Solution of bilinear equations - Space periodic solutions and rogue wave solution of DNLS equat

Y, Θ ! ffiffiffiffiffiffiffiffi

5. Space periodic solutions and rogue wave solution of DNLS equation DNLS equation is one of the most important nonlinear integrable equations in

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 f and g in (189) are cut off, up to the 2’th power order ofϵ, and have the following formal form:

f ¼ f₀�1þϵf₁þϵ²f₂�

; g¼g₀�1þϵg₁þϵ²g₂�

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

u^{½ �}¹ðx, tÞ ¼f^{½ �}¹g^{½ �}¹=f^{½ �2}¹ (198) with

g^{½ �}¹ ¼ρe^iωt�1þa₁e^pxþΩtþ^ϕ⁰þa₂e^�pxþΩtþ^ϕ⁰þMa₁a₂eð^ΩþΩÞ^tþϕ⁰^þϕ⁰�

(199) f^{½ �}¹ ¼e^iβx�1þb₁e^pxþΩtþ^ϕ⁰þb₂e^�pxþΩtþ^ϕ⁰þMb₁b₂eð^ΩþΩÞ^tþϕ⁰^þϕ⁰�

(200) where

ω¼3ρ⁴=16;β¼ρ²=4 (201)

a₁¼b₁2Ωþ2ip²�pρ²

2Ω�2ip²�pρ²; a₂¼b₂2Ωþ2ip²þpρ²

2Ω�2ip²þpρ² (202) b₂¼b1�Ωþip²�pρ²�

Ω�ip²�pρ² ; M¼1þ 4p⁴ ΩþΩ

� �2 (203)

Notice thatρand M are real; b1andφ₀are complex constants, so there are two restrictions for a valid calculation: (1) the wave number p must be a pure imaginary number; (2) the angular frequencyΩmust not be purely imaginary number and must furthermore satisfy the quadratic dispersion relation:

4Ω²þ4pρ²Ωþ4p⁴þ3p²ρ⁴¼0 (204) According to the test rule for a one-variable quadratic, there is a threshold condition under whichΩwill not be a pure imaginary number:

2p⁴þp²ρ⁴<0 (205)

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 frequencyΩto have two solutions:

Ωþ¼ �pρ²þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2pð ⁴þp²ρ⁴Þ

� q �

=2 (206)

Ω�¼ �pρ²�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2pð ⁴þp²ρ⁴Þ

� q �

=2 (207)

If we setΩ¼Ωþ, because ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�ð2p⁴þp²ρ⁴Þ

p >0, then t! �∞ will lead to:

g^{½ �}¹ !ρexp iωtð Þ (208)

f^{½ �}¹ ! exp iðβxÞ (209)

u^{½ �}¹ !ρexp ið�3βxþωtÞ (210) And t!∞ will lead to:

g^{½ �}¹ !ρMa1a2exp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2pð ⁴þp²ρ⁴Þ q

þϕ₀þϕ₀þiωt

� �

(211) f^{½ �}¹ !Mb1b2exp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2pð ⁴þp²ρ⁴Þ q

þϕ₀þϕ₀þiβx

� �

(212) u^{½ �}¹ !ρexp ið�3βxþωtþφÞ (213) whereφis the phase shift across the breather:

exp iφð Þ ¼a1a2=b1b2 (214) and due to∣a1a₂∣¼∣b1b₂∣, thus the above phase shiftφis real and does not affect the module of the breather u^{½ �}¹ when t!∞. As for the other choiceΩ¼Ω�, further algebra computation shows the antithetical asymptotic behavior of g^{½ �}¹, f^{½ �}¹, and u^{½ �}¹ when∣t∣!∞. In a nutshell, u^{½ �}¹ will degenerate into a plane wave.

a₁¼b₁2Ωþ2ip²�pρ²

2Ω�2ip²�pρ²; a₂¼b₂2Ωþ2ip²þpρ²

2Ω�2ip²þpρ² (202) b₂¼b1�Ωþip²�pρ²�

Ω�ip²�pρ² ; M¼1þ 4p⁴ ΩþΩ

� �2 (203)

4Ω²þ4pρ²Ωþ4p⁴þ3p²ρ⁴¼0 (204) According to the test rule for a one-variable quadratic, there is a threshold condition under whichΩwill not be a pure imaginary number:

2p⁴þp²ρ⁴<0 (205)

Ωþ¼ �pρ²þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2pð ⁴þp²ρ⁴Þ

� q �

=2 (206)

Ω�¼ �pρ²�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2pð ⁴þp²ρ⁴Þ

� q �

=2 (207)

If we setΩ¼Ωþ, because ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�ð2p⁴þp²ρ⁴Þ

p >0, then t! �∞ will lead to:

g^{½ �}¹ !ρexp iωtð Þ (208)

f^{½ �}¹ ! exp iðβxÞ (209)

u^{½ �}¹ !ρexp ið�3βxþωtÞ (210) And t!∞ will lead to:

g^{½ �}¹ !ρMa1a2exp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2pð ⁴þp²ρ⁴Þ q

þϕ₀þϕ₀þiωt

� �

(211) f^{½ �}¹ !Mb1b2exp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2pð ⁴þp²ρ⁴Þ q

þϕ₀þϕ₀þiβx

� �

(212) u^{½ �}¹ !ρexp ið�3βxþωtþφÞ (213) whereφis the phase shift across the breather:

Hereto, we have completed the computation of the 1st-order space periodic solution, the space-time evolution of its module is depicted inFigure 6. In what follows, we will take the long-wave limit, that is, p!0, to construct a rogue wave solution. Supposing p¼iq, here q is a real value and q!0, then the asymptotic expansion of the angular frequencyΩis:

Ω¼qρ²ð�iþσÞ=2þO q� �³

(215) whereσ¼ � ffiffiffi

. For the sake of a valid form of the rogue wave solution, we need to set b1¼1 andφ₀¼0 (of course, setting b1¼1 and e^φ⁰ ¼ �1 is alright, all we need is to make sure that the coefficients of the q₀and q₁in the expansions of f^{½ �}¹ and g^{½ �}¹ are annihilated). Therefore, the expansions of g^{½ �}¹ and f^{½ �}¹ in terms of q are given by:

g^{½ �}¹ ¼q²e^iωt�8 7ið þ5σÞ þ16x 1ð �2iσÞρ²þ3ð�iþσÞρ⁴ð4x²�4ρ²tx�8itþ3ρ⁴t²Þ

12ð�iþσÞρ³ þO q� �³

(216) f^{½ �}¹ ¼q²eⁱ^β^x8ð�iþσÞ þ16xρ²þ �ið þσÞρ⁴ð4x²�4ρ²tx�8itþ3ρ⁴t²Þ

4ð�iþσÞρ⁴ þO q� �³ (217) Consequently, the rogue wave solution can be derived according to Eq. (198):

uRW ¼ρeⁱ^ð^�3^β^xþ^ω^t^Þ�g⁰ f⁰�

=f⁰² (218)

where

g⁰¼ �8 7ið þ5σÞ þ16x 1ð �2iσÞρ²þ3ð�iþσÞρ⁴�4x²�4ρ²tx�8itþ3ρ⁴t²�

; f⁰¼24ð�iþσÞ þ48xρ²þ3ð�iþσÞρ⁴�4x²�4ρ²tx�8itþ3ρ⁴t²�

Hereωandβare given by Eq. (201),ρis an arbitrary real constant. The module of rogue wave solution Eq. (218) is shown inFigure 7.

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:

υ_cðx, tÞ ¼g^{½ �}¹=f^{½ �}¹ (219)

Figure 6.

The space-time evolution of the module of the 1st order space periodic solution in (198) with pffiffiffi ¼i,ρ¼ p2

, b1¼i andΩ¼Ωþ, complex constantφ₀is set to zero.

where, g^{½ �}¹, f^{½ �}¹, 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 turnυ_cinto a rogue wave solution of the CLL equation:

υ_c,RW ¼ρeⁱ^ð^�βxþωt^Þg⁰=f⁰ (220) which has the same parameters as uRW. And this solutionυ_c,RW has 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 ofϵ:

g¼g₀�1þϵg₁þϵ²g₂þϵ³g₃þϵ⁴g₄�

(221) f ¼ f₀�1þϵf₁þϵ²f₂þϵ³f₃þϵ⁴f₄�

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

u^{½ �}²ðx, tÞ ¼f^{½ �}²g^{½ �}²=f^{½ �2}² (223) with

g^{½ �}² ¼ρe^iωt�1þg₁þg₂þg₃þg₄�

(224) f^{½ �}² ¼e^iβx�1þf₁þf₂þf₃þ f₄�

(225) β¼ρ²=4;ω¼3ρ⁴=16;λ¼ρ⁴=16 (226)

g₁¼X

aie^ϕⁱ; f₁¼X

bie^ϕⁱ (227)

Figure 7.

The space-time evolution of the module of the rogue wave solution withρ¼1 andσ¼pffiffiffi2. The max amplitude is equal to 3 at the point xð ¼ �pffiffiffi2, t¼ �2pffiffiffi2=3Þ.

υ_c,RW ¼ρeⁱ^ð^�βxþωt^Þg⁰=f⁰ (220) which has the same parameters as uRW. And this solutionυ_c,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 ofϵ:

g¼g₀�1þϵg₁þϵ²g₂þϵ³g₃þϵ⁴g₄�

(221) f ¼ f₀�1þϵf₁þϵ²f₂þϵ³f₃þϵ⁴f₄�

u^{½ �}²ðx, tÞ ¼f^{½ �}²g^{½ �}²=f^{½ �2}² (223) with

g^{½ �}² ¼ρe^iωt�1þg₁þg₂þg₃þg₄�

(224) f^{½ �}² ¼e^iβx�1þf₁þf₂þf₃þf₄�

(225) β¼ρ²=4;ω¼3ρ⁴=16;λ¼ρ⁴=16 (226)

g₁¼X

aie^ϕⁱ; f₁¼X

bie^ϕⁱ (227)

Figure 7.

The space-time evolution of the module of the rogue wave solution withρ¼1 andσ¼pffiffiffi2. The max amplitude is equal to 3 at the point xð ¼ �pffiffiffi2, t¼ �2pffiffiffi2=3Þ.

g₂¼X

i<j

M_ija_ia_je^ϕⁱ^þ^ϕ^j; f₂¼X

i<j

M_ijb_ib_je^ϕⁱ^þ^ϕ^j (228) g₃¼ X

i<j<k

T_ijka_ia_ja_ke^ϕⁱ^þ^ϕ^j^þ^ϕ^k; f₃¼ X

i<j<k

T_ijkb_ib_jb_ke^ϕⁱ^þ^ϕ^j^þ^ϕ^k (229) g₄¼Aa₁a₂a₃a₄e^ϕ¹^þ^ϕ²^þ^ϕ³^þ^ϕ⁴; f₄¼Ab₁b₂b₃b₄e^ϕ¹^þ^ϕ²^þ^ϕ³^þ^ϕ⁴ (230) where i, j, k¼1, 2, 3, 4, and the above parameters and coefficients are given respectively by:

p₂¼p₁; p₄¼p₃;Ω2¼Ω1;Ω4¼Ω3 (231) ϕ_i¼p_ixþΩitþϕ_0i; ai¼bi2Ωiþ2ip²_i �p_iρ²=2Ωi�2ip²_i �p_iρ² (232)

b₂¼b₁Ω2þip²₂þp₂ρ²

Ω2�ip²₂þp₂ρ²; b₄¼b₃Ω4þip²₄þp₄ρ²

Ω4�ip²₄þp₄ρ² (233)

Mij¼�Ωip_j�Ωjp_i�₂

þp²_ip²_j�p_i�p_j�₂ Ωip_j�Ωjp_i

� �₂

þp²_ip²_j�p_iþp_j�₂ (234) Tijk¼MijMjkMki; A¼Y

i<j

Mij (235)

Of course, for a valid and complete calculation, we are faced with the same situation as the 1st-order breather:ρis real, b1, b3and allφ_0iare complex constants.

Certainly, each wave number p_imust be a pure imaginary number and each angular frequencyΩihas to satisfy the quadratic dispersion relation:

4Ω²_i þ4p_iρ²Ωiþ4p⁴_i þ3p²_iρ⁴¼0, ið ¼1, 2, 3, 4Þ (236) And the threshold conditions for each complex-valuedΩishare the same form as Eq. (205):

2p⁴_i þp²_iρ⁴<0 (237)

Figure 8.

The space-time evolution of the module of the 2nd order space periodic solution with

p₁¼0:4i, p₃¼0:75i, b1¼i, b3¼1 andρ¼1:6. Other phase factorsφ₁andφ₃are set to zero.

The space-time evolution of the module of the 2nd order space periodic solution (223) is shown inFigure 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 p₃¼p₁. Under this condition, M₁₃¼M₂₄¼0, thus the higher order interaction coefficients Tijkand A will vanish. Therefore, g^{½ �}² and

f^{½ �}² will degenerate into the forms of g^{½ �}¹ and f^{½ �}¹, respectively:

g^{½ �}_p²

3¼p₁ ¼g⁰^{½ �}¹ ¼ρe^iωt�1þa⁰₁e^ϕ¹þa⁰₂e^ϕ²þM12a⁰₁a⁰₂e^ϕ¹^þϕ²�

(238) f^{½ �}_p²

3¼p₁ ¼ f⁰^{½ �}¹ ¼e^iβx�1þb⁰₁e^ϕ¹ þb⁰₂e^ϕ²þM₁₂b⁰₁b⁰₂e^ϕ¹^þ^ϕ²�

(239) where b⁰₁¼χb₁, b⁰₂¼χb₂, a⁰₁¼χa₁and a⁰₂¼χa₂withχ¼ðb₁þb₃Þ=b₁. That is how u^{½ �}² can be reduced to u^{½ �}¹. Given to this reduction, a generalized form of these two breathers arises:

u^{½ �}^N ¼f^{½ �}^Ng^{½ �}^N=f^{½ �2}^N ; Nð ¼1, 2Þ (240) g^{½ �}^N ¼ρe^iωt 1þX^2N

r¼1

1≤n1<⋯<nr≤2N

M nð 1,⋯, nrÞYⁿ^r

i¼n1

aie^ϕⁱ

(241)

f^{½ �}^N ¼e^iβx 1þX^2N

r¼1

1≤n1<⋯<nr≤2N

M nð ₁,⋯, nrÞYⁿ^r

i¼n1

b_ie^ϕⁱ

(242) where the coefficient M is defined by:

M ið Þ ¼1 (243)

M nð ₁,⋯, nrÞ ¼Y

i<j

M_{i j}; i, j∈ðn₁,⋯, nrÞ (244)

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 p₁ and p₃. In addition, its asymptotic behaviors are analogical to the 1st-order space periodic solution. Each quadratic dispersion equation has two roots, respectively:

Ω1�¼ �p₁ρ²�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2p� ⁴₁þp²₁ρ⁴�

� q �

=2 (245)

Ω3�¼ �p₃ρ²�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2p� ⁴₃þp²₃ρ⁴�

� q �

=2 (246)

Thus, we will have four combinations ofΩ1andΩ2. Details are numerated in Table 1. The parametersφ₀,φandφ⁰inTable 1are the phase shifts which are all real so that they will not change the module of u^{½ �}² when t!∞. Andφis given in Eq. (214), and others are determined by:

exp iφð ₀Þ ¼a₁a₂a₃a₄=b₁b₂b₃b₄ (247) exp iφð Þ ¼⁰ a3a4=b3b4 (248) FromTable 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 t¼ �∞ to t¼∞, which depended on the choice ofΩ1andΩ2.

f^{½ �}² will degenerate into the forms of g^{½ �}¹ and f^{½ �}¹, respectively:

g^{½ �}_p²

3¼p₁ ¼g⁰^{½ �}¹ ¼ρe^iωt�1þa⁰₁e^ϕ¹þa⁰₂e^ϕ²þM12a⁰₁a⁰₂e^ϕ¹^þϕ²�

(238) f^{½ �}_p²

3¼p₁ ¼ f⁰^{½ �}¹ ¼e^iβx�1þb⁰₁e^ϕ¹þb⁰₂e^ϕ²þM₁₂b⁰₁b⁰₂e^ϕ¹^þ^ϕ²�

u^{½ �}^N ¼f^{½ �}^Ng^{½ �}^N=f^{½ �2}^N ; Nð ¼1, 2Þ (240) g^{½ �}^N ¼ρe^iωt 1þX^2N

r¼1

1≤n1<⋯<nr≤2N

M nð 1,⋯, nrÞYⁿ^r

i¼n1

aie^ϕⁱ

(241)

f^{½ �}^N ¼e^iβx 1þX^2N

r¼1

1≤n1<⋯<nr≤2N

M nð ₁,⋯, nrÞYⁿ^r

i¼n1

b_ie^ϕⁱ

(242) where the coefficient M is defined by:

M ið Þ ¼1 (243)

M nð ₁,⋯, nrÞ ¼Y

i<j

M_{i j}; i, j∈ðn₁,⋯, nrÞ (244)

Ω1�¼ �p₁ρ²�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2p� ⁴₁þp²₁ρ⁴�

� q �

=2 (245)

Ω3�¼ �p₃ρ²�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 2p� ⁴₃þp²₃ρ⁴�

� q �

=2 (246)

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

Dalam dokumen Nonlinear Optics (Halaman 95-101)