Solution of bilinear equations

TOP 1%

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

f ¼ f₀1þϵf₁þϵ²f₂

;g ¼g₀1þϵg₁þϵ²g₂

(197) Substitutingf andginto 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 Š1}ðx,tÞ ¼f^{½ Š1}g^{½ Š1}=f^{½ Š21} (198) with

g^{½ Š1} ¼ρe^iωt1þa1e^pxþΩtþϕ0 þa2e ^pxþΩtþϕ0 þMa1a2eð^ΩþΩÞ^tþϕ0þϕ0

(199) f^{½ Š1} ¼e^iβx1þb₁e^pxþΩtþϕ0 þb₂e ^pxþΩtþϕ0 þMb₁b₂eð^ΩþΩÞ^tþϕ0þϕ0

(200) where

ω¼3ρ⁴=16;β¼ρ²=4 (201)

a1 ¼b12Ωþ2ip² pρ²

2Ω 2ip² pρ²;a2 ¼b22Ωþ2ip²þpρ²

2Ω 2ip²þpρ² (202) b₂¼b₁Ωþip² pρ²

Ω ip² pρ² ;M¼1þ 4p⁴ ΩþΩ

2 (203)

Notice thatρandMare real;b₁andφ₀ are complex constants, so there are two restrictions for a valid calculation: (1) the wave numberpmust be a pure imaginary number; (2) the angular frequencyΩmust not be purely imaginary number and must furthermore satisfy the quadratic dispersion relation:

4Ω²þ4pρ²Ωþ4p⁴þ3p²ρ⁴ ¼0 (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 pis a pure imaginary number, the breather is a periodic function ofx. The quadratic dispersion relation (204) permits the angular frequencyΩto have two solutions:

Ω_þ¼ pρ²þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2pð ⁴þp²ρ⁴Þ

q

=2 (206)

Ω ¼ pρ²

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2pð ⁴þp²ρ⁴Þ

q

=2 (207)

If we setΩ ¼Ω

þ, because ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p⁴þp²ρ⁴

ð Þ

p >0, thent! ∞will lead to:

g^{½ Š1} !ρexpðiωtÞ (208)

f^{½ Š1} ! expðiβxÞ (209)

u^{½ Š1} !ρexpið 3βxþωtÞ (210) Andt!∞will lead to:

g^{½ Š1} !ρMa1a2exp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2pð ⁴þp²ρ⁴Þ q

þϕ₀þϕ₀ þiωt

(211) f^{½ Š1} !Mb₁b₂exp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2pð ⁴þp²ρ⁴Þ q

þϕ₀ þϕ₀þiβx

(212) u^{½ Š1} !ρexpið 3βxþωtþφÞ (213) whereφis the phase shift across the breather:

expð Þ ¼iφ a₁a₂=b₁b₂ (214) and due to∣a₁a₂∣¼∣b₁b₂∣, thus the above phase shiftφis real and does not affect the module of the breatheru^{½ Š1} whent!∞. As for the other choiceΩ¼Ω , further algebra computation shows the antithetical asymptotic behavior ofg^{½ Š1}, f^{½ Š1}, andu^{½ Š1} when∣t∣!∞. In a nutshell,u^{½ Š1} will degenerate into a plane wave.

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. Supposingp¼iq, hereqis a real value andq!0, then the asymptotic expansion of the angular frequencyΩis:

Ω ¼qρ²ð iþσÞ=2þO q³ (215) whereσ ¼ ffiffiffi

p2

. For the sake of a valid form of the rogue wave solution, we need to setb₁¼1 andφ₀ ¼0 (of course, settingb₁ ¼1 ande^φ0 ¼ 1 is alright, all we need is to make sure that the coefficients of theq₀ andq₁in the expansions of f^{½ Š1} andg^{½ Š1} are annihilated). Therefore, the expansions ofg^{½ Š1} and f^{½ Š1} in terms ofqare given by:

g^{½ Š1} ¼q²e^iωt 8 7ið þ5σÞ þ16xð1 2iσÞρ²þ3ð iþσÞρ⁴ð4x² 4ρ²tx 8itþ3ρ⁴t²Þ

12ð iþσÞρ³ þO q³ (216)

f^{½ Š1} ¼q²e^iβx8ð iþσÞ þ16xρ²þð iþσÞρ⁴ð4x² 4ρ²tx 8itþ3ρ⁴t²Þ

4ð iþσÞρ⁴ þO q³ (217)

Consequently, the rogue wave solution can be derived according to Eq. (198):

u_RW ¼ρe^{ið 3βxþωtÞ}g⁰ f⁰

= f⁰² (218)

where

g⁰ ¼ 8 7ið þ5σÞ þ16xð1 2iσÞρ²þ3ð iþσÞρ⁴4x² 4ρ²tx 8itþ3ρ⁴t²

; f⁰ ¼24ð iþσÞ þ48xρ²þ3ð iþσÞρ⁴4x² 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 Š1}=f^{½ Š1} (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 andΩ¼Ω

þ, complex constantφ₀is set to zero.

where,g^{½ Š1}, f^{½ Š1}, 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 ¼ρe^{ið βxþωtÞ}g⁰=f⁰ (220) which has the same parameters asuRW. 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 functionsfandgto have higher order expansions in terms ofϵ:

g¼g₀1þϵg₁þϵ²g₂þϵ³g₃þϵ⁴g₄

(221) f ¼ f₀1þϵf₁þϵ²f₂þϵ³f₃þϵ⁴f₄

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

u^{½ Š2}ðx,tÞ ¼f^{½ Š2}g^{½ Š2}=f^{½ Š22} (223) with

g^{½ Š2} ¼ρe^iωt1þg₁þg₂þg₃þg₄

(224) f^{½ Š2} ¼e^iβx1þ f₁þ f₂þ f₃þ f₄

(225) β¼ρ²=4;ω¼3ρ⁴=16;λ¼ρ⁴=16 (226)

g₁¼X

i

a_ie^ϕi; f₁¼X

i

b_ie^ϕi (227)

Figure 7.

The space-time evolution of the module of the rogue wave solution withρ¼1andσ¼ ffiffiffi p2

. The max amplitude is equal to 3 at the point x¼ pffiffiffi2

,t¼ 2pffiffiffi2

ð =3Þ.

g₂ ¼X

i<j

M_ija_ia_je^ϕiþϕj; f₂ ¼X

i<j

M_ijb_ib_je^ϕiþϕj (228) g₃ ¼ X

i<j<k

T_ijka_ia_ja_ke^{ϕiþϕjþϕk}; f₃ ¼ X

i<j<k

T_ijkb_ib_jb_ke^{ϕiþϕjþϕk} (229) g₄ ¼Aa₁a₂a₃a₄e^{ϕ1þϕ2þϕ3þϕ4}; f₄ ¼Ab₁b₂b₃b₄e^{ϕ1þϕ2þϕ3þϕ4} (230) wherei,j,k¼1, 2, 3, 4, and the above parameters and coefficients are given respectively by:

p₂ ¼p₁;p₄ ¼p₃;Ω₂ ¼Ω₁;Ω₄ ¼Ω₃ (231) ϕ_i¼p_ixþΩ_itþϕ_0i;a_i ¼b_i2Ω_iþ2ip²_i p_iρ²=2Ω_i 2ip²_i p_iρ² (232)

b₂ ¼b₁Ω₂þip²₂þp₂ρ²

Ω₂ ip²₂þp₂ρ²;b₄ ¼b₃Ω₄þip²₄þp₄ρ²

Ω₄ ip²₄þp₄ρ² (233) M_ij ¼

Ω_ip_j Ω_jp_i

2

þp²_ip²_jp_i p_j2

Ω_ip_j Ω_jp_i

2

þp²_ip²_jp_iþp_j2 (234) T_ijk ¼M_ijM_jkM_ki;A¼Y

i<j

M_ij (235)

Of course, for a valid and complete calculation, we are faced with the same situation as the 1st-order breather:ρis real,b₁,b₃ and allφ_0i are complex constants.

Certainly, each wave numberp_imust be a pure imaginary number and each angular frequencyΩ_i has to satisfy the quadratic dispersion relation:

4Ω²

i þ4p_iρ²Ω_iþ4p⁴_i þ3p²_iρ⁴ ¼0,ði ¼1, 2, 3, 4Þ (236) And the threshold conditions for each complex-valuedΩ_ishare 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,p₃ ¼0:75i,b1¼i,b3¼1andρ¼1:6. 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 takep₃ ¼p₁. Under this condition,M₁₃ ¼M₂₄ ¼0, thus the higher order interaction coefficientsT_ijk andAwill vanish. Therefore,g^{½ Š2} and

f^{½ Š2} will degenerate into the forms ofg^{½ Š1} and f^{½ Š1}, respectively:

g^{½ Š}_p²

3¼p₁ ¼g^{0½ Š1} ¼ρe^iωt1þa⁰₁e^ϕ1 þa⁰₂e^ϕ2 þM12a⁰₁a⁰₂e^ϕ1þϕ2

(238) f^{½ Š}_p²

3¼p₁ ¼ f^{0½ Š1} ¼e^iβx1þb⁰₁e^ϕ1þb⁰₂e^ϕ2 þM₁₂b⁰₁b⁰₂e^ϕ1þϕ2

(239) whereb⁰₁¼χb₁,b⁰₂¼χb₂,a⁰₁ ¼χa₁anda⁰₂ ¼χa₂withχ ¼ðb₁þb₃Þ=b₁. That is howu^{½ Š2} can be reduced tou^{½ Š1}. Given to this reduction, a generalized form of these two breathers arises:

u^{½ ŠN} ¼f^{½ ŠN}g^{½ ŠN}=f^{½ Š2N} ;ðN ¼1, 2Þ (240) g^{½ ŠN} ¼ρe^iωt 1þX^2N

r¼1

X

1≤n1<⋯<nr≤2N

M nð 1,⋯,nrÞY^nr

i¼n1

aie^ϕi

!

(241)

f^{½ ŠN} ¼e^iβx 1þX^2N

r¼1

X

1≤n1<⋯<nr≤2N

M nð 1,⋯,nrÞY^nr

i¼n1

bie^ϕi

!

(242) where the coefficientMis defined by:

M ið Þ ¼1 (243)

M nð ₁,⋯,n_rÞ ¼Y

i<j

M_{i j};i,j∈ðn₁,⋯,n_rÞ (244)

On the other hand, this breather possesses the same feature as the former one that it is periodic with respect to variablexdue to the pure imaginary numbersp₁ andp₃. 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 ⁴₁ þp²₁ρ⁴

q

=2 (245)

Ω3 ¼ p₃ρ²

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2p ⁴₃ þp²₃ρ⁴

q

=2 (246)

Thus, we will have four combinations ofΩ₁andΩ₂. Details are numerated in Table 1. The parametersφ₀,φandφ⁰ inTable 1are the phase shifts which are all real so that they will not change the module ofu^{½ Š2} whent!∞. Andφis given in Eq. (214), and others are determined by:

expðiφ₀Þ ¼a₁a₂a₃a₄=b₁b₂b₃b₄ (247) expð Þ ¼iφ⁰ 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 fromt¼ ∞tot¼∞, which depended on the choice ofΩ₁andΩ₂.

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 transforma- tion, 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 solu- tion has multiple wave numbersp_i, we are also interested in seeking an alternative method besides DT that could help us to determine the higher order rogue wave solutions.