is the approach to search for the soliton solution of the DNLS equation under vanishing boundary condition [19, 20]. Here, we set λas a nonzero constant to find solutions under a different boundary condition – a plane-wave background.

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 fand gin (189) are cut off, up to the 2’th power order of ϵ, and have the following formal form:

f=f01+ϵf1+ϵ2f2;g=g01+ϵg1+ϵ2g2E197

Substituting fand ginto Eqs. (194)(196) yields a system of equations at the ascending power orders of ϵ, which allows for determination of its coefficients [14, 19, 20]. We have 15 equations [14, 19, 20] corresponding to the different orders of ϵ. After solving all the equations, then we can obtain the solution of the DNLS equation:

u1xt=f¯1g1/f12E198

with

g1=ρeiωt1+a1epx+Ωt+ϕ0+a2epx+Ω¯t+ϕ¯0+Ma1a2eΩ+Ω¯t+ϕ0+ϕ¯0E199
f1=eiβx1+b1epx+Ωt+ϕ0+b2epx+Ω¯t+ϕ¯0+Mb1b2eΩ+Ω¯t+ϕ0+ϕ¯0E200

where

ω=3ρ4/16;β=ρ2/4E201
a1=b12Ω+2ip2pρ22Ω2ip2pρ2;a2=b22Ω¯+2ip2+pρ22Ω¯2ip2+pρ2E202
b2=b¯1Ω¯+ip2pρ2Ω¯ip2pρ2;M=1+4p4Ω+Ω¯2E203

Notice that ρand Mare real; b1and φ0are complex constants, so there are two restrictions for a valid calculation: (1) the wave number pmust be a pure imaginary number; (2) the angular frequency Ωmust not be purely imaginary number and must furthermore satisfy the quadratic dispersion relation:

4Ω2+4pρ2Ω+4p4+3p2ρ4=0E204

According to the test rule for a one-variable quadratic, there is a threshold condition under which Ωwill not be a pure imaginary number:

2p4+p2ρ4<0E205

The asymptotic behavior of this breather is apparent. Because the wave number p is a pure imaginary number, the breather is a periodic function of x. The quadratic dispersion relation (204) permits the angular frequency Ωto have two solutions:

Ω+=pρ2+22p4+p2ρ4/2E206
Ω=pρ222p4+p2ρ4/2E207

If we set Ω=Ω+, because 2p4+p2ρ4>0, then twill lead to:

g1ρexpiωtE208
f1expiβxE209
u1ρexpi3βx+ωtE210

And twill lead to:

g1ρMa1a2exp22p4+p2ρ4+ϕ0+ϕ¯0+iωtE211
f1Mb1b2exp22p4+p2ρ4+ϕ0+ϕ¯0+iβxE212
u1ρexpi3βx+ωt+φE213

where φis the phase shift across the breather:

exp=a1a2/b1b2E214

and due to a1a2=b1b2, thus the above phase shift φis real and does not affect the module of the breather u1when t. As for the other choice Ω=Ω, further algebra computation shows the antithetical asymptotic behavior of g1, f1, and u1when t. In a nutshell, u1will degenerate into a plane wave.

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 p=iq, here qis a real value and q0, then the asymptotic expansion of the angular frequency Ωis:

Figure 6.

The space-time evolution of the module of the 1st order space periodic solution in (198) with p=i,ρ=2,b1=i and Ω=Ω+, complex constant φ0 is set to zero.

Ω=qρ2i+σ/2+Oq3E215

where σ=±2. For the sake of a valid form of the rogue wave solution, we need to set b1=1and φ0=0(of course, setting b1=1and eφ0=1is alright, all we need is to make sure that the coefficients of the q0and q1in the expansions of f1and g1are annihilated). Therefore, the expansions of g1and f1in terms of qare given by:

g1=q2eiωt87i+5σ+16x12ρ2+3i+σρ44x24ρ2tx8it+3ρ4t212i+σρ3+Oq3E216
f1=q2eiβx8i+σ+16xρ2+i+σρ44x24ρ2tx8it+3ρ4t24i+σρ4+Oq3E217

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

uRW=ρei3βx+ωtgf¯/f2E218

where

g=87i+5σ+16x12ρ2+3i+σρ44x24ρ2tx8it+3ρ4t2;
f=24i+σ+48xρ2+3i+σρ44x24ρ2tx8it+3ρ4t2.

Here ωand βare given by Eq. (201), ρis an arbitrary real constant. The module of rogue wave solution Eq. (218) is shown in Figure 7.

Figure 7.

The space-time evolution of the module of the rogue wave solution with ρ=1 and σ=2. The max amplitude is equal to 3 at the point x=−2t=−22/3.

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:

υcxt=g1/f1E219

where, g1, f1, and other auxiliary parameters are invariant and given by Eqs. (199)(203). The same procedures which are used to derive the rogue wave solution of the KN equation can be used to turn υcinto a rogue wave solution of the CLL equation:

υc,RW=ρeiβx+ωtg/fE220

which has the same parameters as uRW. And this solution υc,RWhas exactly the same form as the result given by ref. [46].

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 ϵ:

g=g01+ϵg1+ϵ2g2+ϵ3g3+ϵ4g4E221
f=f01+ϵf1+ϵ2f2+ϵ3f3+ϵ4f4E222

Similarly, substituting fand ginto the bilinear Eqs. (194)(196) leads to the 27 equations [14, 19, 20] corresponding to different orders of ϵ. Solving these equations is tedious and troublesome but worthy and fruitful. The results are expressed in the following form:

u2xt=f¯2g2/f22E223

with

g2=ρeiωt1+g1+g2+g3+g4E224
f2=eiβx1+f1+f2+f3+f4E225
β=ρ2/4;ω=3ρ4/16;λ=ρ4/16E226
g1=iaieϕi;f1=ibieϕiE227
g2=i<jMijaiajeϕi+ϕj;f2=i<jMijbibjeϕi+ϕjE228
g3=i<j<kTijkaiajakeϕi+ϕj+ϕk;f3=i<j<kTijkbibjbkeϕi+ϕj+ϕkE229
g4=Aa1a2a3a4eϕ1+ϕ2+ϕ3+ϕ4;f4=Ab1b2b3b4eϕ1+ϕ2+ϕ3+ϕ4E230

where i,j,k=1,2,3,4,and the above parameters and coefficients are given respectively by:

p2=p¯1;p4=p¯3;Ω2=Ω¯1;Ω4=Ω¯3E231
ϕi=pix+Ωit+ϕ0i;ai=bi2Ωi+2ipi2piρ2/2Ωi2ipi2piρ2E232
b2=b¯1Ω2+ip22+p2ρ2Ω2ip22+p2ρ2;b4=b¯3Ω4+ip42+p4ρ2Ω4ip42+p4ρ2E233
Mij=ΩipjΩjpi2+pi2pj2pipj2ΩipjΩjpi2+pi2pj2pi+pj2E234
Tijk=MijMjkMki;A=i<jMijE235

Of course, for a valid and complete calculation, we are faced with the same situation as the 1st-order breather: ρis real, b1,b3and all φ0iare complex constants. Certainly, each wave number pimust be a pure imaginary number and each angular frequency Ωihas to satisfy the quadratic dispersion relation:

Figure 8.

The space-time evolution of the module of the 2nd order space periodic solution with p1=0.4i,p3=0.75i,b1=i,b3=1 and ρ=1.6. Other phase factors φ1 and φ3 are set to zero.

4Ωi2+4piρ2Ωi+4pi4+3pi2ρ4=0,i=1234E236

And the threshold conditions for each complex-valued Ωishare the same form as Eq. (205):

2pi4+pi2ρ4<0E237

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 p3=p1. Under this condition, M13=M24=0, thus the higher order interaction coefficients Tijkand Awill vanish. Therefore, g2and f2will degenerate into the forms of g1and f1, respectively:

gp3=p12=g1=ρeiωt1+a1eϕ1+a2eϕ2+M12a1a2eϕ1+ϕ2E238
fp3=p12=f1=eiβx1+b1eϕ1+b2eϕ2+M12b1b2eϕ1+ϕ2E239

where b1=χb1,b2=χ¯b2,a1=χa1and a2=χ¯a2with χ=b1+b3/b1. That is how u2can be reduced to u1. Given to this reduction, a generalized form of these two breathers arises:

uN=f¯NgN/fN2;N=12E240
gN=ρeiωt1+r=12N1n1<<nr2NMn1nri=n1nraieϕiE241
fN=eiβx1+r=12N1n1<<nr2NMn1nri=n1nrbieϕiE242

where the coefficient M is defined by:

Mi=1E243
Mn1nr=i<jMij;i,jn1nrE244

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 p1and p3. In addition, its asymptotic behaviors are analogical to the 1st-order space periodic solution. Each quadratic dispersion equation has two roots, respectively:

Ω1±=p1ρ2±22p14+p12ρ4/2E245
Ω3±=p3ρ2±22p34+p32ρ4/2E246

Thus, we will have four combinations of Ω1and Ω2. Details are numerated in Table 1. The parameters φ0,φand φin Table 1 are the phase shifts which are all real so that they will not change the module of u2when t. And φis given in Eq. (214), and others are determined by:

Choice of ΩiΩ1+,Ω2+Ω1+,Ω2Ω1,Ω2+Ω1,Ω2
tρe3iβx+iωtρe3iβx+iωt+iφρe3iβx+iωt+ρe3iβx+iωt+iφ0
tρe3iβx+iωt+iφ0ρe3iβx+iωt+ρe3iβx+iωt+iφρe3iβx+iωt

Table 1.

Asymptotic behaviors of u2.

expiφ0=a1a2a3a4/b1b2b3b4E247
expiφ=a3a4/b3b4E248

From Table 1, we could draw the conclusion that this breather will also degenerate into the background plane wave as t. Furthermore, there is a phase shift across the breather from t=to t=, which depended on the choice of Ω1and Ω2.

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 pi, we are also interested in seeking an alternative method besides DT that could help us to determine the higher order rogue wave solutions.

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 A1and A2are N×1matrices, Ais a regular N×Nmatrix, then

A1TA1A2=detA+A2A1T/detA1EA1

A2, Binet-Cauchy formula: For a squared N×Nmatrix B

detI+B=1+r=1N1n1<n2<<nrNBn1n2nrEA2

whereBn1n2nris a r’th-order principal minor of B.

A3, For a N × N matrix Q1and a N × N matrix Q2,

detI+Q1Q2=1+r=1N1n1<n2<<nrNΩrn1n2nr

=1+r=1N1n1<<nrN1m1<<mrNQ1n1n2nrm1m2mrQ2m1m2mrn1n2nrEA3

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).

The above formula also holds for the case of detI+Ω1Ω2With Ω1to be a N×N+1matrix and Ω2a N+1×Nmatrix.

A4, For a squared matrix Cwith elements Cjk=fjgkxjyk1,

detC=jfjgjj<j,k<kxjxjykykj,kxjyk1EA4

A5, Some useful blinear derivative formulae.

ABx=DxA·BB2EA5
ABxx=Dx2A·BB2ABDx2B·BB2EA5.1
Dxab·cd=bdDxa·c+acDxb·d=bcDxa·d+adDxb·cEA5.2
Dx2ab·cd=bdDx2a·c+2Dxa·cDxb·d+acDx2b·dEA5.3
DtnDxmexpη1·expη2=Ω1Ω2nΛ1Λ2mexpη1+η2,EA5.4

where ηi=Ωit+Λix+η0ii=1,2; Ωi, Λi, η0iare complex constants.

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Zhou Guo-Quan (September 22nd 2020). Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation Part 2 [Online First], IntechOpen, DOI: 10.5772/intechopen.93450. Available from:

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