A multi-soliton solution of the DNLS equation based upon pure Marchenko formalism

Y, Θ ! ffiffiffiffiffiffiffiffi

3. A simple method to derive and solve Marchenko equation for DNLS equation

3.4 A multi-soliton solution of the DNLS equation based upon pure Marchenko formalism

When there are N simple polesλ₁,λ₂,⋯,λ_Nin the first quadrant of the complex plane ofλ, the Marchenko equation will give a N-solition solution to the DNLS equation with VBC in the reflectionless case. We can assume that

f xð þyÞ ¼F21ðxþyÞ ¼X^N

n¼1

g_nðx, tÞhnð Þ �y G x, tð ÞH yð Þ^T (146)

where g_nðx, tÞ �Cnð Þet ^iλ2nx, hnð Þ �y e^iλ2ny, n¼1, 2,⋯, N, and G x, tð Þ ��g₁ðx, tÞ, g₂ðx, tÞ,⋯, g_Nðx, tÞ�

; H yð Þ^T�ðh₁ð Þy, h₂ð Þy,⋯, hNð ÞyÞ^T (147) Here and hereafter the superscript T represents transposing of a matrix. On the other hand, we assume that

N11ðx, yÞ ¼N11ð ÞH yx ð Þ^T, N12ðx, yÞ ¼N12ð ÞH yx ð Þ^T (148)

owing to the symmetry properties of N^oðx, yÞand N^dðx, yÞ, the function f xð þyÞin (128) and (129) can only has off-diagonal elements, we write

F xð þyÞ ¼ 0 �f xð þyÞ f xð þyÞ 0

!

; G zð þyÞ ¼ 0 �h zð þyÞ h zð þyÞ 0

!

¼iσ3F⁰ðzþyÞ (144) Considering the dependence of the Jost solutions on the squared spectral parameterλ², in the reflectionless case, we choose

f xð þyÞ ¼X^N

n¼1

C_nð Þt e^{iλ2nðxþyÞ} (145)

where C_nð Þt contains a time-dependent factor e^i4λ4nt, which can be introduced by a standard procedure [29], due to a fact of the Lax operator M! �i2λ⁴σ₃ as x! �∞.

As is well known, Lax equations are linear equation so that a constant factor can be introduced in its solution, that is, Cn¼e^βnþiαne^i4λ4nt. It means thatβ_nis related to the center of soliton andα_nexpresses the initial phase up to a constant factor. Thus, the time-independent part of Cnis 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 lim_x!∞e^iλ2nx¼0, so we only consider the N zero points of að Þλ in the first quadrant of complex plane ofλ(also in the upper half part of the complex plane ofλ²), that is, the discrete spectrum forλ₁, λ₂,

⋯⋯λN, although�λ_n, nð ¼12,⋯, NÞin the third quadrant of the complex plane of λare also the zero points of að Þλ due 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 equa- tion (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 polesλ₁,λ₂,⋯,λ_Nin the first quadrant of the complex plane ofλ, the Marchenko equation will give a N-solition solution to the DNLS equation with VBC in the reflectionless case. We can assume that

f xð þyÞ ¼F21ðxþyÞ ¼X^N

n¼1

g_nðx, tÞhnð Þ �y G x, tð ÞH yð Þ^T (146)

where g_nðx, tÞ �Cnð Þet ^iλ2nx, hnð Þ �y e^iλ2ny, n¼1, 2,⋯, N, and G x, tð Þ ��g₁ðx, tÞ, g₂ðx, tÞ,⋯, g_Nðx, tÞ�

; H yð Þ^T�ðh₁ð Þy , h₂ð Þy,⋯, hNð Þy Þ^T (147) Here and hereafter the superscript T represents transposing of a matrix. On the other hand, we assume that

N11ðx, yÞ ¼N11ð ÞH yx ð Þ^T, N12ðx, yÞ ¼N12ð ÞH yx ð Þ^T (148)

Then

F12ðxþyÞ ¼ �F12ðxþyÞ ¼ �G xð ÞH yð Þ^T_; F⁰₁₂ðxþyÞ ¼iλ²_nCne^{�iλ2nðxþyÞ}

¼ �G⁰ð ÞH yx ð Þ^T (149)

Substituting (146)–(149) into the Marchenko equation (128) and (129), we have N11ð ÞH yx ð Þ^TþN12ð ÞxÐ_∞

xdzH zð Þ^TG zð ÞH yð Þ^T¼0 N12ð ÞH yx ð Þ^T�G xð ÞH yð Þ^T�iN11ð ÞxÐ_∞

xdzH zð Þ^TG⁰ð ÞH yz ð Þ^T¼0 (

(150)

or

N11ð Þ þx N12ð ÞΔx 1ð Þ ¼x 0, (151) N₁₂ð Þ �x iN₁₁ð ÞΔx 2ð Þ ¼x G xð Þ (152) here

Δ1ð Þ ¼x ð_∞

xH zð Þ^TG zð Þdz,Δ2ð Þ ¼x ð_∞

xH zð Þ^TG⁰ð Þdzz (153) Both of them are N�N matrices and their matrix element are, respectively, expressed as

Δ1ð Þx _mn¼ ð_∞

xe^�iλ2mzCne^iλ2nzdz¼hmð Þx �i

λ²_m�λ²_ng_nð Þx (154) Δ2ð Þx _mn¼

ð_∞

xe^�iλ2mzCn��iλ²_n�

e^�iλ2nzdz¼hmð Þx λ²_n

λ²_m�λ²_ng_nð Þx (155) From (151) and (152), we immediately get

N₁₁ð Þ ¼ �G xx ð Þ½1þiΔ1ð ÞΔx 2ð Þx�^�1Δ1ð Þx (156) N12ð Þ ¼x G xð Þ½1þiΔ1ð ÞΔx 2ð Þx �^�1 (157) from (148), (156), and (157), we have

N₁₁ðx, yÞ ¼ �G xð Þ½IþiΔ1ð ÞΔx 2ð Þx �^�1Δ1ð ÞH yx ð Þ^T (158) N12ðx, yÞ ¼G xð Þ½IþiΔ1ð ÞΔx 2ð Þx �^�1H yð Þ^T (159) then

N11ðx, xÞ ¼iTr iΔn 1ð ÞH xx ð Þ^TG xð Þ½IþiΔ1ð ÞΔx 2ð Þx �^�1o

¼i det Ih þiΔ1ð ÞΔx 2ð Þ þx iΔ1ð ÞH xx ð Þ^TG xð Þi det I½ þiΔ1ð ÞΔx 2ð Þx � �1 8<

:

9=

; (160)

and

N12ðx, xÞ ¼Tr H xn ð Þ^TG xð Þ½IþiΔ1ð ÞΔx 2ð Þx�^�1o

¼det Ih þiΔ1ð ÞΔx 2ð Þ þx H^Tð ÞG xx ð Þi det I½ þiΔ1ð ÞΔx 2ð Þx� �1

(161)

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

uNðx, tÞ ¼ �2det I� þiΔiΔ2þH^TG�

�det Ið þiΔ1Δ2Þ

det I� �iΔ1Δ2�iΔ1H^TG� �det I� �iΔ1Δ2�

det Ið þiΔ1Δ2Þ¼ �2CD=D² (162) where

D�det Ið þiΔ1Δ2Þ, C�det I� þiΔ1Δ2þH^TG�

�det Ið þiΔ1Δ2Þ, (163) and we will prove that in (136)

det I� �iΔ1Δ2�iΔ1H^TG�

¼det Ið þiΔ1Δ2Þ �D (164) By means of some linear algebraic techniques, especially the Binet-Cauchy for- mula 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.

det I� �iΔ1Δ2�iΔ1H^TG�

¼det Iþ �iΔ� 1�

Δ2þH^TG

� �

h i

≔det Ið þM₁M₂Þ (165) where

Μ1

ð Þ_nm� �iΔ� 1�

mn¼hn 1

λ²_n�λ²_mg_m, Mð 2Þ_mn��Δ2þH^TG�

mn¼hm λ²_m λ²_m�λ²_ng_n

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

g_nðx, tÞhnð Þ ¼x Cnð Þet ^i2λ2nx¼cnoe^i4λ4nte^i2λ2nx�e^�θne^iφn (167) hereλ_n¼μ_nþivn, and

θn�4μ_nvn x�xn0þ4�μ²_n�v²_n�

� t�

¼4κnðx�xn0�υ_ntÞ;κ_n¼4μ_nν_n;υ_n

¼ �4�μ²_n�v²_n�

; φ_n�2�μ²_n�v²_n�

xþ 4�μ²_n�v²_n�2

�16μ²_nv²_n

h i

•tþα_no; cn0

�e^4κnxnoe^iαno; n¼1, 2,⋯, N (168)

det I� �iΔ1Δ2�iΔ1H^TG�

¼1þX^N

r¼1

X

1≤n1<n2<⋯<nr≤N

X

1≤m1<m2<⋯<mr≤N

M1ðn1, n2,⋯, nr; m1, m2,⋯mrÞM2ðm1, m2,⋯, mr, n1, n2,⋯, nrÞ

(169) where M1ðn1, n2,⋯, nr; m1, m2,⋯mrÞdenotes 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).

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

uNðx, tÞ ¼ �2det I� þiΔiΔ2þH^TG�

�det Ið þiΔ1Δ2Þ

det I� �iΔ1Δ2�iΔ1H^TG� �det I� �iΔ1Δ2�

det Ið þiΔ1Δ2Þ¼ �2CD=D² (162) where

D�det Ið þiΔ1Δ2Þ, C�det I� þiΔ1Δ2þH^TG�

�det Ið þiΔ1Δ2Þ, (163) and we will prove that in (136)

det I� �iΔ1Δ2�iΔ1H^TG�

¼det Ið þiΔ1Δ2Þ �D (164) By means of some linear algebraic techniques, especially the Binet-Cauchy for- mula 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.

det I� �iΔ1Δ2�iΔ1H^TG�

¼det Iþ �iΔ� 1�

Δ2þH^TG

� �

h i

≔det Ið þM₁M₂Þ (165) where

Μ1

ð Þ_nm� �iΔ� 1�

mn¼hn 1

λ²_n�λ²_mg_m, Mð 2Þ_mn��Δ2þH^TG�

mn¼hm λ²_m λ²_m�λ²_ng_n

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

g_nðx, tÞhnð Þ ¼x Cnð Þet ^i2λ2nx¼cnoe^i4λ4nte^i2λ2nx�e^�θne^iφn (167) hereλ_n¼μ_nþivn, and

θn�4μ_nvn x�xn0þ4�μ²_n�v²_n�

� t�

¼4κnðx�xn0�υ_ntÞ;κ_n¼4μ_nν_n;υ_n

¼ �4�μ²_n�v²_n�

; φ_n�2�μ²_n�v²_n�

xþ 4�μ²_n�v²_n�2

�16μ²_nv²_n

h i

•tþα_no; cn0

�e^4κnxnoe^iαno; n¼1, 2,⋯, N (168)

det I� �iΔ1Δ2�iΔ1H^TG�

¼1þX^N

r¼1

X

1≤n1<n2<⋯<nr≤N

X

1≤m1<m2<⋯<mr≤N

M1ðn1, n2,⋯, nr; m1, m2,⋯mrÞM2ðm1, m2,⋯, mr, n1, n2,⋯, nrÞ

(169) where M1ðn1, n2,⋯, nr; m1, m2,⋯mrÞdenotes 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).

M1ðn₁, n2,⋯, nr; m1, m2,⋯mrÞ ¼Y

n, m

hng_m λ²_n�λ²_m

Y

n<n⁰, m<m⁰

λ²_n�λ²_n0

� �

λ²_m0�λ²_m

� �

(170) M₂ðm₁, m₂,⋯, mr; n₁, n₂,⋯nrÞ ¼Y

n, m

hmg_nλ²_m λ²_m�λ²_n

Y

n<n⁰, m<m⁰

λ²_m�λ²_m0

� �

λ²_n0�λ²_n

� �

(171)

where n, n⁰∈fn1, n2,⋯, nrg, m, m⁰∈fm1, m2,⋯, mrg, then M1ðn1, n2,⋯, nr; m1, m2,⋯mrÞM2ðm1, m2,⋯, mr; n1, n2,⋯nrÞ

¼ �1ð Þ^rY

n, m

e^�θne^iφne^�θme^�iφmλ²_m λ²_n�λ²_m

� �₂ Y

n<n⁰, m<m⁰

λ²_m�λ²_m0

� �2

λ²_n�λ²_n0

� �2

(172)

If we define matrices Q₁¼iΔ1and Q₂¼Δ2, then we can similarly attain D¼det Ið þiΔ1Δ2Þ ¼det Ið þQ₁Q₂Þ ¼1þX^N

r¼1

X

1≤n1<n2<⋯<nr≤N

X

1≤m1<m2<⋯<mr≤N

Q₁ðn₁, n₂,⋯, n_r; m₁, m₂,⋯m_rÞQ₂ðm₁, m₂,⋯, m_r; n₁, n₂,⋯, n_rÞ

(173) and

Q₁ðn₁, n₂,⋯, n_r; m₁, m₂,⋯m_rÞQ₂ðm₁, m₂,⋯, m_r; n₁, n₂,⋯, n_rÞ

¼ �1ð Þ^rY

n, m

e^�θme^iφme^�θne^�iφnλ²_n λ²_m�λ²_n

� �₂ Y

n<n⁰, m<m⁰

λ²_m�λ²_m0

� �2�λ²_n�λ²_n0�₂ (174)

where n, n⁰∈fn1, n2,⋯, nrg, m, m⁰∈fm1, m2,⋯, mrg. Comparing (172) and (174), we find the following permutation symmetry between them

M1ðn1, n2,⋯, nr; m1, m2,⋯mrÞM2ðm1, m2,⋯, mr; n1, n2,⋯, nrÞ

¼Q₁ðm1, m2,⋯, mr; n1, n2,⋯nrÞQ₂ðn1, n2,⋯, nr; m1, m2,⋯, mrÞ 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 C in (163). In order to calculate det I� þiΔ1Δ2þH^TG�

, we introduce an N�ðNþ1ÞmatrixΩ1and an Nþ1

ð Þ �N matrixΩ2

Ω1

ð Þ_nm¼ðiΔ1Þ_nm,ð ÞΩ1 _n0¼h_n¼ h_nλ²_n

λ²_n�0²;ð ÞΩ2 _mn¼ð ÞΔ2 _mn, Ω2

ð Þ_0n¼g_n¼ �λ²_ng_n 0²�λ²_n

(175)

with n, m = 1, 2, … , N. We thus have

det Ið þΩ1Ω2Þ ¼1þX^N

r¼1

X

1≤n1≤n2<⋯<nr≤N

X

0≤m1<m2<⋯<mr≤N

Ω1ðn₁, n₂,⋯, n_r; m₁, m₂,⋯m_rÞΩ2ðm₁, m₂,⋯, m_r; n₁, n₂,⋯, n_rÞ

(176)

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 m1¼0, m2≥1). Due to (175), we have

C¼det Ið þΩ1Ω2Þ �det Ið þiΔ1Δ2Þ

¼X^N

r¼1

X

1≤n1<n2<⋯<nr≤N

X

1≤m2<m3<⋯<mr≤N

Ω1ðn₁, n₂,⋯, nr; 0, m₂,⋯, mrÞΩ2ð0, m₂,⋯, mr; n₁, n₂,⋯, nrÞ (177) Ω1ðn1, n2,⋯, nr; 0, m2,⋯mrÞ ¼Y

n hn

Y

m g_m Y

n<n⁰, m<m⁰

λ²_n�λ²_n0

� �

λ²_m0�λ²_m

� �Y

n, m

1 λ²_n�λ²_m

(178) Ω2ð0, m2,⋯, mr; n1, n2,⋯nrÞ ¼Y

n g_nY

m hm

Y

n<n⁰, m<m⁰

λ²_n�λ²_n0

� �

λ²_m0�λ²_m

� �Y

n, m

1 λ²_m�λ²_n

� �1ð Þ^rþ1Y

m

λ²_m,

(179) which leads to

Ω1ðn₁, n₂,⋯, nr; 0, m₂,⋯mrÞΩ2ð0, m₂,⋯, mr; n₁, n₂,⋯, nrÞ

¼ �1ð Þ^rþ1Y

n e^�θne^�iφnY

me^�θme^iφm Y

n<n⁰, m<m⁰

λ²_n�λ²_n0

� �₂

λ²_m0�λ²_m

� �2Y

n, m

1 λ²_m�λ²_n

� �2

Y

m

λ²_m, (180) here n, n⁰∈ðn₁, n₂,⋯, nrÞ, m, m⁰∈ðm₂,⋯, mrÞin (178)–(180). Finally, substitut- ing (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 að Þ. Obviously, the N-soliton solution permits uncertain complexλ constants cn0ðn¼1, 2,⋯, NÞas well as an arbitrary global constant phase factor.