Oscillatory solitons of U(1)-invariant mKdV equations I: Envelope speed and temporal frequency
OOSCILLATORY SOLITONS OF U(1)-INVARIANT MKDV EQUATIONS I:ENVELOPE SPEED AND TEMPORAL FREQUENCY
STEPHEN C. ANCO , ABDUS SATTAR MIA , , MARK R. WILLOUGHBY department of mathematicsbrock universityst. catharines, on canada department of mathematics and statisticsuniversity of saskatchewansaskatoon, sk canadaAbstract. Harmonically modulated complex solitary waves which are a generalized typeof envelope soliton (herein called oscillatory solitons ) are studied for the two U (1)-invariantintegrable generalizations of the modified Korteweg-de Vries equation, given by the Hirotaequation and the Sasa-Satsuma equation. A bilinear formulation of these two equations isused to derive the oscillatory 1-soliton and 2-soliton solutions, which are then written outin a physical form parameterized in terms of their speed, modulation frequency, and phase.Depending on the modulation frequency, the speeds of oscillatory waves (1-solitons) can bepositive, negative, or zero, in contrast to the strictly positive speed of ordinary solitons.When the speed is zero, an oscillatory wave is a time-periodic standing wave. Propertiesof the amplitude and phase of oscillatory 1-solitons are derived. Oscillatory 2-solitons aregraphically illustrated to describe collisions between two oscillatory 1-solitons in the casewhen the speeds are distinct. In the special case of equal speeds, oscillatory 2-solitons areshown to reduce to harmonically modulated breather waves. Introduction
The modified Korteweg-de Vries (mKdV) equation u t + au u x + bu xxx = 0 (1.1)(where a and b are arbitrary positive constants) is an integrable evolution equation whicharises in many physical applications, such as acoustic waves in anharmonic lattices [1] andAlfven waves in collision-free plasmas [2]. Its well-known integrability properties consistof multi-soliton solutions, a Lax pair, a bi-Hamiltonian structure, an infinite hierarchy ofsymmetries and conservation laws, and a bilinear formulation. Soliton solutions of the mKdVequation are solitary waves u ( t, x ) = (cid:15) (cid:114) ca sech (cid:16)(cid:114) cb ( x − ct ) (cid:17) (1.2)whose shape, wave speed c >
0, and up/down orientation (cid:15) = ± Key words and phrases. mKdV equation, Hirota equation, Sasa-Satsuma equation, solitary wave, envelopesoliton, oscillatory soliton, breather, overtake collision, head-on collision. a r X i v : . [ n li n . S I] N ov ave solutions that are single-peaked, unidirectional, and decaying for large | x | . Collisions oftwo or more mKdV solitary waves are described by multi-soliton solutions that reduce to alinear superposition of distinct solitary waves in the asymptotic past and future. All mKdVsoliton solutions carry mass, momentum, energy, as well as Galilean energy associated withthe motion of center of momentum, which are constants of motion arising from conservationlaws for the mKdV equation (1.1). Remarkably, the only net effect of a collision is to shiftthe asymptotic positions of the solitary waves such that the center of momentum moves ata constant speed throughout the collision.There are exactly two integrable complex generalizations [3, 4] of the mKdV equation(1.1). One is the Hirota equation [5] u t + a | u | u x + bu xxx = 0 , (1.3)and the other is the Sasa-Satsuma equation [6] u t + a ( u ¯ u x + 3 u x ¯ u ) u + bu xxx = 0 . (1.4)These two equations share the same scaling symmetry t → λ t, x → λx, u → λ − u (1.5)admitted by the mKdV equation (1.1), and possess an additional U (1) phase symmetry u → exp( iφ ) u. (1.6)Both the Hirota equation (1.3) and the Sasa-Satsuma equation (1.4) are interesting physicallyand mathematically. In particular, under a Galilean transformation t → t , x → x − vt combined with a phase-modulation transformation u → exp( i ( kx + ωt )) u , each equationtakes the form of a 3rd order generalization of the nonlinear Schrodinger equation, describingshort wave pulses in optical fibers [7, 8] and deep water waves [9, 10]. Both equations haveintegrability properties similar to those of the mKdV equation, and their solitary wavesolutions have the form of mKdV solitons up to a phase factor u ( t, x ) = exp( iφ ) f mKdV ( x − ct ) (1.7)where c > − π ≤ φ ≤ π is the phase angle, and where f mKdV is theenvelope function f mKdV ( x − ct ) = (cid:114) ca sech (cid:16)(cid:114) cb ( x − ct ) (cid:17) . (1.8)For each equation (1.3) and (1.4), collisions of two or more solitary waves are describedby multi-soliton solutions with the main feature that the net effect on the solitary wavesis a shift in their asymptotic positions, while their asymptotic phases stay unchangedin the case of the Hirota equation (1.3) but undergo a shift in the case of the Sasa-Satsuma equation (1.4). The actual nonlinear interaction of these solitary waves dur-ing a collision exhibits interesting features which depend on the speed ratios and rela-tive phase angles of the waves, as studied in recent work [11]. (See the animations athttp://lie.math.brocku.ca/ ~ sanco/solitons/mkdv solitons.php)Most interestingly, both the Hirota equation (1.3) and the Sasa-Satsuma equation (1.4)possess a more general type of soliton solution [12] u ( t, x ) = exp( iφ ) exp( i ( κx + ωt )) f ( kx + wt ) (1.9) hich has the form of a solitary wave exp( iφ ) f ( kx + wt ), with speed c = − w/k and phaseangle φ , modulated by a harmonic wave exp( i ( κx + ωt )), with frequency ω/ (2 π ) and wavelength 2 π/κ , satisfying the algebraic relations w = − bk ( k − κ ) , ω = − bκ (3 k − κ ) , κ (cid:54) = 0 . (1.10)The envelope function f in this solution differs from f mKdV , specifically f H ( kx + wt ) = (cid:114) ba | k | sech (cid:0) kx + wt (cid:1) (1.11)in the case of the Hirota equation (1.3), and f SS ( kx + wt ) = (cid:114) ba | k | (cid:0) k + κ + ( κ + ik )( κ/
2) exp (cid:0) kx + wt ) (cid:1)(cid:1) ( k + κ ) cosh (cid:0) kx + wt (cid:1) + ( κ /
8) exp (cid:0) kx + wt ) (cid:1) (1.12)in the case of the Sasa-Satsuma equation (1.4). If κ = 0 (and hence ω = 0) then theseenvelope functions (1.11) and (1.12) reduce to f mKdV , whereby the soliton solution (1.9)reduces to the solitary wave (1.7). In contrast to an ordinary soliton (1.7), the envelopespeed c = − w/k = b ( k − κ ) (1.13)can be positive, negative, or zero, depending on whether | κ | is less than, greater than, orequal to | k | / √
3, respectively. Consequently, harmonically modulated solitons can have threedifferent types of collisions: (1) right-overtake — where a faster right-moving soliton over-takes a slower right-moving soliton or a stationary soliton; (2) left-overtake — where a fasterleft-moving soliton overtakes a slower left-moving soliton or a stationary soliton; (3) head-on — where a right-moving soliton collides with a left-moving soliton. All of these collisionscan be expected to exhibit highly interesting new features compared to collisions of ordinarysolitons. However, very little seems to be known about the explicit behaviour of collidingharmonically modulated soliton solutions in the literature on the Hirota equation (1.3) andthe Sasa-Satsuma equation (1.4), although general formulas yielding the harmonically mod-ulated multi-soliton solutions for both equations have been known for some time [5, 6, 13].The present paper and a sequel paper will be devoted to studying the basic properties ofharmonically modulated complex solitons and their nonlinear interactions. One new aspectof the analysis is that it will introduce a direct physical parameterization for these solitons,which will greatly help in understanding their interaction properties.In section 2, we use a bilinear formulation of general U (1)-invariant complex mKdV equa-tions to derive explicit expressions for the harmonically modulated 2-soliton solutions u ( t, x ) = exp( iφ ) exp( i ( κ x + ω t )) f ( k x + w t, k x + w t )+ exp( iφ ) exp( i ( κ x + ω t )) f ( k x + w t, k x + w t ) (1.14)of the Hirota equation (1.3) and the Sasa-Satsuma equation (1.4). The envelope functions f and f turn out to be complex functions, which implies that the solitary wave envelopes aregiven by | f | and | f | while their modulation comes from the overall phases κ x + ω + arg( f )and κ x + ω + arg( f ) . Consequently, for the purpose of analytically and graphicallyunderstanding these solutions, it is mathematically and physically preferable to rewrite theexpressions (1.14) in a different parameterization given by the speed of the envelopes andthe frequency of the modulations. n section 3, we first express the harmonically modulated 1-soliton solutions (1.9) in thedirect physical parameterization u ( t, x ) = exp( iφ ) exp( iνt ) ˜ f ( x − ct ) (1.15)involving only the envelope speed c = − w/k , a temporal modulation frequency ν = ω + cκ ,and the phase angle φ . We show that c and ν obey a simple kinematic relation which givesa direct way to classify the cases for which c is positive, negative, or zero, depending onlyon ν . In particular, when the envelope speed is c = 0, these solutions describe time-periodicstanding waves u ( t, x ) = exp( iφ ) exp( iνt ) ˜ f ( x ) . (1.16)We next express the harmonically modulated 2-soliton solutions (1.14) for the Hirota equa-tion (1.3) and the Sasa-Satsuma equation (1.4) in an analogous physical parameterization,given by u ( t, x ) = exp( iφ ) exp( iν t ) ˜ f ( x − c t, x − c t ) + exp( iφ ) exp( iν t ) ˜ f ( x − c t, x − c t ) (1.17)if the envelope speeds c = − w /k and c = − w /k are distinct, or u ( t, x ) = exp( iφ ) exp( iν t ) ˜ f ( x − ct, ( ν − ν ) t )+exp( iφ ) exp( iν t ) ˜ f ( x − ct, ( ν − ν ) t ) (1.18)with c = − w /k = − w /k if the envelope speeds are equal.We will call solitary wave solutions of the form (1.15) an oscillatory soliton , and solutionsof the form (1.17) an oscillatory -soliton . The special case (1.18) describes a solitary wavesolution that has speed c and involves two temporal frequencies ν and ν . When thesefrequencies are related by ν + ν = 0, the resulting wave is a breather , which has theequivalent general form u ( t, x ) = exp( iφ ) ˜ f ( x − ct, νt + φ ) (1.19)involving the frequency ν = ν = − ν and the phase angles φ = ( φ − φ ) / φ =( φ + φ ) /
2. When the frequencies are independent, ν + ν (cid:54) = 0, we will write the solitarywave solution (1.18) in a similar physical form u ( t, x ) = exp( i ( ν t + φ )) ˜ f ( x − ct, νt + φ ) (1.20)which we call an oscillatory breather , with an envelope frequency ν = ( ν − ν ) / φ = ( φ − φ ) /
2, and a temporal modulation frequency ν = ( ν + ν ) / (cid:54) = 0 and phaseangle φ = ( φ + φ ) / | u | and phase arg( u ) for the oscillatory 1-solitons (1.15) of the Hirota equation (1.3)and the Sasa-Satsuma equation (1.4). Next we graphically illustrate that the oscillatory 2-solitons (1.17) of these two equations describe collisions of oscillatory 1-solitons with distinctspeeds c (cid:54) = c , and that the oscillatory breathers (1.20) of the equations describe solitarywaves whose amplitude displays time-periodic oscillations with frequency ν and speed c .Finally, in section 5 we make some concluding remarks. ll computations in the paper have been carried out by use of Maple. Hereafter, by scalingvariables t, x, u , we will put a = 24 , b = 1 (1.21)for convenience.2. Derivation of harmonically modulated soliton solutions
Consider a general U (1)-invariant complex mKdV equation u t + ( αu ¯ u x + βu x ¯ u ) u + γu xxx = 0 (2.1)where α, β, γ are constants. A bilinear formulation of this equation can be obtained by thefollowing steps [14, 12]. First, we convert equation (2.1) into a rational form through thestandard transformation u = G/F, ¯ u = ¯ G/F (2.2)where F ( t, x ) is a real function and G ( t, x ) is a complex function. Second, we express allderivatives of F , G and ¯ G in terms of Hirota’s bilinear operator defined by D ( f, g ) = gDf − f Dg (2.3) D ( f, g ) = gD f + f D g − Df ) Dg (2.4) D ( f, g ) = gD f − Dg ) D f + 3( Df ) D g − f D g (2.5)where D denotes a total derivative. Last, we split the resulting rational equation0 = F ( γD x ( G, F )+ D t ( G, F )) − αGF D x ( G, ¯ G ) − (3 γD x ( F, F ) − ( α + β ) G ¯ G ) D x ( G, F ) (2.6)into a system of bilinear equations D t ( G, F ) + γD x ( G, F ) = GH (2.7a)3 γD x ( F, F ) − ( α + β ) G ¯ G = λGH (2.7b) αD x ( G, ¯ G ) + λD x ( G, F ) =
F H (2.7c)where H ( t, x ) is an auxiliary function and λ is a complex constant.When the original equation (2.1) is integrable, N -soliton solutions can be derived via theHirota ansatz λ = 0 (2.8) G = complex polynomial of odd degree in e Θ i , e ¯Θ j (2.9) F = real polynomial of even degree in e Θ i , e ¯Θ j (2.10) H = complex polynomial of even degree in e Θ i , e ¯Θ j (2.11)with Θ i = k i x + w i t, ¯Θ i = ¯k i x + ¯w i t, i = 1 , . . . , N (2.12)where k i and w i are complex constants when harmonically modulated solitons are sought orreal constants if ordinary solitons are sought instead. Substitution of this ansatz (2.8)–(2.11)into the bilinear system (2.7) yields a system of algebraic equations given by polynomialsin e Θ i , e ¯Θ j , whose monomial coefficients must separately vanish. This system can be solveddegree by degree, starting at the respective degrees 1, 2, 2 in equations (2.7a), (2.7b), (2.7c),and stopping at some degree such that the corresponding highest-degree coefficients in the olynomials G, F, H are found to vanish. (This termination generally will not occur unlessthe original equation is integrable.)2.1.
Harmonically modulated Hirota solitons.
The bilinear system (2.7) combined withthe ansatz equation (2.8) applied to the Hirota equation u t + 24 | u | u x + u xxx = 0 (2.13)gives H = 0 (2.14a) D t ( G, F ) + D x ( G, F ) = 0 (2.14b) D x ( F, F ) − G ¯ G = 0 . (2.14c)To set up the soliton ansatz (2.9)–(2.10), let all monomial terms of fixed degree n ≥ G, F be denoted by G ( n ) , F ( n ) , so thus the ansatz is written as G = G (1) + G (3) + · · · , F = 1 + F (2) + F (4) + · · · (2.15)(where F (0) has been normalized to 1 without loss of generality). Then it is straightforwardto split the bilinear system (2.14) into a hierarchy of equations indexed by degree: D t ( G (1) ,
1) + D x ( G (1) ,
1) = 0 (2.16a) D x ( F (2) , − G (1) ¯ G (1) = 0 (2.16b) D t ( G (3) ,
1) + D x ( G (3) ,
1) = − D t ( G (1) , F (2) ) − D x ( G (1) , F (2) ) (2.16c) D x ( F (4) , − G (3) ¯ G (1) + G (1) ¯ G (3) ) = − D x ( F (2) , F (2) ) (2.16d)etc.The hierarchy of bilinear equations (2.16) is solved in Appendix A to obtain the harmon-ically modulated 1-soliton and 2-soliton solutions for the Hirota equation (2.13). Proposition 1.
The harmonically modulated -soliton (1.9) for the Hirota equation (2.13) has an explicit rational cosh form given by the envelope function (up to space translationson x and phase shifts on φ ) f H ( θ ) = | k | θ ) , θ = k ( x − ( k − κ ) t ) (2.17) which is invariant under reflection k → − k . Proposition 2.
When w /k (cid:54) = w /k , (2.18) the harmonically modulated -soliton (1.14) for the Hirota equation (2.13) has an explicitrational cosh form given by the envelope functions (up to space-time translations on t, x ) f ( θ , θ ) = X ( θ , θ ) /Y H ( θ , θ ) , f ( θ , θ ) = X ( θ , θ ) /Y H ( θ , θ ) (2.19) with X ( θ , θ ) = | k |√ Γ cosh( θ + iγ ) , X ( θ , θ ) = | k |√ Γ cosh( θ + iγ ) (2.20) Y H ( θ , θ ) = cosh( θ − θ ) + Γ cosh( θ + θ ) − | Γ − | cos( µ θ − µ θ + φ − φ ) (2.21) here w , w , ω , ω are given by equation (A.29) and µ , µ are given by equation (A.52) ,and where Γ = ( k − k ) + ( κ − κ ) ( k + k ) + ( κ − κ ) , (2.22) γ = arg (cid:0) ( k + k )( k − k ) − ( κ − κ ) + i k ( κ − κ ) (cid:1) ,γ = arg (cid:0) ( k + k )( k − k ) − ( κ − κ ) − i k ( κ − κ ) (cid:1) , (2.23) θ = k x + w t = k ( x − ( k − κ ) t ) ,θ = k x + w t = k ( x − ( k − κ ) t ) . (2.24)The kinematic condition (2.18) will be seen later to have the interpretation that theharmonically modulated 2-soliton describes a collision of harmonically modulated 1-solitonswith distinct speeds c = − w /k (cid:54) = c = − w /k . These explicit expressions (2.19)–(2.24)for the harmonically modulated Hirota 2-soliton (1.14) have not previously appeared in theliterature. They reduce to the ordinary 2-soliton solution [15, 11] for the Hirota equation inthe case κ = κ = 0.We remark that in the case of equal speeds c = c , Proposition 2 remains valid if µ θ − µ θ is replaced by ϑ − ϑ through equation (A.51), and if k x and k x are respectivelyreplaced by k x + χ and k x − χ through the equation k ( x + a )+ w t = − k ( x + a ) − w t = − χ , where χ = ( a − a ) k k / ( k + k ) is a shift which cannot be absorbed by a space-timetranslation (A.45).Finally, we examine the properties of the Hirota envelope functions (2.19)–(2.21) underreflections k → ∓ k , k → ± k (2.25)which will be important when expressing the Hirota 2-soliton solution in oscillatory form.We first note that θ → ∓ θ , θ → ± θ and that w → ∓ w , w → ± w while ω → ω , ω → ω from equation (A.29). We then have µ → − µ , µ → − µ (2.26)Γ → / Γ , γ → − γ , γ → − γ (2.27)from equations (A.52), (2.22), (2.23), and thus X → (1 / Γ) X , X → (1 / Γ) X , Y H → (1 / Γ) Y H . (2.28)These transformations establish the following reflection property. Lemma 1.
The Hirota envelope functions (2.19) are invariant under reflections (2.25) . Harmonically modulated Sasa-Satsuma solitons.
For the Sasa-Satsuma equation u t + 6( u ¯ u x + 3 u x ¯ u ) u + u xxx = 0 (2.29)the bilinear system (2.7) combined with the ansatz equation (2.8) gives D t ( G, F ) + D x ( G, F ) = GH (2.30a) D x ( F, F ) − G ¯ G = 0 (2.30b)6 D x ( G, ¯ G ) = F H (2.30c) hich is more complicated than in the case of the Hirota equation. To set up the solitonansatz (2.9)–(2.10), let all monomial terms of fixed degree n ≥ G, F, H be denoted by G ( n ) , F ( n ) , H ( n ) . The ansatz is thus written as G = G (1) + G (3) + · · · , F = 1 + F (2) + F (4) + · · · , H = H (2) + H (4) + · · · (2.31)(where F (0) has been normalized to 1 without loss of generality). Then the bilinear system(2.30) splits into a hierarchy of equations indexed by degree: D t ( G (1) ,
1) + D x ( G (1) ,
1) = 0 (2.32a) D x ( F (2) , − G (1) ¯ G (1) = 0 (2.32b) H (2) = 6 D x ( G (1) , ¯ G (1) ) (2.32c) D t ( G (3) ,
1) + D x ( G (3) ,
1) = − D t ( G (1) , F (2) ) − D x ( G (1) , F (2) ) + G (1) H (2) (2.32d) D x ( F (4) , − G (3) ¯ G (1) + G (1) ¯ G (3) ) = − D x ( F (2) , F (2) ) (2.32e) H (4) = 6 D x ( G (1) , ¯ G (3) ) + 6 D x ( G (3) , ¯ G (1) ) − F (2) H (2) (2.32f) D t ( G (5) ,
1) + D x ( G (5) ,
1) = − D t ( G (3) , F (2) ) − D x ( G (3) , F (2) ) + G (3) H (2) − D t ( G (1) , F (4) ) − D x ( G (1) , F (4) ) + G (1) H (4) (2.32g) D x ( F (6) , − G (5) ¯ G (1) + G (1) ¯ G (5) ) = 4 G (3) ¯ G (3) − D x ( F (4) , F (2) ) (2.32h)etc.Note that H (2) , H (4) , etc. can be successively eliminated through equations (2.32c), (2.32f),and so on.The hierarchy of bilinear equations (2.32) is solved in Appendix B to obtain the harmon-ically modulated 1-soliton and 2-soliton solutions for the Sasa-Satsuma equation (2.29). Proposition 3.
The harmonically modulated -soliton (1.9) for the Sasa-Satsuma equation (2.29) has an explicit rational cosh form when κ (cid:54) = 0 given by the envelope function (up tospace translations on x and phase shifts on φ ) f SS ( θ ) = | k | (2 | κ | ) / ( k + κ ) / cosh( θ + iλ/ | κ | cosh(2 θ ) + ( k + κ ) / , θ = k ( x − ( k − κ ) t ) (2.33) where λ is given by equation (B.14) . This function is invariant under reflections k → − k . Proposition 4.
When w /k (cid:54) = w /k , κ (cid:54) = 0 , κ (cid:54) = 0 (2.34) the harmonically modulated -soliton (1.14) for the Sasa-Satsuma equation (2.29) has anexplicit rational cosh form given by the envelope functions (up to space-time translations on t, x ) f ( θ , θ ) = X ( θ , θ ) /Y SS ( θ , θ ) , f ( θ , θ ) = X ( θ , θ ) /Y SS ( θ , θ ) (2.35) ith X ( θ , θ ) = | k | ( k + κ ) / | κ | / × (cid:18) | κ | (cid:16) √ ∆Γ cosh( θ + 2 θ + i ( α + γ )) + 1 √ ∆Γ cosh( θ − θ + i ( υ − γ )) (cid:17) + ( k + κ ) / (cid:16) − k | κ | √ ΩΥ cosh( θ + i ( ρ + ρ + (cid:36) + γ ))+ √ Γ √ ∆ cosh( θ + i ( α − γ )) + √ ∆ √ Γ cosh( θ + i ( υ + γ )) (cid:17)(cid:19) − k | k | ( k + κ ) / | κ | / × (cid:18) √ Ω √ Υ exp( i ( µ θ − µ θ + φ − φ )) cosh( θ − i ( ρ + ρ + (cid:36) − γ )) (cid:19) , (2.36) X ( θ , θ ) = | k | ( k + κ ) / | κ | / × (cid:18) | κ | (cid:16) √ ∆Γ cosh( θ + 2 θ + i ( α + γ )) + 1 √ ∆Γ cosh( θ − θ + i ( υ − γ )) (cid:17) + ( k + κ ) / (cid:16) − k | κ | √ ΩΥ cosh( θ + i ( ρ + ρ + (cid:36) + γ ))+ √ Γ √ ∆ cosh( θ + i ( α − γ )) + √ ∆ √ Γ cosh( θ + i ( υ + γ )) (cid:17)(cid:19) − k | k | ( k + κ ) / | κ | / × (cid:18) √ Ω √ Υ exp( i ( µ θ − µ θ + φ − φ )) cosh( θ − i ( ρ + ρ + (cid:36) − γ )) (cid:19) , (2.37) Y SS ( θ , θ ) =( k + κ ) / ( k + κ ) / (cid:16) Γ∆ + ∆Γ + 64 k k κ κ (cid:17) + | κ κ | (cid:16) ∆Γ cosh(2( θ + θ )) + 1∆Γ cosh(2( θ − θ )) (cid:17) + 2 | κ | ( k + κ ) / cosh(2 θ ) + 2 | κ | ( k + κ ) / cosh(2 θ )+ 4 k k ΩΥ cos(2( µ θ − µ θ ) + 2( φ − φ ) + ρ − ρ ) − | k k || κ κ | / ( k + κ ) / ( k + κ ) / Re (cid:16) exp( i ( µ θ − µ θ + φ − φ )) × (cid:16) √ Γ √ Υ cosh( θ + θ + i ( (cid:36) − (cid:36) )) + 1 √ ΓΥ cosh( θ − θ + i ( (cid:36) + (cid:36) )) (cid:17)(cid:17) , (2.38) where w , w , ω , ω are given by equation (A.29) and µ , µ are given by equation (A.52) ,and where Ω = (cid:113)(cid:0) ( k + k ) + ( κ + κ ) (cid:1)(cid:0) ( k − k ) + ( κ + κ ) (cid:1) , (2.39)Υ = (cid:0) ( k − k ) + ( κ − κ ) (cid:1)(cid:0) ( k + k ) + ( κ − κ ) (cid:1) , (2.40) = (cid:115) ( k − k ) + ( κ + κ ) ( k + k ) + ( κ + κ ) , (2.41)Γ = ( k − k ) + ( κ − κ ) ( k + k ) + ( κ − κ ) , (2.42) α = ( λ + δ ) / , α = ( λ + δ ) / , (2.43) υ = ( λ − δ ) / , υ = ( λ − δ ) / , (2.44) (cid:36) = ( λ − δ ) / , (cid:36) = ( λ − δ ) / , (2.45) γ = arg( k − k − ( κ − κ ) + i k ( κ − κ )) ,γ = arg( k − k − ( κ − κ ) − i k ( κ − κ )) , (2.46) δ = arg( k − k + ( κ + κ ) + i k ( κ + κ )) ,δ = arg( k − k + ( κ + κ ) + i k ( κ + κ )) , (2.47) λ = arg( κ ( κ + ik )) , λ = arg( κ ( κ + ik )) , (2.48) θ = k x + w t = k ( x − ( k − κ ) t ) ,θ = k x + w t = k ( x − ( k − κ ) t ) . (2.49)It will be useful to write out the half-angle expressions for λ / λ / δ / δ / λ / (cid:16)(cid:114)(cid:113) k /κ + 1 + iε (cid:114)(cid:113) k /κ − (cid:17) , (2.50) λ / (cid:16)(cid:114)(cid:113) k /κ + 1 + iε (cid:114)(cid:113) k /κ − (cid:17) , (2.51)where ε = sgn( κ ) , ε = sgn( κ ) , (2.52)and δ / (cid:16) ( (cid:15) (cid:15) − + (1 − (cid:15) (cid:15) − )sgn( κ + κ )) (cid:113) k − k + ( κ + κ ) + Ω+ iε (1 + (cid:15) (cid:15) − (sgn( κ + κ ) − (cid:113) k − k + ( κ + κ ) − Ω (cid:17) , (2.53) δ / (cid:16) ( (cid:15) (cid:15) + + (1 − (cid:15) (cid:15) + )sgn( κ + κ )) (cid:113) k − k + ( κ + κ ) + Ω+ iε (1 + (cid:15) (cid:15) + (sgn( κ + κ ) − (cid:113) k − k + ( κ + κ ) − Ω (cid:17) , (2.54)where (cid:15) ± = (1 ± (cid:15) ) / , (cid:15) = sgn( | k | − | k | ) = , | k | > | k |− , | k | < | k | , | k | = | k | . (2.55)Then, combining expressions (2.53) and (2.54), we have( δ ± δ ) / (cid:16) ε ( κ + κ + i ( k ± k )) (cid:17) (2.56) here ε = (cid:15) + (1 − (cid:15) )sgn( κ + κ ) = (cid:40) , | k | (cid:54) = | k | sgn( κ + κ ) , | k | = | k | . (2.57)Additionally, we noteexp( iρ ) = ε , exp( iρ ) = ε , exp( iρ ) = ε. (2.58)These explicit expressions (2.35)–(2.49) and (2.50)–(2.58) for the harmonically modulatedSasa-Satsuma 2-soliton (1.14) have not previously appeared in the literature. The kine-matic condition (2.34) will be seen later to imply that this solution describes a collision ofharmonically modulated 1-solitons with distinct speeds c = − w /k (cid:54) = c = − w /k .In the case of equal speeds c = c , we remark that Proposition 4 remains valid if µ θ − µ θ is replaced by ϑ − ϑ through equation (A.51), and if k x and k x are respectively replaced by k x + χ and k x − χ through the equation k ( x + a )+ w t = − k ( x + a ) − w t = − χ , where χ = ( a − a ) k k / ( k + k ) is a shift which cannot be absorbed by a space-time translation(A.45). The solution in this particular case has previously appeared in an equivalent envelopeform in Ref. [16, 17].For expressing the Sasa-Satsuma 2-soliton in oscillatory form, the properties of the en-velope functions (2.35)–(2.38) under reflections (2.25) will be important. As in the Hirotacase, we have θ → ∓ θ , θ → ± θ , (2.59) µ → − µ , µ → − µ . (2.60)Also, from equations (2.39)–(2.48), we haveΩ → Ω , Υ → Υ , (2.61)Γ → / Γ , ∆ → / ∆ , (2.62) λ → ∓ λ , λ → ± λ , (2.63) γ → ∓ γ , γ → ± γ , (2.64) δ → ∓ δ , δ → ± δ . (2.65)These transformations yield X → X , X → X , Y SS → Y SS . (2.66)which establishes the following reflection property. Lemma 2.
The Sasa-Satsuma envelope functions (2.35) are invariant under reflections (2.25) . Oscillatory parameterization
The harmonically modulated 1-soliton solutions shown in Proposition 1 for the Hirotaequation (2.13) and Proposition 3 for the Sasa-Satsuma equation (2.29) will now be expressedin the more physical oscillatory form (1.15).We write kx + wt = k ( x − ct ) , κx + ωt = κ ( x − ct ) + νt, (3.1)where c = − w/k, ν = ω − wκ/k. (3.2) rom relations (1.10), (1.21) for w and ω , we get c = k − κ , (3.3) ν = − κ ( k + κ ) , (3.4)with κ (cid:54) = 0 . (3.5)By combining these equations (3.3) and (3.4), we have a cubic equation that determines κ ,8 κ + 2 κc + ν = 0 (3.6)and an elementary quadratic equation that determines | k | , k = 3 κ + c. (3.7)The discriminant of the cubic equation (3.6) is ∆ = − c + ˜ ν ) (3.8)where ˜ c = c/ , ˜ ν = ν/ (cid:54) = 0 . (3.9)There are three cases to consider.First, if ∆ <
0, i.e. ˜ c > − ˜ ν , then equation (3.6) has only one real root, κ = β − − β + β ± = (cid:112) √ ˜ c + ˜ ν ± ˜ ν (3.11)which is defined as the real cube root. Equation (3.7) then becomes k = 3( κ + ˜ c ) = 3( β − + β + ) | k | = √ β − + β + )2 (3.13)where β + > − β − > ν > β − > − β + > ν < ∆ = 0, i.e. ˜ c = − ˜ ν , then equation (3.6) has three real roots, two of which arerepeated, κ = β − , κ = β + / β ± = ± √ ˜ ν (cid:54) = 0. The single root coincides with the previous real root (3.10), whilethe repeated roots violate equation (3.7) because0 ≤ k = 3( κ + ˜ c ) = 3(( β + / − β ) = − (3 β + / < . (3.15)Last, if ∆ >
0, i.e. ˜ c < − ˜ ν , then equation (3.6) has three distinct real roots, κ = | β + | cos( ψ ) ,ψ = arg β + , ψ = arg β + + 2 π/ , ψ = arg β + − π/ | β ± | = √− ˜ c (cid:54) = 0 and tan(arg β ± ) = ± (cid:112) | ˜ c + ˜ ν | / ˜ ν (cid:54) = 0. But all three roots violateequation (3.7), 0 ≤ k = 3( κ + ˜ c ) = 3 | β + | (cid:0) cos ( ψ ) − (cid:1) < ue to | cos( ψ ) | (cid:54) = 1 which is the condition for no roots to be repeated.Hence we have established the following main identities. Lemma 3. (i) Let w = − k ( k − κ ) , ω = − κ (3 k − κ ) , and κ (cid:54) = 0 . Then | w | = − c | k | , ω = cκ + ν (3.18) is an identity, where κ and | k | are given by equations (3.9) , (3.10) , (3.13) in terms of c and ν . (ii) Let the function f ( kx + wt ) = f ( k ( x − ct )) be invariant under reflection k → − k .Then the harmonically modulated function exp( i ( κx + ωt )) f ( kx + wt ) is reflection invariant,in which case it can be expressed in the equivalent form exp( i ( κx + ωt )) f ( kx + wt ) = exp( iνt ) ˜ f ( x − ct ) , ˜ f ( ξ ) = exp( iκξ ) f ( kξ ) (3.19) in terms of k = √ (cid:16) (cid:113)(cid:112) ( c/ + ( ν/ − ν/ (cid:113)(cid:112) ( c/ + ( ν/ + ν/ (cid:17) , (3.20) κ = 12 (cid:16) (cid:113)(cid:112) ( c/ + ( ν/ − ν/ − (cid:113)(cid:112) ( c/ + ( ν/ + ν/ (cid:17) , (3.21) where ( c/ + ( ν/ ≥ . (3.22)Applying Lemma 3 to the Hirota and Sasa-Satsuma harmonically modulated 1-solitons,which are given by the reflection-invariant envelope functions (2.17) and (2.33), we obtainthe following result. Theorem 1.
The equivalent oscillatory form (1.15) of a harmonically modulated -soliton (1.9) is parameterized by a phase angle φ , a temporal frequency ν , and a speed c , which satisfythe kinematic relation (3.22) , where k, κ are given in terms of c, ν by equations (3.20) , (3.21) .For the Hirota equation (2.13) and the Sasa-Satsuma equation (2.29) , the oscillatory form ofthe harmonically modulated -soliton solutions (2.17) and (2.33) expressed using a travellingwave coordinate ξ = x − ct is given by u ( t, x ) = exp( iφ ) exp( iνt ) ˜ f ( ξ ) (3.23) in terms of the respective functions ˜ f H ( ξ ) = k exp( iκξ )2 cosh( kξ ) , (3.24)˜ f SS ( ξ ) = k (2 | κ | ) / ( k + κ ) / exp( iκξ ) cosh( kξ + iλ/ | κ | cosh(2 kξ ) + ( k + κ ) / , κ (cid:54) = 0 , (3.25) where λ is given by equation (B.14) . When c = 0 (and κ (cid:54) = 0 ), these -soliton solutions arestanding waves (i.e. harmonically modulated stationary solitons). We emphasize that, due to the invariance of the Hirota and Sasa-Satsuma envelope func-tions (2.17) and (2.33) under reflection k → − k , the parameters ( κ, k ) and ( κ, − k ) correspondto the same harmonically-modulated 1-soliton. Thus, the change of parameterization from( κ, ± k ) to ( ν, c ) in Theorem 1 is one-to-one, such that the parameter range0 ≤ | κ | < ∞ , < k < ∞ (3.26) orresponds to the kinematic range0 ≤ | ν | < ∞ , − (3 / √ (cid:112) | ν | ) < c < ∞ (3.27)through the relations (3.3)–(3.4) and (3.20)–(3.21).From relations (3.21) and (3.20), we note that κ = 0 iff ν = 0 and that k = 0 iff( c/ + ( ν/ = 0. Consequently, Theorem 1 implies the following result. Corollary 1.
For the Hirota and Sasa-Satsuma equations, a harmonically modulated -soliton solution u ( t, x ) = exp( iφ ) exp( i ( κx + ωt )) f ( kx + wt ) = exp( iφ ) exp( iνt ) ˜ f ( x − ct ) (3.28) is distinguished from an ordinary -soliton solution u ( t, x ) = exp( iφ ) f ( x − ct ) (3.29) by the kinematic conditions ν (cid:54) = 0 , ( c/ + ( ν/ > , (3.30) or equivalently c > − (3 / √ √ ν ) (cid:54) = 0 . (3.31)Next, the harmonically modulated 2-soliton solutions shown in Proposition 2 for the Hirotaequation (2.13) and Proposition 4 for the Sasa-Satsuma equation (2.29) will be expressed inthe analogous oscillatory form (1.17).From Lemmas 1 and 2, the envelope functions (2.19) and (2.35) in these 2-soliton so-lutions are invariant under reflections (2.25), so thus the parameters ( κ , κ , ± k , k ) and( κ , κ , k , ± k ) correspond to the same solution. This invariance property allows Lemma 3to be applied as follows. We define k = √ β − + β ) / , κ = ( β − − β ) / , (3.32) k = √ β − + β ) / , κ = ( β − − β ) / , (3.33)in terms of β ± = (cid:113)(cid:112) ( c / + ( ν / ± ν / ,β ± = (cid:113)(cid:112) ( c / + ( ν / ± ν / , (3.34)where ( c / + ( ν / ≥ , ( c / + ( ν / ≥ . (3.35)We then have the identities c = − w /k , ν = ω − w κ /k ,c = − w /k , ν = ω − w κ /k , (3.36)holding from the algebraic relations (A.29). Through these expressions, the identity (A.51)–(A.52) becomes( κ − κ ) x + ( ω − ω ) t = ( κ − µ )( x − c t ) − ( κ − µ )( x − c t ) , c (cid:54) = c (3.37)where µ = ( ν − ν ) / ( c − c ) . (3.38) e emphasize that this change of parameterization from ( κ , ± k ) and ( κ , ± k ) to ( ν , c )and ( ν , c ) is one-to-one, such that the parameter range0 ≤ | κ | < ∞ , ≤ | κ | < ∞ , < k < ∞ , < k < ∞ (3.39)corresponds to the kinematic range 0 ≤ | ν | < ∞ , ≤ | ν | < ∞ , − (3 / √ (cid:112) | ν | ) < c < ∞ , − (3 / √ (cid:112) | ν | ) < c < ∞ . (3.40)This leads to the following main result. Theorem 2.
The equivalent oscillatory form (1.17) of a harmonically modulated -soliton (1.14) has parameters φ , φ , ν , ν , c , c , satisfying the kinematic relations (3.35) , where k , κ are given in terms of c , ν by equation (3.32) , and k , κ are given in terms of c , ν by equation (3.33) . As expressed using travelling wave coordinates ξ = x − c t and ξ = x − c t when c (cid:54) = c , the oscillatory form for the -soliton solutions (2.19) – (2.24) for the Hirotaequation (2.13) and (2.35) – (2.49) for the Sasa-Satsuma equation (2.29) is given by u ( t, x ) = exp( iφ ) exp( iν t ) ˜ f ( ξ , ξ ) + exp( iφ ) exp( iν t ) ˜ f ( ξ , ξ ) (3.41) where ˜ f ( ξ , ξ ) = ˜ X ( ξ , ξ ) / ˜ Y ( ξ , ξ ) , ˜ f ( ξ , ξ ) = ˜ X ( ξ , ξ ) / ˜ Y ( ξ , ξ ) (3.42) are the respective functions: ˜ X ( ξ , ξ ) = k exp( iκ ξ ) cosh( k ξ + iγ ) , (3.43)˜ X ( ξ , ξ ) = k exp( iκ ξ ) cosh( k ξ + iγ ) , (3.44)˜ Y H ( ξ , ξ ) = √ Γ cosh( k ξ + k ξ ) + 1 √ Γ cosh( k ξ − k ξ ) − k k √ Υ cos( κ ξ − κ ξ + µ ( ξ − ξ ) + φ − φ ) (3.45) n the Hirota case; and ˜ X ( ξ , ξ ) = exp( iκ ξ ) (cid:18) k ( k + κ ) / | κ | / (cid:18) | κ | (cid:16) √ ∆Γ cosh( k ξ + 2 k ξ + i ( α + γ ))+ 1 √ ∆Γ cosh( k ξ − k ξ + i ( υ − γ )) (cid:17) + ( k + κ ) / (cid:16) − k κ ε √ ΩΥ cosh( k ξ + i ( (cid:36) + γ ))+ (cid:114) Γ∆ cosh( k ξ + i ( α − γ )) + (cid:114) ∆Γ cosh( k ξ + i ( υ + γ )) (cid:17)(cid:19) + k k ε (cid:114) ΩΥ (cid:18) k ( k + κ ) / | κ | / ε cosh( k ξ + i ( (cid:36) − γ )) − k ( k + κ ) / | κ | / ε Re (cid:16) cosh( k ξ + i ( γ − (cid:36) )) × exp( i ( κ ξ − κ ξ + µ ( ξ − ξ ) + φ − φ )) (cid:17)(cid:19)(cid:19) , (3.46)˜ X ( ξ , ξ ) = exp( iκ ξ ) (cid:18) k ( k + κ ) / | κ | / (cid:18) | κ | (cid:16) √ ∆Γ cosh( k ξ + 2 k ξ + i ( α + γ ))+ 1 √ ∆Γ cosh( k ξ − k ξ + i ( υ − γ )) (cid:17) + ( k + κ ) / (cid:16) − k κ ε √ ΩΥ cosh( k ξ + i ( (cid:36) + γ ))+ (cid:114) Γ∆ cosh( k ξ + i ( α − γ )) + (cid:114) ∆Γ cosh( k ξ + i ( υ + γ )) (cid:17)(cid:19) + k k ε (cid:114) ΩΥ (cid:18) k ( k + κ ) / | κ | / ε cosh( k ξ + i ( (cid:36) − γ )) − k ( k + κ ) / | κ | / ε Re (cid:16) cosh( k ξ + i ( γ − (cid:36) )) × exp( i ( κ ξ − κ ξ + µ ( ξ − ξ ) + φ − φ )) (cid:17)(cid:19)(cid:19) , (3.47) Y SS ( ξ , ξ ) = | κ κ | (cid:16) ∆Γ cosh(2( k ξ + k ξ )) + 1∆Γ cosh(2( k ξ − k ξ )) (cid:17) + 2( k + κ ) / | κ | cosh(2 k ξ ) + 2( k + κ ) / | κ | cosh(2 k ξ )+ 4 k k ε ε ΩΥ cos(2( κ ξ − κ ξ + µ ( ξ − ξ )) + 2( φ − φ ))+ ( k + κ ) / ( k + κ ) / (cid:18) Γ∆ + ∆Γ + 64 k k κ κ − k k | κ κ | / Re (cid:16) exp( i ( κ ξ − κ ξ + µ ( ξ − ξ ) + φ − φ )) × (cid:16)(cid:114) ΓΥ cosh( k ξ + k ξ + i ( (cid:36) − (cid:36) ))+ 1 √ ΓΥ cosh( k ξ − k ξ + i ( (cid:36) + (cid:36) )) (cid:17)(cid:17)(cid:19) (3.48) in the Sasa-Satsuma case when κ (cid:54) = 0 and κ (cid:54) = 0 . In both cases, µ is given by equation (3.38) , Ω , Υ , ∆ , Γ are given by equations (2.39) – (2.42) , α , α , υ , υ , (cid:36) , (cid:36) , γ , γ aregiven by equations (2.43) – (2.46) , and λ , λ , δ , δ , ε , ε , ε are given by equations (2.50) – (2.57) . Similarly to the 1-soliton case, Theorem 2 implies the following result.
Corollary 2.
For the Hirota and Sasa-Satsuma equations, a harmonically modulated -soliton solution u ( t, x ) = exp( iφ ) exp( i ( κ x + ω t )) f ( k x + w t, k x + w t )+ exp( iφ ) exp( i ( κ x + ω t )) f ( k x + w t, k x + w t )= exp( iφ ) exp( iν t ) ˜ f ( x − c t, x − c t )+ exp( iφ ) exp( iν t ) ˜ f ( x − c t, x − c t ) , c (cid:54) = c (3.49) is distinguished from an ordinary -soliton solution u ( t, x ) = exp( iφ ) f ( x − c t, x − c t ) + exp( iφ ) f ( x − c t, x − c t ) , c (cid:54) = c (3.50) by the kinematic conditions ν (cid:54) = 0 , ( c / + ( ν / > , (3.51) ν (cid:54) = 0 , ( c / + ( ν / > , (3.52) or equivalently c > − (3 / √ √ ν ) (cid:54) = 0 , c > − (3 / √ √ ν ) (cid:54) = 0 . (3.53)3.1. Breathers.
From the remarks made after Proposition 2 for the Hirota equation (2.13)and Proposition 4 for the Sasa-Satsuma equation (2.29), we will now state a counterpart ofTheorem 2 for harmonically modulated breather solutions.Let β ± = (cid:113)(cid:112) ( c/ + (( ν + ν ) / ± ( ν + ν ) / β ± = (cid:113)(cid:112) ( c/ + (( ν − ν ) / ± ( ν − ν ) / ith ( c/ + (( | ν | + | ν | ) / ≥ . (3.55) Theorem 3.
The equivalent oscillatory form (1.20) of a harmonically modulated breather (1.18) has parameters χ, φ , ν , c, φ, ν (cid:54) = 0 , satisfying the kinematic relation (3.55) . As ex-pressed using a travelling wave coordinate ξ = x − ct and an oscillation coordinate τ = νt + φ ,the oscillatory form for the breather solutions for the Hirota equation (2.13) and the Sasa-Satsuma equation (2.29) is given by u ( t, x ) = exp( i ( ν t + φ )) ˜ f ( ξ, τ )˜ f ( ξ, τ ) = exp( iτ ) ˜ X ( ξ, τ ) / ˜ Y ( ξ, τ ) + exp( − iτ ) ˜ X ( ξ, τ ) / ˜ Y ( ξ, τ ) (3.56) in terms of the respective functions: ˜ X ( ξ, τ ) = k exp( iκ ξ ) cosh( k ξ − χ + iγ ) , (3.57)˜ X ( ξ, τ ) = k exp( iκ ξ ) cosh( k ξ + χ + iγ ) , (3.58)˜ Y H ( ξ, τ ) = √ Γ cosh(( k + k ) ξ ) + 1 √ Γ cosh(( k − k ) ξ + 2 χ ) − k k √ Υ cos(( κ − κ ) ξ + 2 τ )(3.59) in the Hirota case; and ˜ X ( ξ, τ ) = exp( iκ ξ ) (cid:18) k ( k + κ ) / | κ | / (cid:18) | κ | (cid:16) √ ∆Γ cosh(( k + 2 k ) ξ − χ + i ( α + γ ))+ 1 √ ∆Γ cosh(( k − k ) ξ + 3 χ + i ( υ − γ )) (cid:17) + ( k + κ ) / (cid:16) − k κ (cid:15) √ ΩΥ cosh( k ξ + χ + i ( (cid:36) + γ ))+ (cid:114) Γ∆ cosh( k ξ + χ + i ( α − γ )) + (cid:114) ∆Γ cosh( k ξ + χ + i ( υ + γ )) (cid:17)(cid:19) + k k ε (cid:114) ΩΥ (cid:18) k ( k + κ ) / | κ | / ε cosh( k ξ + χ + i ( (cid:36) − γ )) − k ( k + κ ) / | κ | / ε Re (cid:16) cosh( k ξ − χ + i ( γ − (cid:36) )) × exp( i (( κ − κ ) ξ + 2 τ )) (cid:17)(cid:19)(cid:19) , (3.60) X ( ξ, τ ) = exp( iκ ξ ) (cid:18) k ( k + κ ) / | κ | / (cid:18) | κ | (cid:16) √ ∆Γ cosh(( k + 2 k ) ξ + χ + i ( α + γ ))+ 1 √ ∆Γ cosh(( k − k ) ξ − χ + i ( υ − γ )) (cid:17) + ( k + κ ) / (cid:16) − k κ ε √ ΩΥ cosh( k ξ − χ + i ( (cid:36) + γ ))+ (cid:114) Γ∆ cosh( k ξ − χ + i ( α − γ )) + (cid:114) ∆Γ cosh( k ξ − χ + i ( υ + γ )) (cid:17)(cid:19) + k k ε (cid:114) ΩΥ (cid:18) k ( k + κ ) / | κ | / ε cosh( k ξ − χ + i ( (cid:36) − γ )) − k ( k + κ ) / | κ | / ε Re (cid:16) cosh( k ξ + χ + i ( γ − (cid:36) )) × exp( i (( κ − κ ) ξ + 2 τ )) (cid:17)(cid:19)(cid:19) , (3.61)˜ Y SS ( ξ, τ ) = | κ κ | (cid:16) ∆Γ cosh(2( k + k ) ξ ) + 1∆Γ cosh(2(( k − k ) ξ + 2 χ )) (cid:17) + 2( k + κ ) / | κ | cosh(2( k ξ − χ )) + 2( k + κ ) / | κ | cosh(2( k ξ + χ ))+ 4 k k ε ε ΩΥ cos(2( κ − κ ) ξ + 4 τ )+ ( k + κ ) / ( k + κ ) / (cid:18) Γ∆ + ∆Γ + 64 k k κ κ − k k | κ κ | / Re (cid:16) exp( i (( κ − κ ) ξ + 2 τ )) × (cid:16)(cid:114) ΓΥ cosh(( k + k ) ξ + i ( (cid:36) − (cid:36) ))+ 1 √ ΓΥ cosh(( k − k ) ξ + 2 χ + i ( (cid:36) + (cid:36) )) (cid:17)(cid:17)(cid:19) (3.62) in the Sasa-Satsuma case when κ (cid:54) = 0 and κ (cid:54) = 0 . In both cases, k , k , κ , κ are givenin terms of c , ν (cid:54) = 0 , ν by equations (3.32) – (3.33) , Ω , Υ , ∆ , Γ are given by equations (2.39) – (2.42) , α , α , υ , υ , (cid:36) , (cid:36) , γ , γ are given by equations (2.43) – (2.46) , and λ , λ , δ , δ , ε , ε , ε are given by equations (2.50) – (2.57) . In the special case ν = 0, the oscillatory breathers (3.56) reduce to ordinary breathers(1.19), which are given by much simpler expressions. Proposition 5.
For the Hirota equation (2.13) and the Sasa-Satsuma equation (2.29) , theordinary breather solutions expressed using a travelling wave coordinate ξ = x − ct and anoscillation coordinate τ = νt + φ are given by u ( t, x ) = exp( iφ ) f ( ξ, τ ) f ( ξ, τ ) = exp( iτ ) X ( ξ, τ ) /Y ( ξ, τ ) + exp( − iτ ) X ( ξ, τ ) /Y ( ξ, τ ) (3.63) n terms of the respective functions X ( ξ, τ ) = k | κ |√ k + κ exp( iκξ ) sinh( kξ − χ − iγ ) , (3.64) X ( ξ, τ ) = k | κ |√ k + κ exp( − iκξ ) sinh( kξ + χ + iγ ) , (3.65) Y H ( ξ, τ ) = κ cosh(2 kξ ) + ( k + κ ) cosh(2 χ ) + k cos(2( κξ + τ )) (3.66) in the Hirota case, and in the Sasa-Satsuma case X ( ξ, τ ) =( k/ Σ) exp( iκξ ) (cid:18) | κ |√ k + κ sinh( kξ − χ − iγ ) − (( k + κ )Λ) / | κ | (1 + Σ) / exp( χ + i ( λ + γ/ kξ + (cid:36) + i ( λ + γ/ (cid:19) , (3.67) X ( ξ, τ ) =( k/ Σ) exp( − iκξ ) (cid:18) | κ |√ k + κ sinh( kξ + χ + iγ ) − (( k + κ )Λ) / | κ | (1 + Σ) / exp( − χ + i ( λ − γ/ kξ + (cid:36) + i ( λ − γ/ (cid:19) , (3.68) Y H ( ξ, τ ) = κ Σ cosh(2 kξ ) + ( k + κ ) cosh(2 χ ) + k cos(2( κξ + τ )) , (3.69) where Λ = sinh(2 | χ | ) , Σ = (cid:112) k /κ )Λ , (cid:36) = ln (cid:16) Σ + 1Σ − (cid:17) (3.70) γ = arg( k + iκ ) , λ = arg(1 + i sgn( κχ )) (3.71) and where k , κ (cid:54) = 0 are given in terms of c , ν (cid:54) = 0 by equations (3.20) and (3.21) , such thatthe kinematic relation ( c/ + ( ν/ > holds. When χ = φ = 0, we remark that both the Hirota and Sasa-Satsuma breather solutionsin Proposition 5 reduce to the well-known mKdV breather solution [18] u ( t, x ) = k | κ |√ k + κ sinh( kξ ) cos( γ ) cos( κξ + τ ) + cosh( kξ ) sin( γ ) sin( κξ + τ ) κ cosh( kξ ) + k cos( κξ + τ ) . (3.72)4. Properties of oscillatory soliton solutions
We begin by discussing some basic properties of the oscillatory 1-soliton solutions fromTheorem 1 for the Hirota equation (2.13) and the Sasa-Satsuma equation (2.29).4.1.
Oscillatory -solitons. An oscillatory wave (3.28) has amplitude | u | = | ˜ f ( ξ ) | where ξ = x − ct is a moving coordinate centered at x = ct . Hence the spatial shape of | u | isdetermined by the properties of the function | ˜ f ( ξ ) | . In both the Hirota and Sasa-Satsumaoscillatory 1-soliton solutions, these functions | ˜ f ( ξ ) | share two main properties, as seen fromexpressions (3.24) and (3.25). First, for large | ξ | , both functions exhibit exponential decay | ˜ f ( ξ ) | ∼ O (exp( − k | ξ | )). Second, both functions exhibit reflection-conjugation invariance˜ f ( − ξ ) = ˜ f ( ξ ) (where a bar denotes complex conjugation), implying that | ˜ f ( ξ ) | is an evenfunction of ξ and thus Re ( ˜ f (cid:48) (0)) = 0. n the case of the Hirota function | ˜ f H ( ξ ) | , from expression (3.24) we find that Re ( ˜ f (cid:48) H ( ξ )) (cid:54) =0 when ξ (cid:54) = 0. Hence the function | ˜ f H ( ξ ) | has a peak at ξ = 0. In contrast, in the case ofthe Sasa-Satsuma function | ˜ f SS ( ξ ) | , from expression (3.25) we find that Re ( ˜ f (cid:48) SS ( ξ )) = 0 hasroots ξ (cid:54) = 0 when (and only when) cosh(2 kξ ) = ( k − κ ) / ( | κ |√ k + κ ), which requires thecondition ( k − κ ) / ( | κ |√ k + κ ) ≥ k, κ . This condition is equivalent to k ≥ κ .Hence in this case the function | ˜ f SS ( ξ ) | has a pair of peaks centered symmetrically around ξ = 0.From these properties, we obtain the following two results about the spatial shape of | u | ,stated in terms of the notation β ± = (cid:113)(cid:112) ( c/ + ( ν/ ± ν/
2. Cases c > c < c = 0are illustrated in Fig. 1a, Fig. 2a, for the Hirota oscillatory 1-soliton, and in Fig. 3a, Fig. 4a,for the Sasa-Satsuma oscillatory 1-soliton. Proposition 6.
For both the Hirota oscillatory -soliton (3.23) , (3.24) , and the Sasa-Satsuma oscillatory -soliton (3.23) , (3.25) , the amplitude | u | is an even function of x − ct and decays exponentially for | x − ct | (cid:29) /k = 2 / ( √ β − + β + )) . The Hirota oscillatory -soliton and the Sasa-Satsuma oscillatory -soliton for c ≤ each have a single peak centeredat x = ct , with the height of the respective peaks given by | u | (cid:12)(cid:12) x = ct = k √
34 ( β − + β + ) (4.1) and | u | (cid:12)(cid:12) x = ct = k | κ | / ( | κ | + ( k + κ ) / ) / = √ β − + β + ) | ν + ( β − − β + ) c | / | ν | / + | ν + ( β − − β + ) c | / ) / . (4.2) For c > , the Sasa-Satsuma oscillatory -soliton instead has a symmetrical pair of peaks at x = ct ± x , where x = k − κ + k ( k + κ ) / | κ | ( k + κ ) / = | β + − β − | c + | β − β − | (3 c ) / + | ν || ν | / | β + − α | / > , (4.3) with the height of the peaks given by | u | (cid:12)(cid:12) x = ct ± x = 12 ( k + κ ) / = | ν | / | β + − β − | . (4.4)In general, an oscillatory wave (3.28) can be factorized into a harmonic wave partexp( i ( κx + ωt )) and a travelling wave part f ( kx + wt ) through equation (3.19). The en-velope function f in this factorization will be real-valued when (and only when) it satisfies f = | f | or arg( f ) = 0, corresponding to arg( u ) = φ + κx + ωt being linear in x, t (in whichcase u is a harmonically modulated travelling wave). This condition is equivalent to therelation arg( ˜ f ( ξ )) = κξ given in terms of the function ˜ f ( ξ ) in the oscillatory wave (3.28). Itis thereby convenient to define ϕ ( u ) = arg( u ) − κx − ( ν − κc ) t mod 2 π. (4.5)Since an oscillatory wave (3.28) has arg( u ) = φ + νt + arg( ˜ f ( ξ )), we then see that thecondition arg( ˜ f ( ξ )) = κξ can be formulated simply as ϕ ( u ) = φ is constant. Therefore,the property ϕ ( u ) (cid:54) = const . distinguishes a general oscillatory wave from a harmonically odulated travelling wave with a real envelope function. Correspondingly, it is natural todefine (cid:96) = 12 π (cid:90) ∞−∞ ϕ ( u ) x dx = ϕ ( u )2 π (cid:12)(cid:12)(cid:12) x = ∞ x = −∞ (4.6)which can be regarded as measuring the net winding contributed by the envelope functionof an oscillatory wave (3.28). (Note that the net winding will be (cid:96) = 0 when the envelopefunction is real-valued.) We will thus refer to the expressions (4.5) and (4.6) as the envelopephase and envelope winding number , respectively.It is straightforward to derive the following phase properties of the Hirota and Sasa-Satsuma oscillatory 1-soliton solutions. Proposition 7.
For both the Hirota oscillatory -soliton (3.23) , (3.24) , and the Sasa-Satsuma oscillatory -soliton (3.23) , (3.25) , the envelope phase ϕ ( u ) is an even functionof x − ct and equals φ at x = ct . Away from x = ct , the envelope phase has the features ϕ ( u ) = φ, ± ( x − ct ) > (cid:96) = 0 (4.8) for the Hirota -soliton, and ϕ ( u ) ∼ φ ± λ/ , ± ( x − ct ) (cid:29) /k = 2 / ( √ β − + β + )) (4.9) (cid:96) = λ/ (2 π ) (4.10) for the Sasa-Satsuma -soliton, where λ is given by equation (B.14) . Cases c > c < c = 0 are shown in Fig. 1b, Fig. 2b, for the Hirota oscillatory 1-soliton,and in Fig. 3b, Fig. 4b, for the Sasa-Satsuma oscillatory 1-soliton. (a) amplitude (b) envelope phase Figure 1.
Hirota oscillatory 1-solitons with c = 4, | ν | = 15 , ,
250 and ν = 0 (dotted line), φ = π/ (a) amplitude (b) envelope phase Figure 2.
Hirota oscillatory 1-solitons with c = − | ν | = 4 , , , φ = π/ a) amplitude (b) envelope phase Figure 3.
Sasa-Satsuma oscillatory 1-solitons with c = 4, | ν | = 15 , , ν = 1 (dotted line), φ = π/ (a) amplitude (b) envelope phase Figure 4.
Sasa-Satsuma oscillatory 1-solitons with c = − | ν | =4 , , , φ = π/ Oscillatory -solitons. We now illustrate some properties of the oscillatory 2-solitonsolutions from Theorem 2, shown graphically by the amplitude | u | and the phase gradientarg( u ) x .An oscillatory 2-soliton (3.49) is parameterized by phases φ , φ , frequencies ν , ν , andspeeds c (cid:54) = c , satisfying the kinematic relations (3.35). These solitons are symmetric undersimultaneously interchanging c ←→ c , ν ←→ ν , φ ←→ φ , and thus we can assume c > c without loss of generality.As seen from Fig. 5–Fig. 7 for the Hirota equation (2.13) and Fig. 8–Fig. 10 for the Sasa-Satsuma equation (2.29), oscillatory 2-solitons describe collisions between two oscillatorywaves at t = t , , − t . The collision is a right-overtake when c > c ≥
0, a left-overtakewhen 0 ≥ c > c , and a head-on when c > > c . (a) amplitude (b) phase gradient Figure 5.
Hirota oscillatory 2-soliton right-overtake with c = 4, c = 2, ν = 2, ν = 5, φ = 0, φ = π/ t = − a) amplitude (b) phase gradient Figure 6.
Hirota oscillatory 2-soliton left-overtake with c = − c = − ν = 2, ν = 5, φ = 0, φ = π/ t = − (a) amplitude (b) phase gradient Figure 7.
Hirota oscillatory 2-soliton head-on with c = 4, c = − ν = 2, ν = 5, φ = 0, φ = π/ t = − . (a) amplitude (b) phase gradient Figure 8.
Sasa-Satsuma oscillatory 2-soliton right-overtake with c = 4, c =2, ν = 2, ν = 5, φ = 0, φ = π/ t = − (a) amplitude (b) phase gradient Figure 9.
Sasa-Satsuma oscillatory 2-soliton left-overtake with c = − c = − ν = 2, ν = 5, φ = 0, φ = π/ t = − a) amplitude (b) phase gradient Figure 10.
Sasa-Satsuma oscillatory 2-soliton head-on with c = 4, c = − ν = 2, ν = 5, φ = 0, φ = π/ t = − . Oscillatory breathers.
Last we discuss a few aspects of the oscillatory breather solu-tions from Theorem 3.An oscillatory breather (3.56) is parameterized by a speed c , an envelope frequency ν (cid:54) = 0and phase φ , and a modulation frequency ν and phase φ , where the speed and frequenciessatisfy the kinematic relation ( c/ + ( ν/ >
0. Depending on the combined frequency | ν | + | ν | , the speed of a breather can be positive, negative, or zero. The additional parameter χ controls the size of the oscillations, such that the oscillations disappear in the limit | χ | (cid:29) | u | = | ˜ f ( ξ, τ ) | hasexponential decay in the moving coordinate | ξ | = | x − ct | and is periodic in the oscillationcoordinate τ = νt + φ . In particular, these breathers are distinguished from oscillatory wavesby having an oscillating amplitude with a temporal period of T = π/ν . We also see that thephase arg( u ) of these breathers is time-periodic if either the frequency ν vanishes, or the twofrequencies ν (cid:54) = 0 and ν (cid:54) = 0 are commensurate, where the condition ν (cid:54) = 0 distinguishesan oscillatory breather from an ordinary breather.The amplitude and phase gradient of the breathers solutions (3.57)–(3.59) and (3.60)–(3.62) are illustrated in Fig. 11–Fig. 16 for the Hirota equation (2.13) and Fig. 17–Fig. 22 forthe Sasa-Satsuma equation (2.29). Each figure shows t = t (= t + T ) , t + T / , t + 2 T / (a) amplitude (b) phase gradient Figure 11.
Hirota stationary breather with χ = 0 . c = 0, ν = 2, φ = 0, φ = 0 ( t = − a) amplitude (b) phase gradient Figure 12.
Hirota moving breather with χ = 0 . c = 3, ν = 2, φ = 0, φ = 0( t = − (a) amplitude (b) phase gradient Figure 13.
Hirota moving breather with χ = 0 . c = − ν = 3, φ = 0, φ = 0 ( t = − (a) amplitude (b) phase gradient Figure 14.
Hirota stationary oscillatory breather with χ = 0, c = 0, ν = 2, φ = 0, ν = 3, φ = 0 ( t = − (a) amplitude (b) phase gradient Figure 15.
Hirota moving oscillatory breather with χ = 0, c = 3, ν = 2, φ = 0, ν = 3, φ = 0 ( t = − a) amplitude (b) phase gradient Figure 16.
Hirota moving oscillatory breather with χ = 0, c = − ν = 1, φ = 0, ν = 9, φ = 0 ( t = − (a) amplitude (b) phase gradient Figure 17.
Sasa-Satsuma stationary breather with χ = 0 . c = 0, ν = 2, φ = 0, φ = 0 ( t = − (a) amplitude (b) phase gradient Figure 18.
Sasa-Satsuma moving breather with χ = 0 . c = 3, ν = 2, φ = 0, φ = 0 ( t = − (a) amplitude (b) phase gradient Figure 19.
Sasa-Satsuma moving breather with χ = 0 . c = − ν = 3, φ = 0, φ = 0 ( t = − a) amplitude (b) phase gradient Figure 20.
Sasa-Satsuma stationary oscillatory breather with χ = 0, c = 0, ν = 2, φ = 0, ν = 3, φ = 0 ( t = − (a) amplitude (b) phase gradient Figure 21.
Sasa-Satsuma moving oscillatory breather with χ = 0, c = 3, ν = 2, φ = 0, ν = 3, φ = 0 ( t = − (a) amplitude (b) phase gradient Figure 22.
Sasa-Satsuma moving oscillatory breather with χ = 0, c = − ν = 1, φ = 0, ν = 9, φ = 0 ( t = − Concluding remarks
In the literature, harmonically modulated solitons (1.9) and (1.14) are commonly calledan envelope soliton . Strictly speaking, however, the factorization of such solitons (1.9) into asolitary wave part f ( kx + wt ) and a harmonic wave part exp( i ( κx + ωt )) is well-defined onlyif the function f is real-valued, so that the modulation of the solitary wave envelope is fullycontained in the phase arg( u ( t, x )) = κx + ωt , as seen in the Hirota envelope soliton (1.11).Otherwise, when the function f is complex-valued, as happens in the Sasa-Satsuma harmon-ically modulated soliton (1.12) as well as in the harmonically modulated 2-solitons for boththe Hirota and Sasa-Satsuma equations, the solitary wave envelope is given by the modulus | f ( kx + wt ) | while its modulation comes from the combined phase κx + ωt + arg( f ( kx + wt )).In general the only mathematically and physically meaningful way to decompose this phase s to write it as a travelling wave part arg( ˜ f ( kx + wt )) plus a temporal part νt , correspond-ing to the oscillatory forms (1.15) and (1.17). Thus the oscillatory parameterization that wehave introduced in this paper for harmonically modulated solitons is mathematically clearerand physically simpler than the usual envelope parameterization (1.9) and (1.14).In a sequel paper [19], we will study the main features of the amplitude and phase ofcolliding oscillatory waves as described by the oscillatory 2-soliton solutions (3.41)–(3.48)presented in Theorem 2 for the Hirota equation (2.13) and the Sasa-Satsuma equation (2.29). Acknowledgement
S. Anco is supported by an NSERC research grant. The authors thank the referee forremarks which have helped to improve this paper. Nestor Tchegoum Ngatat is thanked forassistance in an early stage of this work.Email: [email protected], sattar [email protected], [email protected]
AWe will first review the derivation of the harmonically modulated 1-soliton solution fromthe split bilinear system (2.16) for the Hirota equation (2.13). For N =1, the lowest degreeterms in the ansatz (2.15) are given by G (1) = Ae Θ , F (2) = Be Θ+ ¯Θ (A.1)with Θ = k x + w t, ¯Θ = ¯k x + ¯w t (A.2)where k , w , A are complex constants and B is a real constant. The two lowest degree equa-tions (2.16a) and (2.16b) yield w = − k , (A.3) B = 4 A ¯ A (k + ¯k) . (A.4)The inhomogeneous terms in the next equation (2.16c) turn out to vanish, which determines G (3) = 0. Likewise, the next equation (2.16d) determines F (4) = 0. Hence the ansatz (2.15)terminates at degrees 1 and 2, respectively. This yields the 1-soliton solution u = Ae Θ | A | / Re k) e Θ+ ¯Θ , Θ = k( x − k t ) . (A.5)It can be written in the form of an harmonically modulated soliton (1.9), (1.10), (1.11) inthe following way. By puttingRe k = k, Im k = κ, Re w = w, Im w = ω, (A.6)we see that the algebraic relations (1.10) and (A.3) match. Next expressingexp(Θ) = exp( i Im Θ) exp(Re Θ) = exp( i ( κx + ωt )) exp( kx + wt ) ,A = | A | exp( iφ ) , φ = arg A, (A.7)we see that the solution (A.5) has the general harmonically modulated form (1.9) with f = | A | exp( kx + wt ) / (1 + ( | A | /k ) exp(2 kx + 2 wt )) . (A.8) hen writing | A/k | = e − ak (A.9)and using the identity sech θ = 2 exp θ/ (1 + exp 2 θ ) where θ = Re Θ − ak = k ( x − a ) + wt, (A.10)we find f matches the Hirota envelope function (1.11) up to a space translation x → x − a .Hence we have u ( t, x ) = exp( i ( ϕ + κ ( x − a ) + ωt ))( | k | /
2) sech( k ( x − a ) + wt ) (A.11)after putting φ = ϕ − aκ . This yields the rational cosh form shown in Proposition 1.We will next derive the explicit form of the harmonically modulated 2-soliton solution.For N = 2, the lowest degree terms in the ansatz (2.15) are given by G (1) = A e Θ + A e Θ , F (2) = B e Θ + ¯Θ + B e Θ + ¯Θ + Ce Θ + ¯Θ + ¯ Ce Θ + ¯Θ (A.12)with Θ = k x + w t, Θ = k x + w t, ¯Θ = ¯k x + ¯w t, ¯Θ = ¯k x + ¯w t, (A.13)where k , k , w , w , A , A , C are complex constants and B , B are real constants. Similarlyto the N = 1 case, the two lowest degree equations (2.16a) and (2.16b) in the split bilinearsystem yield w = − k , w = − k , (A.14) B = 4 A ¯ A (k + ¯k ) , B = 4 A ¯ A (k + ¯k ) , C = 4 A ¯ A (k + ¯k ) . (A.15)The inhomogeneous terms in the next two equations (2.16c) and (2.16d) no longer vanish.Instead, equation (2.16c) now contains monomial terms e Θ +Θ + ¯Θ and e Θ +Θ + ¯Θ , whileequation (2.16d) contains a single monomial term e Θ +Θ + ¯Θ + ¯Θ . To balance these degree 3and 4 terms, the ansatz (2.15) needs to contain the corresponding monomial terms G (3) = D e Θ +Θ + ¯Θ + D e Θ +Θ + ¯Θ , F (4) = Ee Θ +Θ + ¯Θ + ¯Θ (A.16)where D , D are complex constants and E is a real constant. Equations (2.16c) and (2.16d)then yield D = 4 A A ¯ A (k − k ) (¯k + k ) (k + ¯k ) , D = 4 A A ¯ A (k − k ) (k + ¯k ) (k + ¯k ) (A.17)and E = 16 A A ¯ A ¯ A (k − k ) (¯k − ¯k ) (k + ¯k ) (k + ¯k ) (¯k + k ) (k + ¯k ) . (A.18)The next two higher degree equations in the split bilinear system (2.16) are given by D t ( G (5) ,
1) + D x ( G (5) ,
1) = − D t ( G (3) , F (2) ) − D x ( G (3) , F (2) ) − D t ( G (1) , F (4) ) − D x ( G (1) , F (4) ) (A.19a) D x ( F (6) , − G (5) ¯ G (1) + G (1) ¯ G (5) ) = 4 G (3) ¯ G (3) − D x ( F (4) , F (2) ) . (A.19b) he inhomogeneous terms in these equations are found to vanish, which determines G (5) = 0and F (6) = 0. Hence the ansatz (2.15) terminates at degrees 3 and 4, respectively. Thisyields the 2-soliton solution u = A e Θ (1 + V ) + A e Θ (1 + V )1 + W (A.20)with V = | A | Γ e , V = | A | Γ e (A.21) W = | A | Ω e + | A | Ω e + | A | | A | Ω Ω Γ e +Θ ) + 8Re (cid:0) A ¯ A Φ e i Im (Θ − Θ ) (cid:1) e Re (Θ +Θ ) (A.22)where Θ = k ( x − k t ) , Θ = k ( x − k t ) , (A.23)Γ = D A | A | = (k − k ) (Re k ) (¯k + k ) , Γ = D A | A | = (k − k ) (Re k ) (k + ¯k ) , (A.24)Ω = B | A | = 1(Re k ) , Ω = B | A | = 1(Re k ) , (A.25)Φ = C A ¯ A = 1(k + ¯k ) , (A.26)Γ = √ E √ B B = | k − k | | k + ¯k | . (A.27)To write this solution (A.20) in the form of an harmonically modulated soliton, we proceedsimilarly to the 1-soliton case. First we putRe k = k , Im k = κ , Re w = w , Im w = ω , Re k = k , Im k = κ , Re w = w , Im w = ω , (A.28)so thus the algebraic relations (1.10) become w = − k ( k − κ ) , ω = − κ (3 k − κ ) ,w = − k ( k − κ ) , ω = − κ (3 k − κ ) . (A.29)Next we write A = | A | exp( iφ ) , φ = arg A , A = | A | exp( iφ ) , φ = arg A , (A.30)Γ = ΓΩ exp( i γ ) , γ = arg(Γ )2 = arg (cid:16) k − k ¯k + k (cid:17) , Γ = ΓΩ exp( i γ ) , γ = arg(Γ )2 = arg (cid:16) k − k k + ¯k (cid:17) . (A.31)We now express | A | (cid:112) ΓΩ = e − a k , | A | (cid:112) ΓΩ = e − a k (A.32)and Re Θ − a k = k ( x − a ) + w t = θ , Im Θ = κ x + ω t = ϑ , (A.33)Re Θ − a k = k ( x − a ) + w t = θ , Im Θ = κ x + ω t = ϑ . (A.34) he expressions in the numerator and denominator of the 2-soliton solution (A.20) are thengiven by V = e i γ e θ , V = e i γ e θ (A.35) W = e θ + θ ) + 1Γ ( e θ + e θ ) − | Γ − | Γ Re ( e i ( φ − φ − γ + γ ) e i ( ϑ − ϑ ) ) e θ + θ (A.36)where we have used the identities Φ / ¯Φ = (Γ Ω ) / (Γ Ω ) (A.37)Φ ¯Φ = (Γ − Ω Ω /
16 (A.38)arg( − Φ) = γ − γ , | Φ | = | Γ − | (cid:112) Ω Ω / . (A.39)Hence we have u = e iφ e iϑ f + e iφ e iϑ f (A.40)with f = X /Y, f = X /Y (A.41)given by X = (1 / (cid:112) Ω Γ)(1 + V ) e − θ = (2 / (cid:112) Ω Γ) exp( iγ ) cosh( θ + iγ ) (A.42) X = (1 / (cid:112) Ω Γ)(1 + V ) e − θ = (2 / (cid:112) Ω Γ) exp( iγ ) cosh( θ + iγ ) (A.43) Y = (1 + W ) e − θ − θ = (2 / Γ) (cid:0) cosh( θ − θ ) + Γ cosh( θ + θ ) − | Γ − | cos( ϑ − ϑ + φ − φ + γ − γ ) (cid:1) . (A.44)As expressed in this form, the 2-soliton solution (A.40)–(A.44) closely resembles a har-monically modulated 2-soliton (1.14) except for the presence of the shifts a , a on x in θ , θ and the appearance of ϑ , ϑ in the functions f , f . However, under the assumption w /k (cid:54) = w /k , this solution can be converted exactly into the form (1.14). First we applya combined space-time translation x → x − x , t → t − t (A.45)such that 0 = k ( x + a ) + w t = k ( x + a ) + w t (A.46)whereby θ → k x + w t, θ → k x + w t (A.47)absorbs the shifts a , a on x . This transformation (A.45)–(A.46) exists provided k w (cid:54) = k w . It induces a corresponding transformation ϑ → κ x + ω t − ( κ x + ω t ) , ϑ → κ x + ω t − ( κ x + ω t ) (A.48)producing additional phase angles which can be absorbed by shifts φ → φ − ϕ , φ → φ − ϕ (A.49)so that ϑ + φ + γ → κ x + ω t + φ , ϑ + φ + γ → κ x + ω t + φ (A.50)via ϕ = γ − ( κ x + ω t ), ϕ = γ − ( κ x + ω t ). ext we use the identity( κ − κ ) x + ( ω − ω ) t = µ ( k x + w t ) − µ ( k x + w t ) (A.51)with µ = w ( κ − κ ) − k ( ω − ω ) k w − k w , µ = w ( κ − κ ) − k ( ω − ω ) k w − k w (A.52)which holds provided k w (cid:54) = k w . This leads to the rational cosh form presented inProposition 2. Appendix
BWe will first summarize the derivation of the harmonically modulated 1-soliton solutionfrom the split bilinear system (2.32) for the Sasa-Satsuma equation (2.29). For N =1, thelowest degree terms in the ansatz (2.31) are given by G (1) = Ae Θ , F (2) = Be Θ+ ¯Θ , H (2) = Ce Θ+ ¯Θ , (B.1)with Θ = k x + w t, ¯Θ = ¯k x + ¯w t (B.2)where k , w , A, C are complex constants and B is a real constant. The two lowest degreeequations (2.32a) and (2.32b) yield exactly the results (A.3) and (A.4) obtained for theHirota solution, while the next equation (2.32c) determines C = 6 A ¯ A (k − ¯k) . (B.3)In the next two lowest degree equations (2.32d) and (2.32e), the inhomogeneous terms arefound to consist of the respective monomials e and e . These degree 3 and 4 termsneed to be balanced by having the ansatz (2.31) for G and F contain the correspondingmonomial terms G (3) = De , F (4) = Ee (B.4)where D is a complex constant and E is a real constant. Equations (2.32d) and (2.32e) thenyield D = A ¯ A (k − ¯k)k(k + ¯k) , E = − A ¯ A (k − ¯k) k¯k(k + ¯k) . (B.5)Next, the inhomogeneous terms in equation (2.32f) turn out to vanish, which determines H (4) = 0. Likewise, the next two higher degree equations (2.32e) and (2.32g) determine G (5) = 0 and F (6) = 0. Hence the ansatz (2.31) terminates at degrees 3, 4, and 2, respectively.This yields the 1-soliton solution u = Ae Θ (cid:16) | A | (Re k) i Im k2k e Θ+ ¯Θ (cid:17) | A | (Re k) e Θ+ ¯Θ + | A | (Re k) (Im k) | k | e , Θ = k( x − k t ) (B.6)which can be written in the form of a harmonically modulated soliton (1.9), (1.10), (1.12)similarly to the Hirota case. In particular, using expressions (A.6) and (A.7), we see thatthe solution (B.6) has the general harmonically modulated form (1.9) with f = | A | exp( kx + wt ) (cid:0) i ( | A | /k ) Λ exp(2 kx + 2 wt ) (cid:1) kx + 2 wt ) + ( | A | /k ) | Λ | exp(4 kx + 4 wt ) (B.7) here Λ = iκ k + iκ ) = κ ( κ + ik )2( k + κ ) . (B.8)After we simplify this function f in terms of | A/k | = e − ak and θ = Re Θ − ak = k ( x − a ) + wt ,it matches the Sasa-Satsuma envelope function (1.12) up to a space translation x → x − a .Hence we have u ( t, x ) = exp( iφ ) exp( i ( κx + ωt )) | k | exp( k ( x − a ) + wt ) (cid:0) k ( x − a ) + wt )) (cid:1) k ( x − a ) + wt )) + | Λ | exp(4( k ( x − a ) + wt ))= exp( iϕ ) exp( i ( κ ( x − a ) + ωt )) | k | (cid:0) k + κ + ( κ/ κ + ik ) exp(2( k ( x − a ) + wt )) (cid:1) k + κ ) cosh( k ( x − a ) + wt ) + ( κ/ exp(3( k ( x − a ) + wt )) (B.9)where ϕ = φ + aκ .The harmonically modulated 1-soliton solution (B.9) can be converted into a rational coshform. This was carried out in [11] when κ = 0. For κ (cid:54) = 0, the form of the highest-degreemonomial terms in the numerator and denominator of expression (B.6) motivates writing | A | (cid:112) | Λ | / | k | = e − ˜ ak (B.10)˜ θ = Re Θ − ˜ ak = k ( x − ˜ a ) + wt (B.11)so that ˜ a is absorbed into a space translation x → x − ˜ a . The envelope function (B.7) thenbecomes f = ( | k | / (cid:112) | Λ | ) e ˜ θ (1 + e iλ e θ ) / (1 + e θ + (1 / | Λ | ) e θ ) (B.12)where | Λ | = | κ | √ k + κ , (B.13) λ = arg( κ ( κ + ik )) . (B.14)We next use the identity 1 + exp(4˜ θ ) = 2 exp(2˜ θ ) cosh(2˜ θ ), which yields f = ( | k | / (cid:112) | Λ | ) e iλ/ cosh(˜ θ + iλ/ / (cosh(2˜ θ ) + 1 / (2 | Λ | )) . (B.15)By now absorbing λ/ φ → ϕ = φ + ˜ aκ + λ/
2, we obtain u ( t, x ) = exp( iϕ ) exp( i ( κ ( x − ˜ a ) + ωt ))( | k | / (cid:112) | Λ | ) cosh( k ( x − ˜ a ) + wt + iλ/ k ( x − ˜ a ) + wt )) + 1 / (2 | Λ | ) (B.16)This yields the rational cosh form shown in Proposition 3.We will next derive the explicit form of the envelope 2-soliton solution. For N =2, thelowest degree terms in the ansatz (2.31) are given by G (1) = A e Θ + A e Θ , F (2) = B e Θ + ¯Θ + B e Θ + ¯Θ + Ce Θ + ¯Θ + ¯ Ce Θ + ¯Θ , (B.17) H (2) = D e Θ + ¯Θ + D e Θ + ¯Θ + E e Θ + ¯Θ + E e Θ + ¯Θ , (B.18)with Θ = k x + w t, Θ = k x + w t, ¯Θ = ¯k x + ¯w t, ¯Θ = ¯k x + ¯w t, (B.19)where k , k , w , w , A , A , C, D , D , E , E are complex constants and B , B are real con-stants. Similarly to the N = 1 case, the two lowest degree equations (2.16a) and (2.16b) in he split bilinear system yield exactly the results (A.14) and (A.15) obtained for the Hirotasolution, while the next equation (2.32c) determines D = 6 A ¯ A (k − ¯k ) , D = 6 A ¯ A (k − ¯k ) , E = 6 A ¯ A (k − ¯k ) = − ¯ E . (B.20)In contrast the next two lowest degree equations (2.32d) and (2.32e) now contain many moreinhomogeneous monomial terms than in the Hirota case. As a consequence, the ansatz (2.31)continues past degrees 3 and 4 for G and F and finally turns out to terminate at degree 7for G , degree 8 for F , and degree 6 for H . The higher degree equations that determine theseterms are given by H = 6 D x ( G (1) , ¯ G (5) ) + 6 D x ( G (5) , ¯ G (1) ) + 6 D x ( G (3) , ¯ G (3) ) − F (4) H (2) − F (2) H (4) (B.21a) D t ( G (7) ,
1) + D x ( G (7) ,
1) = − D t ( G (5) , F (2) ) − D x ( G (5) , F (2) ) + G (5) H (2) − D t ( G (3) , F (4) ) − D x ( G (3) , F (4) ) + G (3) H (4) − D t ( G (1) , F (6) ) − D x ( G (1) , F (6) ) + G (1) H (2) (B.21b) D x ( F (8) , − G (7) ¯ G (1) + G (1) ¯ G (7) ) =4( G (5) ¯ G (3) + G (3) ¯ G (5) ) − D x ( F (4) , F (4) ) − D x ( F (6) , F (2) ) . (B.21c)Omitting all details, we find that equations (2.32f)–(2.32h) and (B.21a)–(B.21c) in thesplit bilinear system lead to the following results. The higher degree terms in G and F consist of G (3) = F e + ¯Θ + F e + ¯Θ + F e + ¯Θ + F e + ¯Θ + G e Θ +Θ + ¯Θ + G e Θ +Θ + ¯Θ (B.22) G (5) = H e Θ +2Θ +2 ¯Θ + H e +Θ +2 ¯Θ + H e Θ +2Θ +2 ¯Θ + H e +Θ +2 ¯Θ + I e +Θ + ¯Θ + ¯Θ + I e Θ +2Θ + ¯Θ + ¯Θ (B.23) G (7) = J e +2Θ + ¯Θ +2 ¯Θ + J e +2Θ + ¯Θ +2 ¯Θ (B.24)and F (4) = K e +2 ¯Θ + K e +2 ¯Θ + K e +2 ¯Θ + ¯ K e +2 ¯Θ + L e Θ +Θ +2 ¯Θ + ¯ L e + ¯Θ + ¯Θ + L e Θ +Θ +2 ¯Θ + ¯ L e + ¯Θ + ¯Θ + M e Θ +Θ + ¯Θ + ¯Θ (B.25) F (6) = N e +Θ +2 ¯Θ + ¯Θ + N e Θ +2Θ + ¯Θ +2 ¯Θ + Oe +Θ + ¯Θ +2 ¯Θ + ¯ Oe Θ +2Θ +2 ¯Θ + ¯Θ (B.26) F (8) = P e +2Θ +2 ¯Θ +2 ¯Θ (B.27)where F , F , F , F , G , G , H , H , H , H , I , I , J , J are complex constants given by F = A ¯ A (k − ¯k )k (k + ¯k ) , F = A ¯ A (k − ¯k )k (k + ¯k ) , (B.28) F = A ¯ A (k − ¯k )k (k + ¯k ) , F = A ¯ A (k − ¯k )k (k + ¯k ) , (B.29) = 2 A A ¯ A ((k + k )(k − k ) + (k − ¯k )(k + ¯k ) + (k − ¯k )(k + ¯k ) )(k + k )(k + ¯k ) (k + ¯k ) , (B.30) G = 2 A A ¯ A ((k + k )(k − k ) + (k − ¯k )(k + ¯k ) + (k − ¯k )(k + ¯k ) )(k + k )(k + ¯k ) (k + ¯k ) , (B.31) H = − A A ¯ A (k − ¯k )(k − k ) (k − ¯k ) k ¯k (k + k )(k + ¯k ) (k + ¯k ) , (B.32) H = − A A ¯ A (k − ¯k )(k − ¯k ) (k − k ) k ¯k (k + k )(k + ¯k ) (k + ¯k ) , (B.33) H = − A A ¯ A (k − ¯k )(k − k ) (k − ¯k ) k ¯k (k + k )(k + ¯k ) (k + ¯k ) , (B.34) H = − A A ¯ A (k − ¯k )(k − ¯k ) (k − k ) k ¯k (k + k )(k + ¯k ) (k + ¯k ) , (B.35) I = − A A ¯ A ¯ A (k − ¯k )(k − ¯k )(k − k ) × (k − ¯k )(k + ¯k ) + (k − ¯k )(k + ¯k ) − (¯k + ¯k )(¯k − ¯k ) k (k + k )(¯k + ¯k )(k + ¯k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) , (B.36) I = − A A ¯ A ¯ A (k − ¯k )(k − ¯k )(k − k ) × (k − ¯k )(k + ¯k ) + (k − ¯k )(k + ¯k ) − (¯k + ¯k )(¯k − ¯k ) k (k + k )(¯k + ¯k )(k + ¯k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) , (B.37) J = A A ¯ A ¯ A (k − ¯k )(k − ¯k )(k − ¯k ) (k − ¯k ) (¯k − ¯k ) (k − k ) k k ¯k (¯k + ¯k )(k + k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) , (B.38) J = A A ¯ A ¯ A (k − ¯k )(k − ¯k )(k − ¯k ) (k − ¯k ) (¯k − ¯k ) (k − k ) k k ¯k (¯k + ¯k )(k + k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) , (B.39)and where K , K , M, N , N , P are real constants and K , L , L , O are complex constantsgiven by K = − A ¯ A (k − ¯k ) k ¯k (k + ¯k ) , K = − A ¯ A (k − ¯k ) k ¯k (k + ¯k ) , (B.40) K = − A ¯ A (k − ¯k ) k ¯k (k + ¯k ) , (B.41) L = − A A ¯ A (k − ¯k )(k − ¯k )¯k (k + k )(k + ¯k ) (k + ¯k ) , (B.42) L = − A A ¯ A (k − ¯k )(k − ¯k )¯k (k + k )(k + ¯k ) (k + ¯k ) , (B.43) M = 8 A ¯ A A ¯ A ( M − M − M )(k + k )(¯k + ¯k )(k + ¯k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) , (B.44) M = (k + k )(k − k ) (¯k + ¯k )(¯k − ¯k ) , (B.45) M = (k + ¯k ) (k − ¯k )(k + ¯k ) (k − ¯k ) , (B.46) = (k + ¯k ) (k − ¯k )(k + ¯k ) (k − ¯k ) , (B.47) N = 4 A A ¯ A ¯ A (k − ¯k )(k − ¯k )(k − k ) (k − ¯k ) (¯k − ¯k ) k ¯k (k + k )(¯k + ¯k )(k + ¯k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) , (B.48) N = 4 A A ¯ A ¯ A (k − ¯k )(k − ¯k )(k − k ) (k − ¯k ) (¯k − ¯k ) k ¯k (k + k )(¯k + ¯k )(k + ¯k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) , (B.49) O = 4 A A ¯ A ¯ A (k − ¯k )(k − ¯k )(k − k ) (k − ¯k ) (¯k − ¯k ) k ¯k (k + k )(¯k + ¯k )(k + ¯k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) , (B.50) P = A A ¯ A ¯ A (k − ¯k ) (k − ¯k ) (k − ¯k ) (k − ¯k ) (k − k ) (¯k − ¯k ) k k ¯k ¯k (k + k ) (¯k + ¯k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) . (B.51)For completeness, we also list the higher degree terms in H : H (4) = Q e Θ +Θ +2 ¯Θ + Q e Θ +Θ +2 ¯Θ + Q e + ¯Θ + ¯Θ + Q e + ¯Θ + ¯Θ + Re Θ +Θ + ¯Θ + ¯Θ (B.52) H (6) = S e +Θ + ¯Θ +¯2Θ + S e Θ +2Θ +2 ¯Θ + ¯Θ + S e Θ +2Θ + ¯Θ +2 ¯Θ + S e +Θ +2 ¯Θ + ¯Θ (B.53)where Q , Q , Q , Q , R, S , S , S , S are complex constants given by Q = − A A ¯ A (k − ¯k )(k − ¯k )(k − k ) ¯k (k + ¯k ) (k + ¯k ) = − ¯ Q , (B.54) Q = − A A ¯ A (k − ¯k )(k − ¯k )(k − k ) ¯k (k + ¯k ) (k + ¯k ) = − ¯ Q , (B.55) R = − A A ¯ A ¯ A ( R + R + R − ¯ R + R + ¯ R )(k + k )(¯k + ¯k )(k + ¯k ) (k + ¯k ) (k + ¯k ) (k + ¯k ) , (B.56) R = − k ¯k (k − ¯k )(k + ¯k ) (cid:0) (k − ¯k ) + (k − k )(¯k − ¯k ) + (k − ¯k )(¯k − k ) (cid:1) , (B.57) R = − k ¯k (k − ¯k )(k + ¯k ) (cid:0) (k − ¯k ) + (k − k )(¯k − ¯k ) + (k − ¯k )(¯k − k ) (cid:1) , (B.58) R = k k (k + k )(k − k ) (cid:0) (¯k − ¯k ) + (k − ¯k )(k − ¯k ) + (k − ¯k )(k − ¯k ) (cid:1) , (B.59) R = − k ¯k (k − ¯k )(k + ¯k ) (cid:0) (¯k − k ) − (k − ¯k )(k − ¯k ) − (¯k − ¯k )(k − k ) (cid:1) , (B.60) S = 6 A A ¯ A ¯ A (k − ¯k )(k − ¯k )(k − ¯k )(k − k ) (k − ¯k ) (¯k − ¯k ) k ¯k (k + k )(¯k + ¯k )(k + ¯k ) (k + ¯k ) (k + ¯k ) = − ¯ S , (B.61) S = 6 A A ¯ A ¯ A (k − ¯k )(k − ¯k )(k − ¯k )(k − k ) (k − ¯k ) (¯k − ¯k ) k ¯k (k + k )(¯k + ¯k )(k + ¯k ) (k + ¯k ) (k + ¯k ) , (B.62) S = 6 A A ¯ A ¯ A (k − ¯k )(k − ¯k )(k − ¯k )(k − k ) (k − ¯k ) (¯k − ¯k ) k ¯k (k + k )(¯k + ¯k )(k + ¯k ) (k + ¯k ) (k + ¯k ) . (B.63)Expressions (B.28)–(B.51) yield the 2-soliton solution u = A e Θ (1 + V ) + A e Θ (1 + V )1 + W (B.64) ith V = | A | Λ e + | A | (Π + | A | | Λ | Ψ e ) e + | A | | A | Λ Ψ ( ¯Π + | A | | Λ | Ψ e ) e +Θ ) + A ¯ A Σ (1 + | A | ¯Λ Ψ e ) e i Im (Θ − Θ ) e Re (Θ +Θ ) (B.65) V = | A | Λ e + | A | (Π + | A | | Λ | Ψ e ) e + | A | | A | Λ Ψ ( ¯Π + | A | | Λ | Ψ e ) e +Θ ) + A ¯ A Σ (1 + | A | ¯Λ Ψ e ) e i Im (Θ − Θ ) e Re (Θ +Θ ) (B.66) W = | A | (Ω + | A | | Λ | e ) e + | A | (Ω + | A | | Λ | e ) e + | A | | A | (cid:0) Π + | A | Ω | Λ | Ψ e + | A | Ω | Λ | Ψ e (cid:1) e +Θ ) + | A | | A | | Λ | | Λ | Ψ e +Θ ) + 2Re (cid:0) ( A ¯ A ) Σ ¯Σ e i − Θ ) (cid:1) e +Θ ) + 8Re (cid:0) A ¯ A Φ(1 + | A | Λ ¯∆ e + | A | ¯Λ ∆ e ) e i Im (Θ − Θ ) (cid:1) e Re (Θ +Θ ) + 8 | A | | A | Re (cid:0) A ¯ A ¯ΦΛ ¯Ψ ¯Λ Ψ e i Im (Θ − Θ ) (cid:1) e +Θ ) (B.67)where Θ = k ( x − k t ) , Θ = k ( x − k t ) , (B.68)Ω = B | A | = 1(Re k ) , Ω = B | A | = 1(Re k ) , (B.69)Λ = F A | A | = i Im k (Re k ) , Λ = F A | A | = i Im k (Re k ) , (B.70)Σ = F A ¯ A = k − ¯k k (k + ¯k ) , Σ = F A ¯ A = k − ¯k k (k + ¯k ) , (B.71)∆ = ¯ A L ¯ F ¯ C = k − ¯k k + k , ∆ = ¯ A L ¯ F C = k − ¯k k + k , (B.72)Ψ = ¯ A H ¯ F F = | A | H A | F | = (k − ¯k )(k − k ) (k + k )(k + ¯k ) , (B.73)Ψ = ¯ A H ¯ F F = | A | H A | F | = (k − ¯k )(k − k ) (k + k )(k + ¯k ) , (B.74)Ξ = 2 ¯ A F L ¯ F F B = 4 i Im k k + k , Ξ = 2 ¯ A F L ¯ F F B = 4 i Im k k + k , (B.75)Π = G A | A | = F ¯ F ¯ I A ¯ F ¯ H = (k − k ) ) (k + ¯k ) + k − ¯k ) (k + k ) + 4 i Im k (k + k )(k + ¯k ) , (B.76)Π = G A | A | = F ¯ F ¯ I A ¯ F ¯ H = (k − k ) ) (k + ¯k ) + k − ¯k ) (k + k ) + 4 i Im k (k + k )(k + ¯k ) , (B.77) = C A ¯ A = 1(k + ¯k ) , (B.78)Ψ = | Ψ | = | Ψ | = | A || A |√ P | F || F | = | A | N | F | B = | A | N | F | B = A J F H = A J F H = | k − ¯k | | k − k | | k + k | | k + ¯k | , (B.79)∆ = | ∆ | = | ∆ | = | A || L || F || C | = | A || L || F || C | = | k − ¯k || k + k | , (B.80)Ξ = 2Re (Ξ ¯Ξ ) = Re (cid:18) A ¯ A L ¯ L F ¯ F (cid:19) B B = 32Im k Im k | k + k | , (B.81)Π = M | A | | A | = | k − k | Re k ) | k + ¯k | + | k − ¯k | Re k ) | k + k | + 32Im k Im k | k + k | | k + ¯k | . (B.82)This solution (B.64) can be shown to reduce to the ordinary 2-soliton solution [12, 11]for the Sasa-Satsuma equation when κ = κ = 0. For κ (cid:54) = 0 and κ (cid:54) = 0, it can bewritten in the form of a harmonically modulated soliton similarly to the Hirota case by useof the notation (A.28), (A.29), (A.30). To proceed, we first we observe that the highest-degree monomial terms in the numerator and denominator of expression (B.64) consistof | Λ | | Λ | Ψ e e , | Λ | Ψ Λ Ψ e e , | Λ | Ψ Λ Ψ e e . Based onthe form of their coefficients, we write | A | (cid:112) | Λ | Ψ = e − a k , | A | (cid:112) | Λ | Ψ = e − a k , (B.83)andRe Θ − a k = k ( x − a ) + w t = θ , Re Θ − a k = k ( x − a ) + w t = θ , (B.84)Im Θ = κ x + ω t = ϑ , Im Θ = κ x + ω t = ϑ . (B.85)Then, in the denominator of the 2-soliton solution (B.64), we have W = e θ +4 θ + 1Ψ (cid:16) e θ + e θ (cid:17) + Ω Ψ | Λ | (cid:16) e θ (cid:17) e θ + Ω Ψ | Λ | (cid:16) e θ (cid:17) e θ + 1Ψ | Λ || Λ | (cid:16) Π + 2Re (cid:0) Σ ¯Σ e i φ − φ ) e i ϑ − ϑ ) (cid:1)(cid:17) e θ +2 θ + 8∆Ψ (cid:112) | Λ || Λ | Re (cid:16) Φ e i ( φ − φ ) e i ( ϑ − ϑ ) (cid:16) ¯Λ ∆ | Λ | ∆ e θ + Λ ¯∆ | Λ | ∆ e θ (cid:17)(cid:17) e θ + θ + 8Ψ (cid:112) | Λ || Λ | Re (cid:16) e i ( φ − φ ) e i ( ϑ − ϑ ) (cid:16) Φ + ¯Φ ¯Λ Ψ | Λ | Ψ Λ ¯Ψ | Λ | Ψ e θ +2 θ (cid:17)(cid:17) e θ + θ . (B.86) n the numerator of the 2-soliton solution (B.64), we have V = Λ Ψ | Λ | Ψ e θ +4 θ + 1Ψ (cid:16) Λ | Λ | e θ + Ψ Ψ e θ (cid:17) + 1Ψ | Λ | (cid:16) Π + Λ Ψ | Λ | Ψ ¯Π e θ (cid:17) e θ + Σ Ψ (cid:112) | Λ || Λ | (cid:16) Ψ | Λ | Ψ e θ (cid:17) e i ( φ − φ ) e i ( ϑ − ϑ ) e θ + θ (B.87)and V = Λ Ψ | Λ | Ψ e θ +4 θ + 1Ψ (cid:16) Λ | Λ | e θ + Ψ Ψ e θ (cid:17) + 1Ψ | Λ | (cid:16) Π + Λ Ψ | Λ | Ψ ¯Π e θ (cid:17) e θ + Σ Ψ (cid:112) | Λ || Λ | (cid:16) Ψ | Λ | Ψ e θ (cid:17) e i ( φ − φ ) e i ( ϑ − ϑ ) e θ + θ (B.88)where Π = Ω (cid:0) Ψ ∆ + ∆ (cid:1) + Ξ Φ , Π = Ω (cid:0) Ψ ∆ + ∆ (cid:1) + Ξ ¯Φ , (B.89)Π = Ξ | Φ | + Ω Ω (cid:16) Ψ ∆ + ∆ (cid:17) . (B.90)Next we write λ = arg(Λ ) = arg( i ¯k Im k ) , λ = arg(Λ ) = arg( i ¯k Im k ) , (B.91) δ = arg(∆ ) = arg (cid:0) (¯k + ¯k )(k − ¯k ) (cid:1) ,δ = arg(∆ ) = arg (cid:0) (¯k + ¯k )(k − ¯k ) (cid:1) , (B.92) σ = arg(Σ ) = arg (cid:0) ¯k (k − ¯k )(¯k + k ) (cid:1) ,σ = arg(Σ ) = arg (cid:0) ¯k (k − ¯k )(k + ¯k ) (cid:1) , (B.93) ψ = arg(Ψ ) = arg (cid:0) (¯k + ¯k )(¯k − k )(k + ¯k ) (k − k ) (cid:1) ,ψ = arg(Ψ ) = arg (cid:0) (¯k + ¯k )(¯k − k )(¯k + k ) (k − k ) (cid:1) , (B.94) ζ = arg(Ξ ¯Φ) = arg (cid:0) i (¯k + ¯k )(k + ¯k ) Im k (cid:1) ,ζ = arg(Ξ Φ) = arg (cid:0) i (¯k + ¯k )(¯k + k ) Im k (cid:1) , (B.95)and use the identitiesΦ¯Φ = ∆ ¯Ψ Ψ Ψ ¯∆ ∆ , | Φ | = − Φ (cid:16) ∆ Ψ ∆ Ψ (cid:17) / = − ¯Φ (cid:16) ∆ Ψ ∆ Ψ (cid:17) / . (B.96)Hence the 2-soliton solution takes the form u = e iφ e iϑ f + e iφ e iϑ f (B.97)with f = X /Y, f = X /Y (B.98) iven by X = (1 / (cid:112) | Λ | Ψ)(1 + V ) e − θ − θ = 2 (cid:112) | Λ | Ψ (cid:18) exp( i ( λ + ψ )) (cid:16) Ψ cosh( θ + 2 θ + i ( ψ + λ ))+ cosh( θ − θ + i ( λ − ψ )) + | Ξ || Φ || Λ | cosh( θ + i ( ( λ + ψ ) − ζ ))+ Ω | Λ | (cid:16) Ψ∆ cosh( θ + i ( ( λ − ψ ) + δ )) + ∆ cosh( θ + i ( ( λ + ψ ) − δ )) (cid:17)(cid:17) + exp( i ( σ + ( ψ − λ ))) (cid:16) | Σ | (cid:112) | Λ || Λ | exp( i ( ϑ − ϑ + φ − φ )) × cosh( θ + i ( ψ − λ )) (cid:17)(cid:19) (B.99) X = (1 / (cid:112) | Λ | Ψ)(1 + V ) e − θ − θ = 2 (cid:112) | Λ | Ψ (cid:18) exp( i ( λ + ψ )) (cid:16) Ψ cosh( θ + 2 θ + i ( ψ + λ ))+ cosh( θ − θ + i ( λ − ψ )) + | Ξ || Φ || Λ | cosh( θ + i ( ( λ + ψ ) − ζ ))+ Ω | Λ | (cid:16) Ψ∆ cosh( θ + i ( ( λ − ψ ) + δ )) + ∆ cosh( θ + i ( ( λ + ψ ) − δ )) (cid:17)(cid:17) + exp( i ( σ + ( ψ − λ ))) (cid:16) | Σ | (cid:112) | Λ || Λ | exp( i ( ϑ − ϑ + φ − φ )) × cosh( θ + i ( ψ − λ )) (cid:17)(cid:19) (B.100) Y = (1 + W ) e − θ − θ = 2Ψ (cid:18) Ω | Λ | cosh(2 θ ) + Ω | Λ | cosh(2 θ ) + Ψ cosh(2( θ + θ )) + 1Ψ cosh(2( θ − θ )) − | Φ | (cid:112) | Λ || Λ | Re (cid:16) exp( i ( ϑ − ϑ + φ − φ + ( λ − λ + ψ − ψ ))) × (cid:16) cosh( θ + θ + i ( δ − δ + λ − λ ))+ ∆Ψ cosh( θ − θ + i ( λ + λ − δ − δ )) (cid:17)(cid:17) + 1Ψ | Λ || Λ | (cid:16) | Σ || Σ | cos(2( ϑ − ϑ ) + 2( φ − φ ) + σ − σ )+ Ξ | Φ | Ω (cid:16) ∆ + Ψ ∆ (cid:17)(cid:17)(cid:19) . (B.101) n this form, the 2-soliton solution (B.97)–(B.101) closely resembles a harmonically mod-ulated 2-soliton (1.14), except that a , a occur in θ , θ while ϑ , ϑ appear in the functions f , f . The same space-time translation (A.45)—(A.46) used in the Hirota case can be ap-plied here to absorb a , a on x , and then the same identity (A.51) expressing ϑ − ϑ as alinear combination of θ and θ can be used to convert f , f into the proper harmonicallymodulated form. Finally, the phase angles in the resulting expressions can be substantiallysimplified through phase shifts (A.49) given by ϕ = ( λ + ψ ) / , ϕ = ( λ + ψ ) / , (B.102)and through the angle identities ζ − δ − ρ = σ − λ + ρ = arg( i (¯k − k )(k + ¯k ) ) , (B.103) ζ − δ − ρ = σ − λ + ρ = arg( i (¯k − k )(¯k + k ) ) , (B.104)( ψ − δ ) / − k )(k + ¯k )) , (B.105)( ψ − δ ) / − k )(¯k + k )) , (B.106)( δ − δ ) / ρ + arg( i (¯k − k )) , ( δ + δ ) / ρ + arg( i (¯k + ¯k )) , (B.107)where ρ = arg(Im k ) , ρ = arg(Im k ) , (B.108) ρ = (cid:40) , | Re k | (cid:54) = | Re k | arg(Im k + Im k ) , | Re k | = | Re k | . (B.109)This leads to the rational cosh form presented in Proposition 4. References [1] N.J. Zabusky, in
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