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

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4. The one and two-soliton solutions to DNLS + equation with NVBC We give two concrete examples – the one and two breather-type soliton solu-

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

When all the simple poles are on the circleðO,ρÞcentered at the originO, just as shown inFigure 4, our revised IST for DNLS⁺equation with NVBC will give a typical pureN-soliton solution. The discrete part ofa zð Þis of a slightly different form from that of the case for breather-type solution, and it can be expressed as

a zð Þ ¼Y^N

n¼1

z² z²_n z² z²_n

z_n

z_n;a z_ð Þ ¼_n 2zn

z²_n z²_n z_n z_n

Y^N

m¼1,m6¼n

z²_n z²_m z²_n z²_m

z_m

z_m (143)

Figure 3.

Evolution of the square amplitude of a breather-type two-soliton with respect to time and spaceρ¼2;δ1¼0:4; δ3¼0:6;β₁¼π=5:0;β₃ ¼π=2:2; x10¼0; x20¼0=0; x30¼0; x40¼0;φ₁₀¼0;φ₂₀¼0;φ₃₀¼0;φ₄₀¼0.

Figure 4.

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

At the zeros ofa zð Þ, we have

ϕðx,z_nÞ ¼b_nψðx,z_nÞ,a_ð z_nÞ ¼ a z_ð Þ_n ,b_n ¼ b_n (144) On the other hand, the zeros ofa zð Þappear in pairs and can be designed byz_n, (n¼1, 2,⋯,N), in the I quadrant, andz_nþN ¼ zn in the III quadrant. The

Zakharov-Shabat equation for pure soliton case of DNLS+ equation under reflectionless case can be derived immediately

~

ψ₁ðx,zÞ ¼e ^iΛxþλ X^N

n¼1

2z λ_n

1

z² z²_nc_nψ₁ðx,z_nÞe^iΛnx

" #

e ^iΛx (145)

~

ψ₂ðx,zÞ ¼iρz ¹e ^iΛxþλ X^N

n¼1

2zn

λ_n 1

z² z²_nc_nψ₂ðx,z_nÞe^iΛnx

" #

e ^iΛx (146) HereΛ¼κλ, Λ_n ¼κ_nλ_n; Letting z¼ρ²z_m¹,m ¼1, 2,⋯,N, then

ψ₁ðx,z_mÞ ¼ iρz_m¹e^iΛmxþX^N

n¼1

λ_mcn

λ_nz²_m 2ρ³

iρ⁴z_m² z²_nψ₁ðx,z_nÞe^{iðΛnþΛmÞx} (147) ψ₂ðx,zmÞ ¼e^iΛmxþX^N

n¼1

λ_mz_nc_n

λ_nz_m 2ρ

iρ⁴z_m² z²_nψ₂ðx,znÞe^{iðΛnþΛmÞx} (148) Different from that in breather-type case, we definez_n ρe^δne^iβn ¼ρe^iβn, with β_n∈ð0,π=2Þ,δ_n¼0,ði¼1, 2,⋯,NÞ, specially we have

c_n0 ¼b_n0=a z_ð Þ ¼_n ib_n0ρsin 2β_ne^iβn Y^N

k¼1;k6¼n

sinðβ_nþβ_kÞ

sinðβ_n β_kÞ (149) tanh²ðΘ_n Θ_mÞ ¼ sin²ðβ_n β_mÞ=sin²ðβ_nþβ_mÞ (150) An inverse Galileo transformationðx,tÞ !ðx ρ²t,tÞchangesF_nðx,tÞinto

F_n f²_n ρ

i z ²_n ρ⁴z_n²¼ 2ρ

i z ²_n ρ⁴z_n²c_noe ^i4λ3nκnte^i2Λnx

¼ð ibn0Þ Y^N

k¼1;k6¼n

sinðβ_nþβ_kÞ

sinðβ_n β_kÞe^i2Λn½^x ð^2λ2þρ2Þ^tŠ^þiβn (151) Due tob_n0 ¼ b_n0, ib_n0∈, following equations hold:

Fn¼ e^iβn Y^N

k¼1;k6¼n

sinðβ_nþβ_kÞ sinðβ_n β_kÞ

!

ibn0e^i2Λn½^x ð^2λ2nþρ2Þ^tŠ

ε_nEne^iβnexpð θ_nþiφ_nÞ (152) where

ibn0e^i2Λn½^x ð^2λ2nþρ2Þ^tŠ ε_nexpð θ_nþiφ_nÞ (153) θ_nðx,tÞ ν_nðx υ_nt xn0Þ,φ_n¼0, (154) ε_n sgnð ib_n0Þ;j ib_n0j e^νnxn0 (155)

ν_n¼ρ²sin 2β_n,υ_n¼ρ²1þ2 cos²β_n

(156) E_n Y^N

k¼1;k6¼n

sinðβ_nþβ_kÞ

sinðβ_n β_kÞ (157)

whereEnis also a real constant which is only dependent upon the order number n. The constant and positive real numberj ib_n0jhas been absorbed by redefinition of then’th soliton centerxn0 in (155). Thus the determinants in formula (83) for pure soliton solution can be calculated as follows

D_N detðIþBÞ ¼1þX^N

r¼1

X

1≤n1<⋯<nr≤N

B nð ₁,n₂,⋯,n_rÞ (158)

B nð ₁,n₂,⋯,n_rÞ ¼Y

n

f²_n ρ i z ²_n ρ⁴z_n²

" # Y

n<m

i z ²_n z²_m

iρ⁴z_m² ρ⁴z_n² i z ²_n ρ⁴z_m²

i z ²_m ρ⁴z_n²

¼Y

n

F_nY

n<m

tanh²ðΘ_n Θ_mÞ

¼Y

n

ε_nEne^iβne ^θnY

n<m

sin²ðβ_n β_mÞ

sin²ðβ_nþβ_mÞ;n,m∈ðn1,n2,⋯,nrÞ

(159)

C_N ¼detðIþAÞ ¼1þX^N

r¼1

X

1≤n1<n2<⋯<nr≤N

Aðn₁,n₂,⋯,n_rÞ (160)

A n ₁,n₂,⋯,n_f

¼Y

n

z²_n ρ²F_nY

n<m

tanh²ðΘ_n Θ_mÞ

¼Y

n

ε_nE_ne^i3βne ^θnY

n<m

sin²ðβ_n β_mÞ

sin²ðβ_nþβ_mÞ;n,m∈ðn₁,n₂,⋯,n_rÞ (161) Substituting (149)–(157) into (158)–(161), and substituting (158)–(161) into the following formula, we attain the explicit pureN-soliton solution

uN ρCNDN=D²_N oruN ρCNDN=D²_N (162) TheN ¼2 case, that is, the pure two-soliton is also a typical illustration of the general explicitN-soliton formula. According to (158)–(162), it can be calculated as follows

D2¼1þBð Þ þ1 Bð Þ þ2 Bð1, 2Þ

¼1þε₁E1e^iβ1e ^θ1þε₂E2e^iβ2e ^θ2 þε₁ε₂E1E2e^{iðβ1þβ2Þ}e ^ðθ1þθ2Þsin²ðβ₁ β₂Þ=sin²ðβ₁þβ₂Þ

¼1þε₁ sinðβ₁þβ₂Þ

sinðβ₁ β₂Þe^iβ1e ^θ1 ε₂ sinðβ₁ þβ₂Þ

sinðβ₁ β₂Þe^iβ2e ^θ2 ε₁ε₂e^{iðβ1þβ2Þ}e ^ðθ1þθ2Þ

(163) C2 ¼1þAð Þ þ1 Að Þ þ2 Að1, 2Þ

¼1þε₁ sinðβ₁þβ₂Þ

sinðβ₁ β₂Þe^i3β1e ^θ1 ε₂ sinðβ₁þβ₂Þ

sinðβ₁ β₂Þe^i3β2e ^θ2 ε₁ε₂e^{i3ðβ1þβ2Þ}e ^ðθ1þθ2Þ (164)

u₂ðx,tÞ ¼ρC₂D₂=D²₂ (165) The evolution of pure two-soliton solution with respect to time and space is given inFigure 5. It clearly demonstrates the whole process of the elastic collision between pure two solitons. If 0<β₂<β₁<π=2, thenε₁ ¼1, sgnE₁¼1 andε₂¼ 1,

sgnE2 ¼ 1 correspond to double-dark pure 2-soliton solution as inFigure 5a;

ε₁ ¼ 1, sgnE1 ¼1 andε₂¼1, sgnE2 ¼ 1 correspond to a double-bright pure 2-soliton solution inFigure 5c;ε₁¼1, sgnE1¼1 andε₂ ¼1, sgnE2 ¼ 1

correspond to a dark-bright-mixed pure 2-soliton solution inFigure 5b. In the limit of infinite timet! ∞, the pure 2-soliton solution is asymptotically decomposed into two pure 1-solitons.

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 constantE_n, (n¼1, 2,⋯,N), is also absorbed into then’th soliton centerx_n0 just like ib_n0 does in (152)–(154), and replaceβ_n withβ⁰_n¼π=2 β_n∈ð0,π=2Þ, 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 1and rewriting the expression ofa_nð Þz as in Ref. [7, 8, 12], our result can

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.

naturally generate the mixed case with both pure and breather-type multi-soliton solution.

4.2 The asymptotic behaviors of theN-soliton solution

Without loss of generality, we assumeβ₁>β₂>⋯>β_n>⋯>β_N;

υ₁<υ₂<⋯<υ_n<⋯<υ_N in (156), and define then’th neighboring area asϒ_n: x x_no υ_nt0,ðn¼1:2,⋯,NÞ. In the neighboring area ofϒ_n,

θ_j ¼ν_jx xj0 υ_jt

! f^þ∞,_∞, ^for_for^j_j^><ⁿn (166) D≈Bð1, 2,…,n 1Þ þBð1, 2,…,n 1,nÞ (167) C≈Að1, 2,…,n 1Þ þAð1, 2,…,n 1,nÞ (168) where

Bð1, 2,…,nÞ ¼ε_nEne ^{θn iβn}Y^{n 1}

j¼1

sin²β_j β_n

sin²β_jþβ_nBð1, 2,…,n 1Þ (169)

Að1, 2,…,n 1,nÞ ¼ε_nE_ne ^θnþi3βnY^{n 1}

j¼1

sin² β_j β_n

sin² β_jþβ_n

Að1, 2,…,n 1Þ (170)

In the neighboring area ofϒ_n, we have

u≃u₁θ_nþΔθ^{ð Þ}_n

(171) With

Δθ^{ð Þ}_n ¼2X^{n 1}

j¼1

ln sin β_j β_n

sin β_jþβ_n

(172)

Ast! ∞, theNneighboring areas queue up in a descending series ϒ_N,ϒ_{N 1},⋯,ϒ₁, then

u_N≃ X^N

n¼1

u₁θ_nþΔθ^{ð Þ}_n

(173) theN-soliton solution can be viewed asNwell-separated exact pure one solitons, eachu1θ_nþΔθ^{ð Þ}_n

, (1, 2,⋯,n) is a single pure soliton characterized by one parameterβ_n, moving to the positive direction of the x-axis, queuing up in a series with descending order numbern.

Ast!∞, in the neighboring area ofϒ_nwe have θ_j ¼ν_jx xj0 υ_jt

! f_þ∞,^{∞, for}_for^j_j^><n_n (174) D≈B n,ð nþ1,…,NÞ þB nð þ1,nþ2,…,NÞ (175) C≈A n,ð nþ1,…,NÞ þA nð þ1,nþ2,…,NÞ (176)

where

B n,ð nþ1,…,NÞ ¼ε_nE_ne ^{θn iβn} Y^N

j¼nþ1

sin² β_j β_n

sin² β_jþβ_n

B nð þ1,nþ2,…,NÞ (177)

A n,ð nþ1,…,NÞ ¼ε_nE_ne ^θnþi3βn Y^N

j¼nþ1

sin²β_j β_n

sin²β_jþβ_nA nð þ1,nþ2,…,NÞ (178) u≃u₁θ_nþΔθ^{ð Þ}_n^þ

(179)

Δθ^{ð Þ}_n^þ ¼2X^N

nþ1

ln sin β_jþβ_n

sin β_j β_n

(180)

uN≃ X^N

n¼1

u1 θ_nþΔθ^{ð Þ}_n^þ

(181)

That is, theN-soliton solution can be viewed asNwell-separated exact pure one solitons, queuing up in a series with ascending order numbernsuch as

ϒ₁,ϒ₂,⋯,ϒ_N:.

In the process of going fromt! ∞to t!∞, then’th pure single soliton overtakes the solitons from the 1’th ton 1’th and is overtaken by the solitons from nþ1’th toN’th. In the meantime, due to collisions, then’th soliton got a total forward shiftΔθ^{ð Þ}_n =νnfrom exceeding those slower soliton from the 1’th ton 1’th, got a total backward shiftΔθ^{ð Þ}_n^þ =ν_n from being exceeded by those faster solitons fromnþ1’th toN’th, and just equals to the summation of shifts due to each collision between two solitons, that is,

Δx_n ¼Δθ^{ð Þ}_n^þ Δθ^{ð Þ}_n =ν_n (182) 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^þequation with NVBC and normal dispersion, the rigorously proved breather-typeN-soliton solution to the DNLS^þ 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 theN-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 4and rewriting the expression ofa_nð Þz as 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^þequation 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^þequation with NVBC and normal dispersion makes corresponding Jost functions be of regular properties and asymptotic behav- iors, and thus supplies substantial foundation for its direct perturbation theory.

5. Space periodic solutions and rogue wave solution of DNLS equation