The typical examples for one- and two-soliton solutions

Y, Θ ! ffiffiffiffiffiffiffiffi

2. An N-soliton solution to the DNLS equation based on a revised inverse scattering transform

2.4 The typical examples for one- and two-soliton solutions

Then the typical soliton argumentsθ_nandφ_ncan be defined according to f²_n¼2cn0e^i2λ2nxe^i4λ4nt�2cn0e^�θne^iφn (65) whereλ�μ_nþiνn, andθ_n¼4μ_nν_n xþ4�μ²_n�ν²_n�

� t�

¼4κnðx�VntÞ;

φ_n¼2�μ²_n�ν²_n�

xþ 4�μ²_n�ν²_n�₂

�16μ²_nν²_n

h i

�t V_n¼ �4�μ²_n�ν²_n�

,κ_n¼4μ_nν_n (66)

2.3.3 Calculation of determinant of D and A

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

Q₁ðn1;n2;⋯;nr;m1;m2;⋯;mrÞQ₂ðm1;m2;⋯;mr;n1;n2;⋯;nrÞ

¼ �1ð Þ^rY

n,m

2cn

ð Þð2cmÞe^�θne^iφne^�θme^�iφm λ²_n λ²_n�λ²_m

� �2

Y

n<n⁰,m<m⁰

λ²_n�λ²_n0

� �2

λ²_m�λ²_m0

� �₂

(67) with n, n⁰∈fn₁, n₂,⋯, nrgand m, m⁰∈fm₁, m₂,⋯, mrg. 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 A in (52), we introduce a N�(N + 1) matrixΩ1and a (N + 1)�N matrixΩ2defined as

Ω1

ð Þ_nm¼� �B₁

nm¼ðQ₁Þ_nm,ð ÞΩ1 _n0¼f_n, Ω2

ð Þ_mn¼ð ÞB2 _mn¼ðQ₂Þ_mn,ð ÞΩ2 _0n¼f_n (68) with n, m¼1, 2,⋯, N. We thus have

det I� þB1B2þFF^T�

¼det Ið þΩ1Ω2Þ

¼1þX^N

r¼1

X

1≤n₁<⋯<nr≤N

X

0≤m₁<⋯<mr≤NΩ1ðn1;n2;⋯;nr;m1;m2;⋯;mrÞΩ2ðm1;m2;⋯;mr;n1;n2;⋯;nrÞ

(69) The above summation obviously can be decomposed into two parts: one is extended to m1= 0, the other is extended to m₁≥1. Subtracted from (69), the part that is extended to m₁≥1, the remaining parts of (69) is just A in (52) (with m₁¼0 and m₂≥1). Due to (68), we thus have

A¼det Ið þΩ1Ω2Þ �det Ið þQ₁Q₂Þ

¼X^N

r¼1

X

1≤n₁<⋯<nr≤N

X

1≤m2<⋯<mr≤N

Arðn1;n2;⋯;nr;0;m2;⋯;mrÞ

¼X^N

r¼1

X^N

1≤n1<n2<nr≤N

X

1≤m2<m3<⋯<mr≤N

Ω1ðn1;n2;⋯;nr;0;m2;⋯;mrÞΩ2ð0;m2;⋯;mr;n1;n2;⋯;nrÞ (70) with

Ω1ðn₁, n₂,⋯, n_r; 0, m₂,⋯, m_rÞΩ2ð0, m₂,⋯, m_r; n₁, n₂,⋯, n_rÞ

¼ �1ð Þ^r�1Y

n, m

f²_nf²_mλ²_m λ²_n�λ²_m

� �2

Y

n<n⁰, m<m⁰

λ²_n�λ²_n0

� �2

λ²_m�λ²_m0

� �₂

¼ �1ð Þ^r�1Y

n, mð2cnÞð2cmÞe^�θne^iφne^�θme^�iφm λ²_m λ²_n�λ²_m

� �₂ Y

n<n⁰, m<m⁰

λ²_n�λ²_n0

� �₂

λ²_m�λ²_m0

� �₂

(71) Here n, n⁰∈fn1, n2,⋯, nrgand especially m, m⁰∈fm2,⋯, mrg, which completes the calculation of determinant A in formula (52). Substituting the explicit expres- sions 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 con- stant global phase factor, there is also an undetermined constant complex parameter bn0before each of the typical soliton factor e^�θne^iφn, (n = 1,2, … , N). It can be absorbed into e^�θne^iφnby 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.

Ω1ðn₁, n₂,⋯, n_r; 0, m₂,⋯, m_rÞΩ2ð0, m₂,⋯, m_r; n₁, n₂,⋯, n_rÞ

¼ �1ð Þ^r�1Y

n, m

f²_nf²_mλ²_m λ²_n�λ²_m

� �2

Y

n<n⁰, m<m⁰

λ²_n�λ²_n0

� �2

λ²_m�λ²_m0

� �₂

¼ �1ð Þ^r�1Y

n, mð2cnÞð2cmÞe^�θne^iφne^�θme^�iφm λ²_m λ²_n�λ²_m

� �₂ Y

n<n⁰, m<m⁰

λ²_n�λ²_n0

� �₂

λ²_m�λ²_m0

� �₂

(71) Here n, n⁰∈fn1, n2,⋯, nrgand especially m, m⁰∈fm2,⋯, mrg, which completes the calculation of determinant A in formula (52). Substituting the explicit expres- sions 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 con- stant global phase factor, there is also an undetermined constant complex parameter bn0before each of the typical soliton factor e^�θne^iφn, (n = 1,2, … , N). It can be absorbed into e^�θne^iφn by redefinition of soliton center and its initial phase factor.

This kind of arbitrariness is in correspondence with the unfixed initial conditions of the DNLS equation.

2.4 The typical examples for one- and two-soliton solutions

We give two concrete examples – the one- and two-soliton solutions as illustra- tions of the general explicit soliton solution.

In the case of one-soliton solution, N = 1,λ₂¼ �λ₁,λ₁¼ρ₁e^iβ1 ¼μ₁þiν₁, and A1¼Ω1ðn1¼1; m1¼0ÞΩ2ðm1¼0; n1¼1Þ ¼ f²₁ (72) D1¼Q₁ðn1¼1; m1¼1ÞQ₂ðm1¼1; n2¼1Þ ¼1���f₁��⁴λ²₁=�λ²₁�λ²₁�2

(73) f²₁¼2c₁₀e^i2λ21xe^i4λ41t; c₁₀¼b₁₀=a_ð Þ; bλ_{1 10}¼e^4μ1ν1x10e^iα10; b₁₀e^i2λ21xe^i4λ41t�e^�θ1e^iφ1; θ₁¼4μ₁ν₁ x�x₁₀þ4�μ²₁�ν²₁�

� t�

;φ₁¼2�μ²₁�ν²₁�

xþ 4�μ²₁�ν²₁�2

�16μ₁ν²₁

h i

tþα₁₀ (74) It is different slightly from the definition in (66) for that here b10has been absorbed into the soliton center and initial phase. Then

A1¼λ₁�λ²₁�λ²₁�

e^�θ1e^iφ1=λ²₁¼i2ρ₁sin 2β₁e^i3β1e^�θ1e^iφ1

D₁¼1� λ²₁�λ²₁

��

�

��

�² λ₁ j j²

λ²₁ λ²₁�λ²₁

� �₂e^�2θ1 ¼1þe^i2β1e^�2θ1

and

u₁ðx, tÞ ¼ �i2A₁D₁=D²₁¼4ρ₁sin 2β₁e^i3β1�1þe^�i2β1e^�2θ1� 1þe^i2β1e^�2θ1

ð Þ² e^�θ1e^iφ1 (75)

The complex conjugate of one-soliton solution u1ðx, tÞin (75) is u1ðx, tÞ, 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,λ₃¼ �λ₁,λ₄¼ �λ₂and

λ₁¼ρ₁e^iβ1 ¼μ₁þiν1; λ₂¼ρ₂e^iβ2¼μ₂þiν2 (76) c₁₀¼ b₁₀

a_ð Þλ₁ ¼b₁₀λ²₁�λ²₁

2λ₁ �λ²₁�λ²₂ λ²₁�λ²₂�λ²₁

λ²₁�λ²₂ λ²₂ c20¼ b20

a_ð Þλ₂ ¼b20λ²₂�λ₂

2λ₂ �λ²₂�λ²₁ λ²₂�λ²₁�λ²₁

λ²₁�λ²₂ λ²₂

(77)

f²_j¼2cj0e^i2λ2jxe^i4λ4jt, j¼1, 2 C:fð :ð1:62ÞÞ, bj0e^{i2λ2jxþi4λ4jt}�e^�θje^iφj, j¼1, 2 (78) whereθ_j¼4μ_jν_j x�x_j0þ4�μ²_j�ν²_j�

h ti

φ_j¼2�μ²_j�ν²_j�

xþ 4�μ²_j�ν²_j�2

�16μ²_jν²_j

� �

�tþα_j0 (79) and bj0is absorbed into the soliton center and the initial phase by

bj0¼e^4μjνjxj0e^iαj0; j¼1, 2 (80) And we get

A2¼ X n1¼1, 2 m₁¼0

Ω1ðn1;0ÞΩ2ð0;n1Þ þ X n1¼1, n2¼2 m₁¼0, m2¼1, 2

Ω1ðn1;n2;0;m2ÞΩ2ð0;m2;n1;n2Þ

¼Ω1ð1;0ÞΩ2ð0;1Þ þΩ1ð2;0ÞΩ2ð0;2Þ þΩ1ð1;2;0;1ÞΩ2ð0;1;1;2Þ þΩ1ð1;2;0;2ÞΩ2ð0;2;1;2Þ

¼f²₁þf²₂���f₁��⁴f²₂ �λ²₁�λ²₂�2λ²₁ λ²₁�λ²₁

� �2

λ²₁�λ²₂

� �2���f₂��⁴f²₁ �λ²₁�λ²₂�2λ²₂ λ²₂�λ²₁

� �2

λ²₂�λ²₂

� �2

¼λ₁�1�e^�i4β1�λ²₁�λ²₂

λ²₁�λ²₂e^{i4ðβ1þβ2Þ}e^�θ1e^iφ1þλ₂�1�e^�i4β2�λ²₁�λ²₂

λ²₁�λ²₂e^{i4ðβ1þβ2Þ}e^�θ2e^iφ2

þ λ₁�1�e^�i4β1�e^�i2β2λ²₁�λ²₂

λ²₁�λ²₂e^{�θ2�iφ2}þλ₂�1�e^�i4β2�e^�i2β1λ²₁�λ²₂ λ²₁�λ²₂e^{�θ1�iφ1}

" #

•e^{�ðθ1þθ2Þ}e^{iðφ1þφ2Þ}e^{i4ðβ1þβ2Þ}

¼i2λ²₁�λ²₂ λ²₁�λ²₂

��

��

�

��

��

�½ρ₁sin 2β₁e^iðφ�αÞe^{i 3ðβ1þ4β2Þ}e^�θ1þiφ1þρ₂sin 2β₂e^{�iðφþαÞ}e^{i 4ðβ1þ3β2Þ}e^�θ2þiϕ2

þρ₁sin 2β₁e^{�iðφ�αÞ}e^{i 3βð 1þ2β2Þ}e^{�2θ2�θ1}�e^iϕ1þρ₂sin 2β₂e^iðφþαÞe^{i 2βð 1þ3β2Þ}e^{�2θ1�θ2}�e^iϕ2� (81) where

φ¼arg�λ²₁�λ²₂�

¼ arctan�ρ²₁sin 2β₁þρ²₂sin 2β₂�

=�ρ²₁cos 2β₁�ρ²₂cos 2β₂� α¼arg�λ²₁�λ²₂�

¼ arctan�ρ²₁sin 2β₁�ρ²₂sin 2β₂�

=�ρ²₁cos 2β₁�ρ²₂cos 2β₂� (82) and

D2¼1þX²

r¼1

X

1≤n1<n2≤2

X

1≤m1<m2≤2

Q₁ðn1;⋯;nr;m1;⋯;mrÞQ₂ðm1;⋯;mr;n1;⋯;nrÞ

¼1þQ₁ðn1¼1;m1¼1ÞQ₂ðm1¼1;n1¼1Þ þQ₁ðn1¼1;m1¼2ÞQ₂ðm1¼2;n1¼1Þ þQ₁ðn1¼2;m1¼1ÞQ₂ðm1¼1;n1¼2Þ þQ₁ðn1¼2;m1¼2ÞQ₂ðm1¼2;n1¼2Þ þQ₁ðn1¼1;n1¼2;m1¼1;m2¼2ÞQ₂ðm1¼1;m2¼2;n1¼1;n2¼2Þ

¼1���f₁��⁴ λ²₁ λ²₁�λ²₁

� �2���f₂��⁴ λ²₂ λ²₂�λ²₂

� �2�f²₁f²₂ λ²₁ λ²₁�λ²₂

� �2�f²₁f²₂ λ²₂ λ²₁�λ²₂

� �2

þ��f₁f₂��⁴�

λ²₁λ²₂�λ²₁�λ²₂�2

λ²₁�λ²₂

� �2

λ²₁�λ²₁

� �2

λ²₁�λ²₂

� �2

λ²₂�λ²₁

� �2

λ²₂�λ²₂

� �2

(83)

D₂¼1þ λ²₁�λ²₂ λ²₁�λ²₂

��

��

�

��

��

�

2

e^�i2β1e^�2θ1þe^�i2β2e^�2θ2

� �

þ 1� λ²₁�λ²₂ λ²₁�λ²₂

��

��

�

��

��

� 0 2

@

1

Ae^{�ðθ1þθ2Þ}e^{�iðβ1þβ2Þ} ρ₁

ρ₂e^{iðφ2�φ1Þ}þρ₂ ρ₁e^{iðφ1�φ2Þ}

� �

þe^{�i2ðβ1þβ2Þ}e^{�2ðθ1þθ2Þ}

(84)

where

λ²₁�λ²₂ λ²₁�λ²₂

��

��

�

��

��

�

2

¼ðρ₁=ρ₂�ρ₂=ρ₁Þ²þ4 sin²ðβ₁þβ₂Þ ρ₁=ρ₂�ρ₂=ρ₁

ð Þ²þ4 sin²ðβ₁�β₂Þ (85) Substituting (81) and (84) into formula (52), we thus get the two-soliton solu- tion to the DNLS equation with VBC

u2¼ �i2A2D2=D²₂ (86) 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 ofN-soliton solution

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

uN ¼i2ANDN=D²_N (87)

Without the loss of generality,forλ_n¼μ_nþivn, Vn¼ �4�μ²_n�v²_n� ,

n¼1, 2,⋯, N, we assume V₁<V₂<⋯<Vn<⋯VNand define the n’th vicinity area asΓn:x�xno�Vnt�0, nð ¼1:2,⋯, NÞ.

As t! �∞, N vicinity areasΓn, n¼1, 2… , N, queue up in a descending series ΓN,ΓN�1,⋯,Γ1 (88) and in the vicinity ofΓn, we have (note thatκ_j>0)

D2¼1þX²

r¼1

X

1≤n1<n2≤2

X

1≤m1<m2≤2

Q₁ðn1;⋯;nr;m1;⋯;mrÞQ₂ðm1;⋯;mr;n1;⋯;nrÞ

¼1þQ₁ðn1¼1;m1¼1ÞQ₂ðm1¼1;n1¼1Þ þQ₁ðn1¼1;m1¼2ÞQ₂ðm1¼2;n1¼1Þ þQ₁ðn1¼2;m1¼1ÞQ₂ðm1¼1;n1¼2Þ þQ₁ðn1¼2;m1¼2ÞQ₂ðm1¼2;n1¼2Þ þQ₁ðn1¼1;n1¼2;m1¼1;m2¼2ÞQ₂ðm1¼1;m2¼2;n1¼1;n2¼2Þ

¼1���f₁��⁴ λ²₁ λ²₁�λ²₁

� �2���f₂��⁴ λ²₂ λ²₂�λ²₂

� �2�f²₁f²₂ λ²₁ λ²₁�λ²₂

� �2�f²₁f²₂ λ²₂ λ²₁�λ²₂

� �2

þ��f₁f₂��⁴�

λ²₁λ²₂�λ²₁�λ²₂�2

λ²₁�λ²₂

� �2

λ²₁�λ²₁

� �2

λ²₁�λ²₂

� �2

λ²₂�λ²₁

� �2

λ²₂�λ²₂

� �2

(83)

D₂¼1þ λ²₁�λ²₂ λ²₁�λ²₂

��

��

�

��

��

�

2

e^�i2β1e^�2θ1þe^�i2β2e^�2θ2

� �

þ 1� λ²₁�λ²₂ λ²₁�λ²₂

��

��

�

��

��

� 0 2

@

1

Ae^{�ðθ1þθ2Þ}e^{�iðβ1þβ2Þ} ρ₁

ρ₂e^{iðφ2�φ1Þ}þρ₂ ρ₁e^{iðφ1�φ2Þ}

� �

þe^{�i2ðβ1þβ2Þ}e^{�2ðθ1þθ2Þ}

(84)

where

λ²₁�λ²₂ λ²₁�λ²₂

��

��

�

��

��

�

2

¼ðρ₁=ρ₂�ρ₂=ρ₁Þ²þ4 sin²ðβ₁þβ₂Þ ρ₁=ρ₂�ρ₂=ρ₁

ð Þ²þ4 sin²ðβ₁�β₂Þ (85) Substituting (81) and (84) into formula (52), we thus get the two-soliton solu- tion to the DNLS equation with VBC

u2¼ �i2A2D2=D²₂ (86) 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 ofN-soliton solution

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

uN¼i2ANDN=D²_N (87)

Without the loss of generality,forλ_n¼μ_nþivn, Vn¼ �4�μ²_n�v²_n� ,

n¼1, 2,⋯, N, we assume V₁<V₂<⋯<Vn<⋯VNand define the n’th vicinity area asΓn:x�xno�Vnt�0, nð ¼1:2,⋯, NÞ.

As t! �∞, N vicinity areasΓn, n¼1, 2… , N, queue up in a descending series ΓN,ΓN�1,⋯,Γ1 (88) and in the vicinity ofΓn, we have (note thatκ_j>0)

θ_j¼4κj�x�xj0�Vjt�

!n^þ∞,_�∞, ^{for j>}_{for j<}ⁿ_n

(89) Here the complex constant 2c_n0in expression (65) has been absorbed into e^�θne^iφn by redefinition of the soliton center xn0and the initial phaseα_n0.

Introducing a typical factor F_n¼ �e^�2θn=�λ²_n�λ²_n�2

>0, n¼1, 2,⋯, N; then

D_nð1, 2,⋯, nÞ ¼Yⁿ

j¼1

λ²_jF_jY

l<m

λ²_l �λ²_m λ²_l �λ²_m

��

��

�

��

��

�

4

(90)

where l, m∈f1, 2,⋯, ng. Thus

D≃Dn�1ð1, 2,⋯, n�1Þ þDnð1, 2,⋯, nÞ ¼ 1þλ²_nFn

Yn�1 j¼1

λ²_j�λ²_n λ²_j�λ²_n

��

��

��

��

��

�� 0 4

@

1

ADn�1ð1, 2,⋯, n�1Þ (91) and

A≃Anð1, 2,⋯, n; 0, 1, 2,⋯n�1Þ ¼Dn�1e^�θne^�iφnY^n�1

j¼1

λ²_j�λ²_n

� �₂

λ²_j�λ²_n

� �₂e^i4βj (92)

In the vicinity ofΓn,

u x, tð Þ ¼i2AD=D²≃u₁�θ_nþΔθ^{ð Þ}_n^�,φ_nþΔφ^{ð Þ}_n^��

(93) Here

Δθ^{ð Þ}_n^� ¼2X^n�1

j¼1

ln λ²_j�λ²_n λ²_j�λ²_n

��

��

�

��

��

�

2

(94)

Δφ^{ð Þ}_n^� ¼ �X^n�1

j¼1

arg �λ²_j�λ²_n�2

λ²_j�λ²_n

� �₂

2 64

3 75þ4β_j 8>

<

>:

9>

=

>;

¼2X^n�1

j¼1

arg�λ²_j�λ²_n�

�arg�λ²_j�λ²_n�

�2β_j

h i

(95)

then

uN≃X^N

n¼1

u1�θ_nþΔθ^{ð Þ}_n^�,φ_nþΔφ^{ð Þ}_n^��

(96) Each u₁ðθ_n,φ_nÞ, (1, 2,⋯, n) is a one-soliton solution characterized by one parameterλ_n, moving along the positive direction of the x-axis, queuing up in a series with descending order number n as in series (88). As t!∞, in the vicinity of Γn, we have (note thatκ_j>0)

θ_j¼4κ_j�x�xj0�Vjt�

!n^�∞,_þ∞, ^{for j}_{for j}^>n_<n

D≃DN�nðnþ1, nþ2,⋯, NÞ þDN�nþ1ðn, nþ1,⋯, NÞ

¼ 1þλ²_nFn Q^N

j¼nþ1 λ²_j�λ²_n λ²_j�λ²_n

��

��

��

��

4!

DN�nðnþ1, nþ2,⋯, NÞ (97) A≃AN�nþ1ðn, nþ1,⋯, N; 0, nþ1, nþ2,⋯, NÞ

¼DN�nðnþ1, nþ2,⋯, NÞe^�θne^�iφn Y^N

j¼nþ1

λ²_j�λ²_n

� �₂

λ²_j�λ²_n

� �2

λ²_j

λ²_j, (98)

So as t!∞, in the vicinity ofΓn,

u¼i2AD=D²≃u₁�θ_nþΔθ^{ð Þ}_n^þ,φ_nþΔφ^{ð Þ}_n^þ�

(99)

Δθ^{ð Þ}_n^þ ¼2 X^N

j¼nþ1

ln λ²_j�λ²_n λ²_j�λ²_n

��

��

�

��

��

�

2

(100)

Δφ^{ð Þ}_n^þ ¼ � X^N

j¼nþ1

arg �λ²_j�λ²_n�2

λ²_j�λ²_n

� �₂

2 64

3 75þ4β_j 8>

<

>:

9>

=

>;

¼2 X^N

j¼nþ1

arg�λ²_j�λ²_n�

�arg�λ²_j�λ²_n�

�2β_j

h i

, (101)

then as t!∞,

uN≃ X^N

n¼1

u1�θ_nþΔθ^{ð Þ}_n^þ,φ_nþΔφ^{ð Þ}_n^þ�

(102)

That is to say, the N-soliton solution can be viewed as N well-separated exact one- solitons, queuing up in a series with ascending order number n:Γ1,Γ2,⋯,ΓN: In the course going from t! �∞ to t!∞, the n’th one-soliton overtakes the solitons from the first to n�1’th and is overtaken by the solitons from nþ1’th to N’th. In the meantime, due to collisions, the n’th soliton got a total forward shift Δθ^{ð Þ}_n^�=κ_nfrom exceeding those slower soliton from the first to n�1’th, and got a total backward shiftΔθ^{ð Þ}_n^þ=κ_nfrom being exceeded by those faster solitons from nþ1’th to N’th, and just equals to the summation of shifts due to each collision between two solitons, together with a total phase shiftΔφ_n, that is,

Δxn¼��Δθ^{ð Þ}_n^þ �Δθ^{ð Þ}_n^���=κ_n (103) Δφ_n¼Δφ^{ð Þ}_n^þ �Δφ^{ð Þ}_n^� (104) 2.6N-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

θ_j¼4κ_j�x�xj0�Vjt�

!n^�∞,_þ∞, ^{for j>}_{for j<}ⁿ_n

D≃DN�nðnþ1, nþ2,⋯, NÞ þDN�nþ1ðn, nþ1,⋯, NÞ

¼ 1þλ²_nFn Q^N

j¼nþ1 λ²_j�λ²_n λ²_j�λ²_n

��

��

��

��

4!

DN�nðnþ1, nþ2,⋯, NÞ (97) A≃AN�nþ1ðn, nþ1,⋯, N; 0, nþ1, nþ2,⋯, NÞ

¼DN�nðnþ1, nþ2,⋯, NÞe^�θne^�iφn Y^N

j¼nþ1

λ²_j�λ²_n

� �₂

λ²_j�λ²_n

� �2

λ²_j

λ²_j, (98)

So as t!∞, in the vicinity ofΓn,

u¼i2AD=D²≃u₁�θ_nþΔθ^{ð Þ}_n^þ,φ_nþΔφ^{ð Þ}_n^þ�

(99)

Δθ^{ð Þ}_n^þ ¼2 X^N

j¼nþ1

ln λ²_j�λ²_n λ²_j�λ²_n

��

��

�

��

��

�

2

(100)

Δφ^{ð Þ}_n^þ ¼ � X^N

j¼nþ1

arg �λ²_j�λ²_n�2

λ²_j�λ²_n

� �₂

2 64

3 75þ4β_j 8>

<

>:

9>

=

>;

¼2 X^N

j¼nþ1

arg�λ²_j�λ²_n�

�arg�λ²_j�λ²_n�

�2β_j

h i

, (101)

then as t!∞,

uN≃X^N

n¼1

u1�θ_nþΔθ^{ð Þ}_n^þ,φ_nþΔφ^{ð Þ}_n^þ�

(102)

That is to say, the N-soliton solution can be viewed as N well-separated exact one- solitons, queuing up in a series with ascending order number n:Γ1,Γ2,⋯,ΓN: In the course going from t! �∞ to t!∞, the n’th one-soliton overtakes the solitons from the first to n�1’th and is overtaken by the solitons from nþ1’th to N’th. In the meantime, due to collisions, the n’th soliton got a total forward shift Δθ^{ð Þ}_n^�=κ_nfrom exceeding those slower soliton from the first to n�1’th, and got a total backward shiftΔθ^{ð Þ}_n^þ=κ_nfrom being exceeded by those faster solitons from nþ1’th to N’th, and just equals to the summation of shifts due to each collision between two solitons, together with a total phase shiftΔφ_n, that is,

Δxn¼��Δθ^{ð Þ}_n^þ �Δθ^{ð Þ}_n^���=κ_n (103) Δφ_n¼Δφ^{ð Þ}_n^þ �Δφ^{ð Þ}_n^� (104) 2.6N-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

i∂_tυþ∂_xxυþiα∂_xj jυ²υ

þ2β υj j²υ¼0 (105) 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 [4–6] and is related to DNLS equation by a gauge-like transformation [23] formulated as

υðx, tÞ ¼u X, Tð Þe^{i2ρXþi4ρ2T} (106) with x¼α^�1ðXþ4ρTÞ, t¼α^�2T;X¼αx�4βt, T¼α²t;ρ¼βα^�2. Using a method that is analogous to reference [16], and applying above gauge-like trans- formation to Eq. (105), the MNLS equation with VBC can be transformed into DNLS equation with VBC.

i∂_Tuþ∂_XXuþi∂_Xj ju²u

¼0 (107)

with u¼u X, Tð Þ. So according to (106), the N-soliton solution to MNLS