The revised inverse scattering transform and the Zakharov-Shabat equation for DNLS equation with VBC

Y, Θ ! ffiffiffiffiffiffiffiffi

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

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 u x, tð Þis usually expressed as

iu_tþu_xxþi u�j j²u�

x¼0 (1)

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

L¼ �iλ²σ₃þλU, U¼ 0 u

�u 0

� �

(2) and

M¼ �i2λ⁴σ₃þ2λ³U�iλ²U²σ₃�λ��U³þiUxσ₃�

(3) whereλis a spectral parameter, andσ₃is the third one of Pauli matricesσ₁,σ₂,σ₃, and a bar over a letter, (e.g., u in (2)), represents complex conjugate. The first Lax equation is

∂_xf x,ð λÞ ¼L x,ð λÞf x,ð λÞ (4) In the limit of∣x∣!∞, u!0, and

L!L₀¼ �iλ²σ₃; M!M₀¼ �i2λ⁴σ₃ (5) The free Jost solution is a 2�2 matrix.

E x,ð λÞ ¼e^�iλ2xσ3; E•1ðx,λÞ ¼ 1 0

� �

e^�iλ2x, E•2ðx,λÞ ¼ 0 1

� �

e^iλ2x (6) The Jost solutions of (4) are defined by their asymptotic behaviors as x! �∞.

Ψðx,λÞ ¼ðψ~ðx,λÞ,ψðx,λÞÞ !E x,ð λÞ, as x!∞ (7) Φðx,λÞ ¼�ϕðx,λÞ,~ϕðx,λÞ�

!E x,ð λÞ, as x! �∞ (8) whereψðx,λÞ ¼ðψ₁ðx,λÞ,ψ₂ðx,λÞÞ^T,ψ~ðx,λÞ ¼ðψ~₁ðx,λÞ,ψ~₂ðx,λÞÞ^T, 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 Tð Þλ is defined as

Φðx,λÞ ¼Ψðx,λÞ Tð Þ,λ (9) where

Tð Þ ¼λ a t,ð λÞ �~b t,ð λÞ b t,ð λÞ ~a t,ð λÞ

!

(10)

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 u x, tð Þis usually expressed as

iu_tþu_xxþi u�j j²u�

x¼0 (1)

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

L¼ �iλ²σ₃þλU, U¼ 0 u

�u 0

� �

(2) and

M¼ �i2λ⁴σ₃þ2λ³U�iλ²U²σ₃�λ��U³þiUxσ₃�

(3) whereλis a spectral parameter, andσ₃is the third one of Pauli matricesσ₁,σ₂,σ₃, and a bar over a letter, (e.g., u in (2)), represents complex conjugate. The first Lax equation is

∂_xf x,ð λÞ ¼L x,ð λÞf x,ð λÞ (4) In the limit of∣x∣!∞, u!0, and

L!L₀ ¼ �iλ²σ₃; M!M₀¼ �i2λ⁴σ₃ (5) The free Jost solution is a 2�2 matrix.

E x,ð λÞ ¼e^�iλ2xσ3; E•1ðx,λÞ ¼ 1 0

� �

e^�iλ2x, E•2ðx,λÞ ¼ 0 1

� �

e^iλ2x (6) The Jost solutions of (4) are defined by their asymptotic behaviors as x! �∞.

Ψðx,λÞ ¼ðψ~ðx,λÞ,ψðx,λÞÞ !E x,ð λÞ, as x!∞ (7) Φðx,λÞ ¼�ϕðx,λÞ,ϕ~ðx,λÞ�

!E x,ð λÞ, as x! �∞ (8) whereψðx,λÞ ¼ðψ₁ðx,λÞ,ψ₂ðx,λÞÞ^T,ψ~ðx,λÞ ¼ðψ~₁ðx,λÞ,ψ~₂ðx,λÞÞ^T, 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 Tð Þλ is defined as

Φðx,λÞ ¼Ψðx,λÞ Tð Þ,λ (9) where

Tð Þ ¼λ a t,ð λÞ �~b t,ð λÞ b t,ð λÞ ~a t,ð λÞ

!

(10)

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

σ₂ L λ σ₂¼Lð Þ,λ σ₂ T λ σ₂¼Tð Þλ (11) σ₂ Ψx,λ

σ₂¼Ψðx,λÞ, σ₂ Φx,λ

σ₂¼Φðx,λÞ (12) and

σ₃Ψðx,λÞσ₃¼Ψðx,�λÞ, σ₃Φðx,λÞσ₃¼Φðx,�λÞ (13) σ₃Lð Þλ σ₃¼Lð�λÞ, σ₃Tð Þλ σ₃¼Tð�λÞ (14) Then we can get the following reduction relation and symmetry properties

iσ₂ψx,λ

¼ψ~ðx,λÞ (15)

�iσ2φx,λ

¼φ~ðx,λÞ (16)

~a λ ¼að Þ;λ ~b λ ¼bð Þλ (17) and

ψðx,�λÞ ¼ �σ₃ψðx,λÞ (18) ψ~ðx,�λÞ ¼σ₃ψ~ðx,λÞ (19) að�λÞ ¼að Þ;λ  bð�λÞ ¼ �bð Þλ

~að�λÞ ¼~að Þ;λ  ~bð�λÞ ¼ �~bð Þλ (20) 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∣λ∣!∞ can be obtained by simple derivation. Letυ¼ðυ₁,υ₂Þ^T�ψ~ðx,λÞ; Eq. (4) can be rewritten as

υ_1xþiλ²υ₁¼λuυ2,υ_2x�iλ²υ₂¼ �λuυ1 (21) Then we have

υ_1xx�uxυ_1xþiλ²υ₁

=uþλ⁴υ₁þλ²j ju²υ₁¼0 (22) In the limit∣λ∣!∞, we assumeψ~₁ðx,λÞ ¼e^�iλ2xþg, substituting it into Eq. (22), then we have

�iλ²þg_x 2

þg_xx�u_xg_x=uþλ⁴þλ²j ju²¼0 (23) In the limit∣λ∣!∞, g_xcan be expanded as series ofλ^�2j

, j¼1, 2,⋯.

ig_x�μ¼μ₀þμ₂2λ²�1

þ⋯ (24)

and

μ₀¼j ju²=2,μ₂¼ �iu_xu=2�j ju⁴=4,⋯ (25)

Eq. (21) leads to g_xυ₁¼λuυ₂. Considering (25), in the limit ofj j !λ ∞, we find a useful formula

u¼i2 lim

∣λ∣!∞λ~ψ₂ðx,λÞ=ψ~₁ðx,λÞ (26) which expresses the conjugate of solution u in terms of the Jost solutions as j j !λ ∞.

On the other hand, the zeros of að Þλ appear in pairs and can be designed byλ_n, n¼1, 2,⋯, N in the I quadrant, andλ_nþN ¼ �λ_nin the III quadrant. The discrete part of að Þλ is [21–23].

að Þ ¼λ Y^N

n¼1

λ²�λ²_n λ²�λ²_n•λ²_n

λ²_n (27)

where a 0ð Þ ¼1. It comes from our consideration of the fact that, from the sum of two Cauchy integrals

ln að Þλ

λ þ0¼ 1 2πi

ð

Γdλ⁰ ln að Þλ⁰ ~að Þλ⁰

λ⁰ðλ⁰�λÞ ,Γ¼ð0, ∞Þ∪ði∞, i0Þ∪ð0,�∞Þ∪ð�i∞, i0Þ, in order to maintain that ln að Þ !λ 0, as λ!0, and ln að Þλ is finite asj j !λ ∞, we then have to introduce a factorλ²_n=λ²_nin (27). At the zeros of að Þ, we haveλ

ϕðx,λ_nÞ ¼b_nψðx,λ_nÞ,a_ð�λ_nÞ ¼ �a_ð Þ, bλ_{n nþN} ¼ �b_n (28) Due toμ₀6¼0 in (24) and (25), the Jost solutions do not tend to free Jost solutions E x,ð λÞin 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 functionΘðx,λÞcan be introduced as usual

Θðx,λÞ ¼ ϕðx,λÞ=að Þ, asλ λ in I, III quadrants:

ψ~ðx,λÞ, as λ in II, IV quadrants:

�

(29) An alternative form of IST equation is proposed as

1

λ²fΘ1ðx,λÞ �E₁₁ðx,λÞge^iλ2x¼ 1 2πi

ð

Γdλ⁰ 1 λ⁰�λ

1

λ⁰²fΘ1ðx,λ⁰Þ �E₁₁ðx,λ⁰Þge^iλ02x (30) Because in the limit ofj j !λ ∞, lim

∣λ∣!∞e^iλ2x¼0, as x>0, Imλ²>0, ðλin the I, III quadrantsÞ, x<0, Imλ²<0, ðλin the II, IV quadrantsÞ, (

then the integral pathΓshould be chosen as shown inFigure 1, where the radius of big circle tends to infinite, while the radius of small circle tends to zero. And the factorλ^�2is introduced to ensure the contribution of the integral along the big arc is vanishing. Meanwhile,

Eq. (21) leads to g_xυ₁¼λuυ₂. Considering (25), in the limit ofj j !λ ∞, we find a useful formula

u¼i2 lim

∣λ∣!∞λ~ψ₂ðx,λÞ=ψ~₁ðx,λÞ (26) which expresses the conjugate of solution u in terms of the Jost solutions as j j !λ ∞.

On the other hand, the zeros of að Þλ appear in pairs and can be designed byλ_n, n¼1, 2,⋯, N in the I quadrant, andλ_nþN¼ �λ_nin the III quadrant. The discrete part of að Þλ is [21–23].

að Þ ¼λ Y^N

n¼1

λ²�λ²_n λ²�λ²_n•λ²_n

λ²_n (27)

where a 0ð Þ ¼1. It comes from our consideration of the fact that, from the sum of two Cauchy integrals

ln að Þλ

λ þ0¼ 1 2πi

ð

Γdλ⁰ ln að Þλ⁰ ~að Þλ⁰

λ⁰ðλ⁰�λÞ ,Γ¼ð0, ∞Þ∪ði∞, i0Þ∪ð0,�∞Þ∪ð�i∞, i0Þ, in order to maintain that ln að Þ !λ 0, as λ!0, and ln að Þλ is finite asj j !λ ∞, we then have to introduce a factorλ²_n=λ²_nin (27). At the zeros of að Þ, we haveλ

ϕðx,λ_nÞ ¼b_nψðx,λ_nÞ,a_ð�λ_nÞ ¼ �a_ð Þ, bλ_{n nþN}¼ �b_n (28) Due toμ₀6¼0 in (24) and (25), the Jost solutions do not tend to free Jost solutions E x,ð λÞin 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 functionΘðx,λÞcan be introduced as usual

Θðx,λÞ ¼ ϕðx,λÞ=að Þ, asλ λ in I, III quadrants:

ψ~ðx,λÞ, as λ in II, IV quadrants:

�

(29) An alternative form of IST equation is proposed as

1

λ²fΘ1ðx,λÞ �E₁₁ðx,λÞge^iλ2x¼ 1 2πi

ð

Γdλ⁰ 1 λ⁰�λ

1

λ⁰²fΘ1ðx,λ⁰Þ �E₁₁ðx,λ⁰Þge^iλ02x (30) Because in the limit ofj j !λ ∞, lim

∣λ∣!∞e^iλ2x¼0, as x>0, Imλ²>0, ðλin the I, III quadrantsÞ, x<0, Imλ²<0, ðλin the II, IV quadrantsÞ, (

then the integral pathΓshould be chosen as shown inFigure 1, where the radius of big circle tends to infinite, while the radius of small circle tends to zero. And the factorλ^�2is introduced to ensure the contribution of the integral along the big arc is vanishing. Meanwhile,

our modification produces no new poles since Lax operator L!0, asλ!0. In the reflectionless case, the revised IST equation gives

ψ~₁ðx,λÞ ¼e^�iλ2xþX^2N

n¼1

1 λ²_n

λ² λ�λ_n

b_n

a_ð Þλ_n ψ₁ðx,λ_nÞe^iλ2nxe^�iλ2x (31) wherea_ð Þ ¼λ_n dað Þλ=dλjλ¼λn. Similarly, an alternative form of IST equation is proposed as follows:

1

λfΘ2ðx,λÞge^iλ2x¼ 1 2πi

ð

Γdλ⁰ 1 λ⁰�λ

1

λ⁰fΘ2ðx,λ⁰Þge^iλ02x (32) where a factorλ^�1is introduced for the same reason asλ^�2in Eq. (30). Then in the reflectionless case, we can attain

ψ~₂ðx,λÞ ¼X^2N

n¼1

1 λ_n

λ λ�λ_n

b_n

a_ð Þλ_n ψ₂ðx,λ_nÞe^iλ2nxe^�iλ2x (33) 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~₁ðx,λÞ ¼e^�iλ2xþX^N

n¼1

2λ²

λ_n�λ²�λ²_n� bn

a_ð Þλ_n ψ₁ðx1,λ_nÞe^iλ2nxe^�iλ2x (34) ψ~₂ðx,λÞ ¼X^N

n¼1

2λ λ²�λ²_n

bn

a_ð Þλ_n ψ₂ðxs,λ_nÞe^iλ2nxe^�iλ2x (35) 2.2 The raw expression ofN-soliton solution

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

uN¼ �i2UN=VN (36)

Figure 1.

The integral path for IST of the DNLS.

where

UN¼X^N

n¼1

2b_n

a_ð Þλ_n ψ₂ðx,λ_nÞe^iλ2nx (37) V_N¼1þX^N

n¼1

2b_n

λ_na_ð Þλ_n ψ₁ðx,λ_nÞe^iλ2nx (38) Letλ¼λ_m, m = 1, 2, … , N, respectively, in Eqs. (34) and (35), and make use of the symmetry and reduction relation (15), we can attain

ψ₂ðx,λ_mÞ ¼e^�iλ2mxþX^N

n¼1

2λ²_m

λ_n�λ²_m�λ²_n�c_nψ₁ðx,λ_nÞe^iλ2nxe^�iλ2mx (39) ψ₁ðx,λ_mÞ ¼ �X^N

n¼1

2λ²_m

λ²_m�λ²_ncnψ₂ðx,λ_nÞe^iλ2nxe^�iλ2mx; m¼1, 2, … , N: (40) where c_n¼b_n=a_ð Þ. We also defineλ_n

f_n¼ ffiffiffiffiffiffiffi 2cn

p e^iλ2nx, w� �j

n¼ ffiffiffiffiffiffiffi 2cn

p ψ_jð Þλ_n j¼1, 2; and n¼1, 2, … N: (41)

B1

ð Þ_mn¼ f_m λ²_m λ²_m�λ²_n

� �

λ_nf_n, Bð Þ2 _mn¼f_m λ_m

λ²_m�λ²_nf_n; m, n¼1, 2, … , N (42) W1¼�ð Þw1 ₁;ð Þw1 ₂;⋯;ð Þw1 _N�_T

, W2¼�ð Þw2 ₁;ð Þw2 ₂;⋯;ð Þw2 _N�_T , F¼�f₁;f₂;⋯;f_N�T

, G¼�f₁=λ₁;f₂=λ₂;⋯;f_N=λ_N�T (43) where superscript “T” represents transposition of a matrix. Then Eqs. (39) and (40) can be rewritten as

w2

ð Þ_m¼f_mþX^N

n¼1

B1

ð Þ_mnð Þw1 _n (44)

w1

ð Þ_m¼ �X^N

n¼1

B2

ð Þ_mnð Þw2 _n (45) where m = 1, 2, … , N. They can be rewritten in a more compact matrix form.

W2¼FþB1�W1 (46)

W1¼ �B2�W2 (47)

Then

W2¼�IþB1B2��1

F (48)

W₁¼ �B₂�IþB₁B₂��1

F (49)

where I is the N�N identity matrix. On the other hand, from (37) and (38), we know

where

UN¼X^N

n¼1

2b_n

a_ð Þλ_n ψ₂ðx,λ_nÞe^iλ2nx (37) V_N ¼1þX^N

n¼1

2b_n

λ_na_ð Þλ_n ψ₁ðx,λ_nÞe^iλ2nx (38) Letλ¼λ_m, m = 1, 2, … , N, respectively, in Eqs. (34) and (35), and make use of the symmetry and reduction relation (15), we can attain

ψ₂ðx,λ_mÞ ¼e^�iλ2mxþX^N

n¼1

2λ²_m

λ_n�λ²_m�λ²_n�c_nψ₁ðx,λ_nÞe^iλ2nxe^�iλ2mx (39) ψ₁ðx,λ_mÞ ¼ �X^N

n¼1

2λ²_m

λ²_m�λ²_ncnψ₂ðx,λ_nÞe^iλ2nxe^�iλ2mx; m¼1, 2, … , N: (40) where c_n¼b_n=a_ð Þ. We also defineλ_n

f_n¼ ffiffiffiffiffiffiffi 2cn

p e^iλ2nx, w� �j

n¼ ffiffiffiffiffiffiffi 2cn

p ψ_jð Þλ_n j¼1, 2; and n¼1, 2, … N: (41)

B1

ð Þ_mn¼ f_m λ²_m λ²_m�λ²_n

� �

λ_nf_n, Bð Þ2 _mn¼f_m λ_m

λ²_m�λ²_nf_n; m, n¼1, 2, … , N (42) W1¼�ð Þw1 ₁;ð Þw1 ₂;⋯;ð Þw1 _N�_T

, W2¼�ð Þw2 ₁;ð Þw2 ₂;⋯;ð Þw2 _N�_T , F¼�f₁;f₂;⋯;f_N�T

, G¼�f₁=λ₁;f₂=λ₂;⋯;f_N=λ_N�T (43) where superscript “T” represents transposition of a matrix. Then Eqs. (39) and (40) can be rewritten as

w2

ð Þ_m¼f_mþX^N

n¼1

B1

ð Þ_mnð Þw1 _n (44)

w1

ð Þ_m ¼ �X^N

n¼1

B2

ð Þ_mnð Þw2 _n (45) where m = 1, 2, … , N. They can be rewritten in a more compact matrix form.

W2¼FþB1�W1 (46)

W1¼ �B2�W2 (47)

Then

W2¼�IþB1B2��1

F (48)

W₁¼ �B₂�IþB₁B₂��1

F (49)

where I is the N�N identity matrix. On the other hand, from (37) and (38), we know

UN¼X^N

n¼1

f_nw2n¼F^TW2 (50)

VN¼1þX^N

n¼1

f_n=λ_n

� �

w1n ¼1þG^TW1 (51)

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

u_N ¼ �i2 F^TW₂

1þG^TW₁¼ �i2 F^T�IþB₁B₂��1F 1�G^TB2�IþB1B2��1

F

¼ �i2det I� þB1B2þFF^T�

�det I� þB1B2� det Iþ�B₁�FG^T�

B₂

� � �det I� þB1B2�

det I� þB1B2�� �2iA�D D²

(52)

where

A�det I� þB1B2þFF^T�

�det I� þB1B2�

(53) D�det I� þB₁B₂�

(54) In the subsequent chapter, we will prove that

det Iþ�B1�FG^T� B2

� �

¼det I� þB1B2�

(55) 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 ofN-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

P1

ð Þ_nm��B1�FG^T�

nm¼f_n λ_m

λ²_n�λ²_m f_m; Pð Þ2 _mn�� �B2

mn¼ f_m λ_m

λ²_m�λ²_nf_n (56) Q₁

ð Þ_nm�� �B1

nm¼ f_n λ²_n λ²_n�λ²_m

f_m λ_m !

; Qð ₂Þ_mn�ð ÞB2 _mn¼f_m λ_m

λ²_m�λ²_nf_n (57) Then

D¼det Ið þQ₁Q₂Þ ¼1þX^N

r¼1

X

1≤n1<n2<⋯<nr≤N

D_rðn₁;n₂;⋯nrÞ

¼1þX^N

r¼1

X

1≤n1<⋯<nr≤N

X

1≤m1<⋯<mr≤N

Q₁ðn1;n2;⋯nr;m1;m2;⋯;mrÞQ₂ðm1;m2;⋯mr;n1;n2⋯;nrÞ (58)

where Q₁ðn₁, n2,⋯nr; m1, m2,⋯, mrÞdenotes a minor, which is the determinant of a submatrix of Q1consisting 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

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

¼ Q

n, m

f_nf_m λ²_n�λ²_m

λ²_n λ_m

Y

n<n⁰, m<m⁰

λ²_n�λ²_n0

� �

λ²_m0�λ²_m

� �Y

m, n

f_mf_n

λ²_m�λ²_nλ_m Y

n<n⁰, m<m⁰

λ²_m�λ²_m0

� �

λ²_n0�λ²_n

� �

¼ �ð 1Þ^rQ

m, n

λ²_nf²_nf²_m λ²_n�λ²_m

� �₂ Y

n<n⁰, m<m⁰

λ²_n�λ²_n0

� �2

λ²_m�λ²_m0

� �2

(59)

where

n, n⁰∈fn₁, n₂,⋯, nrg, m, m⁰∈fm₁, m₂,⋯, mrg (60) Similarly,

P1ðn1, n2,⋯, nr; m1, m2,⋯, mrÞP2ðm1, m2,⋯, mr; n1, n2,⋯, nrÞ

¼ �1ð Þ^rY

n, m

f²_nf²_mλ²_m λ²_n�λ²_m

� �₂ Y

n<n⁰, m<m⁰

λ²_m�λ²_m0

� �2

λ²_n�λ²_n0

� �2

(61)

where

n, n⁰∈fn1;n2;⋯;nrg; m, m⁰∈fm1;m2;⋯;mrg, and det Iþ�B₁�FG^T�

B₂

� �

¼det Ið þP₁P₂Þ

¼1þX^N

r¼1

X

1≤n1<⋯<nr≤N

X

1≤m1<⋯<mr≤N

P1ðn1;⋯;nr;m1;⋯;mrÞP2ðm1;⋯;mr;n1;⋯;nrÞ (62) It is easy to find a kind of permutation symmetry existed between expressions (59) and (61), that is,

P1ðn1,⋯, nr; m1,⋯, mrÞP2ðm1,⋯, mr; n1,⋯, nrÞ

¼Q₁ðm₁,⋯, m_r; n₁,⋯, n_rÞQ₂ðn₁,⋯, n_r; m₁,⋯, m_rÞ (63) Comparing (58) with (62) and making use of (63), we thus complete verifica- tion 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 M! �i2λ⁴σ₃in the limit of∣x∣!∞, it is easy to derive the time dependence of scattering date.

dλn=dt¼0, dað Þλ_n =dt¼0; cnð Þ ¼t cn0e^i4λ4nt, cn0¼bn0=a_ð Þ, bλ_n nð Þ ¼t bn0e^i4λ4nt (64)

where Q₁ðn₁, n2,⋯nr; m1, m2,⋯, mrÞdenotes a minor, which is the determinant of a submatrix of Q1consisting 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

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

¼ Q

n, m

f_nf_m λ²_n�λ²_m

λ²_n λ_m

Y

n<n⁰, m<m⁰

λ²_n�λ²_n0

� �

λ²_m0�λ²_m

� �Y

m, n

f_mf_n

λ²_m�λ²_nλ_m Y

n<n⁰, m<m⁰

λ²_m�λ²_m0

� �

λ²_n0�λ²_n

� �

¼ �ð 1Þ^rQ

m, n

λ²_nf²_nf²_m λ²_n�λ²_m

� �₂ Y

n<n⁰, m<m⁰

λ²_n�λ²_n0

� �2

λ²_m�λ²_m0

� �2

(59)

where

n, n⁰∈fn₁, n₂,⋯, nrg, m, m⁰∈fm₁, m₂,⋯, mrg (60) Similarly,

P1ðn1, n2,⋯, nr; m1, m2,⋯, mrÞP2ðm1, m2,⋯, mr; n1, n2,⋯, nrÞ

¼ �1ð Þ^rY

n, m

f²_nf²_mλ²_m λ²_n�λ²_m

� �₂ Y

n<n⁰, m<m⁰

λ²_m�λ²_m0

� �2

λ²_n�λ²_n0

� �2

(61)

where

n, n⁰∈fn1;n2;⋯;nrg; m, m⁰∈fm1;m2;⋯;mrg, and det Iþ�B₁�FG^T�

B₂

� �

¼det Ið þP₁P₂Þ

¼1þX^N

r¼1

X

1≤n1<⋯<nr≤N

X

1≤m1<⋯<mr≤N

P1ðn1;⋯;nr;m1;⋯;mrÞP2ðm1;⋯;mr;n1;⋯;nrÞ (62) It is easy to find a kind of permutation symmetry existed between expressions (59) and (61), that is,

P1ðn1,⋯, nr; m1,⋯, mrÞP2ðm1,⋯, mr; n1,⋯, nrÞ

¼Q₁ðm₁,⋯, m_r; n₁,⋯, n_rÞQ₂ðn₁,⋯, n_r; m₁,⋯, m_rÞ (63) Comparing (58) with (62) and making use of (63), we thus complete verifica- tion 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 M! �i2λ⁴σ₃in the limit of∣x∣!∞, it is easy to derive the time dependence of scattering date.

dλn=dt¼0, dað Þλ_n =dt¼0; cnð Þ ¼t cn0e^i4λ4nt, cn0¼bn0=a_ð Þ, bλ_n nð Þ ¼t bn0e^i4λ4nt (64)

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.