Oscillatory solitons of U(1)-invariant mKdV equations II: Asymptotic behavior and constants of motion
OOSCILLATORY SOLITONS OF U(1)-INVARIANT MKDV EQUATIONS II:ASYMPTOTIC BEHAVIOR AND CONSTANTS OF MOTION
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. The Hirota equation and the Sasa-Satsuma equation are U (1)-invariant inte-grable generalizations of the modified Korteweg-de Vries equation. These two generalizationsadmit oscillatory solitons, which describe harmonically modulated complex solitary wavesparameterized by their speed, modulation frequency, and phase. Depending on the modula-tion frequency, the speeds of oscillatory waves (1-solitons) can be positive, negative, or zero,in contrast to the strictly positive speed of ordinary solitons. When the speed is zero, anoscillatory wave is a time-periodic standing wave. Oscillatory 2-solitons with non-zero wavespeeds are shown to describe overtake collisions of a fast wave and a slow wave moving inthe same direction, or head-on collisions of two waves moving in opposite directions. Whenone wave speed is zero, oscillatory 2-solitons are shown to describe collisions in which amoving wave overtakes a standing wave. An asymptotic analysis using moving coordinatesis carried out to show that, in all collisions, the speeds and modulation frequencies of theindividual waves are preserved, while the phases and positions undergo a shift such that thecenter of momentum of the two waves moves at a constant speed. The primary constantsof motion as well as some other features of the nonlinear interaction of the colliding wavesare discussed. Introduction
Complex U (1)-invariant modified Korteweg-de Vries (mKdV) equations u t + ( αu ¯ u x + βu x ¯ u ) u + γu xxx = 0 (1.1)(where α, β, γ are real constants) arise in many physical applications, such as short wavepulses in optical fibers [1, 2] and deep water waves [3, 4]. Of particular mathematical andphysical interest are the two integrable equations in this class, given by the Hirota equation[5] u t + β | u | u x + γu xxx = 0 , (1.2)and the Sasa-Satsuma equation [6] u t + α ( u ¯ u x + 3 u x ¯ u ) u + γu xxx = 0 . (1.3) Key words and phrases. mKdV equation, Hirota equation, Sasa-Satsuma equation, solitary wave, envelopesoliton, oscillatory soliton, breather, overtake collision, head-on collision, position shift, phase shift. a r X i v : . [ n li n . S I] N ov he integrability properties of these two equations consist of multi-soliton solutions, a Laxpair, a bilinear formulation, a bi-Hamiltonian structure, and an infinite hierarchy of symme-tries and conservation laws.Both equations possess ordinary soliton solutions of the form u ( t, x ) = exp( iφ ) f ( x − ct ) (1.4)which are solitary waves with speed c > − π ≤ φ ≤ π . Collisions of two ormore solitary waves are described by multi-soliton solutions. In all collisions, the net effecton the solitary waves is to shift in their asymptotic positions, while their asymptotic phaseangles stay unchanged in the case of the Hirota equation (2.1) but become shifted in thecase of the Sasa-Satsuma equation (2.2). The actual nonlinear interaction of these solitarywaves during a collision exhibits interesting features which depend on the speed ratios andrelative phase angles of the waves, as studied in previous work [7]. (See the animations athttp://lie.math.brocku.ca/ ~ sanco/solitons/mkdv solitons.php)In a recent paper [8], we began a comprehensive study of a more general type of solitonsolution with the form u ( t, x ) = exp( iφ ) exp( iνt ) ˜ f ( x − ct ) (1.5)which is a harmonically modulated solitary wave, called an oscillatory soliton , where thetemporal modulation frequency ν (cid:54) = 0 and the speed c obey the kinematic relation( c/ + ( ν/ > . (1.6)This relation is required for the oscillatory soliton to be a solution of the the Hirota equation(2.1) and the Sasa-Satsuma equation (2.2). In contrast to an ordinary soliton, the speed c of an oscillatory soliton can be positive, negative, or zero. Consequently, these solitons havethree different types of collisions: (1) right-overtake — where a faster right-moving solitonovertakes a slower right-moving soliton or a stationary soliton; (2) left-overtake — where afaster left-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 thesecollisions are described by oscillatory 2-soliton solutions 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 ) , c (cid:54) = c (1.7)whose temporal frequencies ν , ν and speeds c , c satisfy the kinematic relations( c / + ( ν / > , ( c / + ( ν / > . (1.8)In the present paper we will study the asymptotic features of colliding oscillatory solitons(1.7) for both the Hirota equation (2.1) and the Sasa-Satsuma equation (2.2). These collisionscan be expected to exhibit highly interesting new features compared to collisions of ordinarysolitons.In section 2, we recall some details of the oscillatory 1-soliton and 2-soliton solutionsderived in Ref. [8] for the Hirota equation and the Sasa-Satsuma equation.In section 3, we carry out an asymptotic analysis using moving coordinates to show thatthe 2-soliton solutions (1.7) for the Hirota equation and the Sasa-Satsuma equation reduce toa superposition of two 1-soliton solutions (1.5) with speeds c , c and temporal frequencies ν , ν in the asymptotic past and future. This analysis rigorously establishes that these2-soliton solutions describe collisions between two oscillatory waves when both c and c are on-zero, or collisions of an oscillatory wave with a standing wave, when one of the speeds c or c is zero.In section 4, for the 1-soliton and 2-soliton solutions, we discuss the primary constantsof motion arising from the conservation laws for momentum, energy, and Galilean energyadmitted by [9] the Hirota equation and the Sasa-Satsuma equation. In particular, fromconservation of Galilean energy, we show that the center of momentum for the 2-solitonsolutions moves at a constant speed throughout a collision.Our main results are obtained in section 5. For overtake and head-on collisions described bythe 2-soliton solutions, we first show that the net effect of a collision is to shift the asymptoticpositions and phases of the two waves while the speed and the temporal frequency of eachwave remains unchanged. Explicit formulas for these asymptotic shifts are presented in termsof the speeds and temporal frequencies of the two waves in the collision. Next, from theseformulas, we find that for overtake collisions the faster wave gets shifted forward relativeto its direction of motion while the slower or stationary wave gets shifted in the backwarddirection. In contrast, for head-on collisions, we find that both waves get shifted forwardrelative to their directions of motion. Finally, for all collisions, we show that the positionshifts of the two oscillatory waves are related by the property that the center of momentumof the waves is preserved in the collision.In section 6, we discuss a few interesting features of the nonlinear interactions that occurfor oscillatory waves and standing waves during collisions, such as the appearance of nodes,phase coils, and phase reversals. We also make some concluding remarks.Previous work on soliton solutions of the Hirota equation and Sasa-Satsuma equationappears in Ref. [10, 11, 12, 13]. This work amounts to deriving the 1-soliton and 2-solitonformulas in a mathematically equivalent but less physically useful envelope form, withoutany analysis of the asymptotic behaviour and the constants of motion for these solutions.All computations in the present paper have been carried out by use of Maple. Hereafter,by scaling variables t, x, u , we will put α = 6 , β = 24 , γ = 1 (1.9)for convenience. 2. Oscillatory soliton solutions
For the Hirota equation u t + 24 | u | u x + u xxx = 0 (2.1)and the Sasa-Satsuma equation u t + 6( u ¯ u x + 3 u x ¯ u ) u + γu xxx = 0 (2.2)we will first summarize the expressions for the respective travelling-wave functions ˜ f ( x − ct )in the oscillatory 1-soliton solutions (1.5).Let k = √ (cid:16) (cid:113)(cid:112) ( c/ + ( ν/ − ν/ (cid:113)(cid:112) ( c/ + ( ν/ + ν/ (cid:17) , (2.3) κ = 12 (cid:16) (cid:113)(cid:112) ( c/ + ( ν/ − ν/ − (cid:113)(cid:112) ( c/ + ( ν/ + ν/ (cid:17) , (2.4) here c and ν (cid:54) = 0 obey the relation (1.6), which corresponds to the properties k > κ (cid:54) = 0. Proposition 1.
The Hirota and Sasa-Satsuma oscillatory -soliton solutions u ( t, x ) = exp( iφ ) exp( iνt ) ˜ f ( ξ ) , ξ = x − ct (2.5) expressed using a travelling wave coordinate are given by ˜ f ( ξ ) = exp( iκξ ) U ( ξ ) (2.6) in terms of the respective envelope functions U H ( ξ ) = k kξ ) , (2.7) U SS ( ξ ) = k (2 | κ | ) / ( k + κ ) / cosh( kξ + iλ/ | κ | cosh(2 kξ ) + ( k + κ ) / , λ = arg( κ ( κ + ik )) , (2.8) with arg( U H ) = 0 and arg( U SS ) = arctan(tanh( kξ ) tan( λ/ . When c = 0 , these -solitonsolutions are harmonically modulated standing waves. We next summarize the expressions for the travelling-wave functions ˜ f ( x − c t, x − c t )and ˜ f ( x − c t, x − c t ) in oscillatory 2-soliton solutions (1.7) for the Hirota equation (2.1)and the Sasa-Satsuma equation (2.2).Let k = √ β − + β ) / , κ = ( β − − β ) / , (2.9) k = √ β − + β ) / , κ = ( β − − β ) / , (2.10)with β ± = (cid:113)(cid:112) ( c / + ( ν / ± ν / , β ± = (cid:113)(cid:112) ( c / + ( ν / ± ν / , (2.11)where c , c , ν (cid:54) = 0, ν (cid:54) = 0 obey the relations (1.8), corresponding to the properties k > , k > , (2.12) κ (cid:54) = 0 , κ (cid:54) = 0 . (2.13) Proposition 2.
As expressed using travelling wave coordinates ξ = x − c t and ξ = x − c t when c (cid:54) = c , the Hirota and Sasa-Satsuma oscillatory -soliton solutions u ( t, x ) = exp( iφ ) exp( iν t ) ˜ f ( ξ , ξ ) + exp( iφ ) exp( iν t ) ˜ f ( ξ , ξ ) (2.14) are given by ˜ f ( ξ , ξ ) = exp( iκ ξ ) V ( ξ , ξ ) /W ( ξ , ξ ) , ˜ f ( ξ , ξ ) = exp( iκ ξ ) V ( ξ , ξ ) /W ( ξ , ξ )(2.15) in terms of the respective envelope functions V ( ξ , ξ ) = k cosh( k ξ + iγ ) , (2.16) V ( ξ , ξ ) = k cosh( k ξ + iγ ) , (2.17) 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 ) , (2.18) in the Hirota case, and V ( ξ , ξ ) = 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 s √ 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 s (cid:114) ΩΥ (cid:18) k ( k + κ ) / | κ | / s cosh( k ξ + i ( (cid:36) − γ )) − k ( k + κ ) / | κ | / s 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) , (2.19) V ( ξ , ξ ) = 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 s √ 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 s (cid:114) ΩΥ (cid:18) k ( k + κ ) / | κ | / s cosh( k ξ + i ( (cid:36) − γ )) − k ( k + κ ) / | κ | / s 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) , (2.20) 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 s s 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) (2.21) in the Sasa-Satsuma case. In both cases, µ = ( ν − ν ) / ( c − c ) , (2.22)Ω = (cid:113)(cid:0) ( k + k ) + ( κ + κ ) (cid:1)(cid:0) ( k − k ) + ( κ + κ ) (cid:1) , (2.23)Υ = (cid:0) ( k − k ) + ( κ − κ ) (cid:1)(cid:0) ( k + k ) + ( κ − κ ) (cid:1) , (2.24)∆ = (cid:115) ( k − k ) + ( κ + κ ) ( k + k ) + ( κ + κ ) , (2.25)Γ = ( k − k ) + ( κ − κ ) ( k + k ) + ( κ − κ ) , (2.26) α = ( λ + δ ) / , α = ( λ + δ ) / , (2.27) υ = ( λ − δ ) / , υ = ( λ − δ ) / , (2.28) (cid:36) = ( λ − δ ) / , (cid:36) = ( λ − δ ) / , (2.29) γ = arg( k − k − ( κ − κ ) + i k ( κ − κ )) ,γ = arg( k − k − ( κ − κ ) − i k ( κ − κ )) , (2.30) δ = arg( k − k + ( κ + κ ) + i k ( κ + κ )) ,δ = arg( k − k + ( κ + κ ) + i k ( κ + κ )) , (2.31) λ = arg( κ ( κ + ik )) , λ = arg( κ ( κ + ik )) . (2.32) s = sgn( κ ) , s = sgn( κ ) , (2.33) s = (cid:40) , | k | (cid:54) = | k | sgn( κ + κ ) , | k | = | k | . (2.34) or convenience, we also write out the half-angle expressions needed in equations (2.27)–(2.29): λ / (cid:16)(cid:114)(cid:113) k /κ + 1 + is (cid:114)(cid:113) k /κ − (cid:17) , (2.35) λ / (cid:16)(cid:114)(cid:113) k /κ + 1 + is (cid:114)(cid:113) k /κ − (cid:17) , (2.36)and δ / (cid:16) ( (cid:15) (cid:15) − + (1 − (cid:15) (cid:15) − )sgn( κ + κ )) (cid:113) k − k + ( κ + κ ) + Ω+ is (1 + (cid:15) (cid:15) − (sgn( κ + κ ) − (cid:113) k − k + ( κ + κ ) − Ω (cid:17) , (2.37) δ / (cid:16) ( (cid:15) (cid:15) + + (1 − (cid:15) (cid:15) + )sgn( κ + κ )) (cid:113) k − k + ( κ + κ ) + Ω+ is (1 + (cid:15) (cid:15) + (sgn( κ + κ ) − (cid:113) k − k + ( κ + κ ) − Ω (cid:17) , (2.38)where (cid:15) ± = (1 ± (cid:15) ) / , (cid:15) = sgn( | k | − | k | ) = , | k | > | k |− , | k | < | k | , | k | = | k | . (2.39)3. Asymptotic analysis
An oscillatory wave (1.5) having phase angle φ , temporal frequency ν , and speed c reducesto an ordinary travelling wave (1.4) when (and only when) ν = 0. In this case the kinematicrelation (1.6) implies c >
0, showing that all ordinary travelling wave solutions of the Hirotaequation (2.1) and the Sasa-Satsuma equation (2.2) are right-moving.In contrast, when ν (cid:54) = 0, the kinematic relation (1.6) states c > − (3 / √ √ ν ) (cid:54) = 0 (3.1)which allows c < c = 0, in addition to allowing c >
0. Correspondingly, oscillatory1-soliton solutions from Proposition 1 for the Hirota equation (2.1) and the Sasa-Satsumaequation (2.2) consist of right-moving oscillatory waves when c >
0, left-moving oscillatorywaves when c <
0, and standing waves when c = 0.We will now show that the oscillatory 2-soliton solutions with c (cid:54) = c from Proposition 2 forthe Hirota equation (2.1) and the Sasa-Satsuma equation (2.2) reduce in both the asymptoticpast ( t → −∞ ) and future ( t → + ∞ ) to a linear superposition of oscillatory 1-solitonsolutions whose speeds are precisely c and c . Since the solutions are symmetric undersimultaneously interchanging c ←→ c , ν ←→ ν , φ ←→ φ , we will assume c > c (3.2)hereafter without loss of generality.To proceed, we first note ξ = x − c t, ξ = x − c t (3.3)are moving coordinates centered at positions x = c t and x = c t , respectively. Consider ε = ξ − ξ = ( c − c ) t (3.4) ith c − c > ξ is the rightmost coordinate and ξ is the leftmost coordinate.Asymptotic expansions for t → ±∞ then correspond to asymptotic expansions given by ε → ±∞ . In each expansion, we separately hold fixed the coordinates ξ and ξ .3.1. Moving-coordinate expansion of the Hirota oscillatory -soliton. We begin byholding the rightmost coordinate fixed, and expressing the leftmost coordinate in terms ofthe expansion parameter ε from equation (3.4), so thus ξ = const ., ξ = ξ + ε → ±∞ (3.5)as ε → ±∞ . In the Hirota oscillatory 2-soliton solution from Proposition 2, applying theexpansion (3.5) to the functions (2.16), (2.17), (2.18) and neglecting subdominant terms, wefind e iκ ξ V ( ξ , ξ + ε ) ∼ e k | ε | (cid:16) k e ± iγ e iκ ξ e ± k ξ (cid:17) + O (1) (3.6) e iκ ( ξ + ε ) V ( ξ , ξ + ε ) ∼ O (1) (3.7) W H ( ξ , ξ + ε ) ∼ e k | ε | (cid:16) k e ± k ξ (cid:16) √ Γ e ± k ξ + 1 √ Γ e ∓ k ξ (cid:17)(cid:17) + O (1) (3.8)(with k > e k | ε | e ± k ξ , the asymptoticfunctions (3.6) and (3.8) resemble the form of the numerator and denominator in the oscil-latory 1-soliton solution (2.6) and (2.7) for the Hirota equation. Specifically, the expansion(3.8) can be expressed as e − k | ε | e ∓ k ξ W H ( ξ , ξ + ε ) ∼ cosh( k ξ ± ) , ε → ±∞ (3.9)where ξ ± = ξ ∓ x , x = − ln Γ2 k > e − k | ε | e ∓ k ξ e iκ ξ V ( ξ , ξ + ε ) ∼ k e i ( ± γ ± κ x ) e iκ ξ ± , ε → ±∞ . (3.11)Hence we have e iκ ξ V ( ξ , ξ + ε ) /W H ( ξ , ξ + ε ) = ˜ f ( ξ , ξ + ε ) ∼ e i ( ± γ ± κ x ) e iκ ξ ± U H ( ξ ± ) , ε → ±∞ (3.12)where ˜ f is the function (2.6) and U H is the function (2.7). Finally, the phase factor e i ( ± γ ± κ x ) can be combined with e iφ to get a shifted phase angle φ ± = φ ± η , η = γ + κ x . (3.13)In a similar way, the expansion (3.7) gives e − k | ε | e ∓ k ξ e iκ ( ξ + ε ) V ( ξ , ξ + ε ) ∼ , ε → ±∞ (3.14)whence e iκ ( ξ + ε ) V ( ξ , ξ + ε ) /W H ( ξ , ξ + ε ) = ˜ f ( ξ , ξ + ε ) ∼ , ε → ±∞ . (3.15) ombining equations (3.12) and (3.15) with equation (2.15), we see that the asymptoticexpansion of the Hirota oscillatory 2-soliton solution with respect to its leftmost coordinateis given by u ( t, x ) ∼ e iφ ± e iν t e iκ ξ ± U H ( ξ ± ) = u ± ( t, x ) , ξ = const ., ε = ξ − ξ → ±∞ . (3.16)Next, we hold the leftmost coordinate fixed, and express the rightmost coordinate in termsof the expansion parameter ε from equation (3.4), so now ξ = const ., ξ = ξ − ε → ∓∞ (3.17)as ε → ±∞ . Applying this expansion (3.17) to the functions (2.16), (2.17), (2.18), we obtain e iκ ( ξ − ε ) V ( ξ − ε, ξ ) ∼ O (1) (3.18) e iκ ξ V ( ξ − ε, ξ ) ∼ e k | ε | (cid:16) k e ∓ iγ e iκ ξ e ∓ k ξ (cid:17) + O (1) (3.19) W H ( ξ − ε, ξ ) ∼ e k | ε | (cid:16) k e ∓ k ξ (cid:16) √ Γ e ∓ k ξ + 1 √ Γ e ± k ξ (cid:17)(cid:17) + O (1) (3.20)(with k > e − k | ε | e ± k ξ W H ( ξ − ε, ξ ) ∼ cosh( k ξ ± ) , ε → ±∞ (3.21)and e − k | ε | e ± k ξ e iκ ξ V ( ξ − ε, ξ ) ∼ k e i ( ∓ γ ± κ x ) e iκ ξ ± , ε → ±∞ (3.22)where ξ ± = ξ ∓ x , x = ln Γ2 k < e iκ ξ V ( ξ − ε, ξ ) /W H ( ξ − ε, ξ ) = ˜ f ( ξ − ε, ξ ) ∼ e i ( ∓ γ ± κ x ) e iκ ξ ± U H ( ξ ± ) , ε → ±∞ . (3.24)The phase factor e i ( ∓ γ ± κ x ) can be combined with e iφ to get a shifted phase angle φ ± = φ ± η , η = − γ + κ x . (3.25)Similarly, the expansion (3.18) gives e − k | ε | e ± k ξ e iκ ( ξ − ε ) V ( ξ − ε, ξ ) ∼ , ε → ±∞ (3.26)whence e iκ ( ξ − ε ) V ( ξ − ε, ξ ) /W H ( ξ − ε, ξ ) = ˜ f ( ξ − ε, ξ ) ∼ , ε → ±∞ . (3.27)Combining equations (3.27) and (3.24) with equation (2.15), we see that the asymptoticexpansion of the Hirota oscillatory 2-soliton solution with respect to its rightmost coordinateis given by u ( t, x ) ∼ e iφ ± e iν t e iκ ξ ± U H ( ξ ± ) = u ± ( t, x ) , ξ = const ., ε = ξ − ξ → ±∞ . (3.28) .2. Moving-coordinate expansion of the Sasa-Satsuma oscillatory -soliton. Forthe Sasa-Satsuma oscillatory 2-soliton solution from Proposition 2, we apply the asymptoticexpansions (3.5) and (3.17) to the functions (2.19), (2.20), (2.21).First using the expansion (3.5) and neglecting subdominant terms for ε → ±∞ , we find e iκ ξ V ( ξ , ξ + ε ) ∼ e k | ε | (cid:18) k ( k + κ ) / | κ / | / | κ | e iκ ξ e ± k ξ × (cid:16) √ ∆Γ e i ( α + γ ) e k ξ + 1 √ ∆Γ e − k ξ e i ( γ − υ ) (cid:17)(cid:19) + O ( e k | ε | )(3.29) e iκ ( ξ + ε ) V ( ξ , ξ + ε ) ∼ O ( e k | ε | ) (3.30) W SS ( ξ , ξ + ε ) ∼ e k | ε | (cid:18) | κ | e ± k ξ (cid:16) | κ | (cid:16) ∆Γ e ± k ξ + 1∆Γ e ∓ k ξ (cid:17) + ( k + κ ) / (cid:17)(cid:19) + O ( e k | ε | ) (3.31)(with k > e − k | ε | e ∓ k ξ W SS ( ξ , ξ + ε ) ∼ | κ | (cid:0) cosh(2 k ξ ± ) + ( k + κ ) / (cid:1) , ε → ±∞ (3.32)where ξ ± = ξ ∓ x , x = − ln(∆Γ)2 k > e − k | ε | e ∓ k ξ e iκ ξ V ( ξ , ξ + ε ) ∼ k ( k + κ ) / | κ | / | κ | e ± i ( γ + δ + κ x ) e iκ ξ ± cosh( k ξ ± + iλ ) , ε → ±∞ . (3.34)These functions (3.32) and (3.34) resemble the denominator and numerator in the oscillatory1-soliton solution (2.6) and (2.8) for the Sasa-Satsuma equation. Specifically, we have e iκ ξ V ( ξ , ξ + ε ) /W SS ( ξ , ξ + ε ) = ˜ f ( ξ , ξ + ε ) ∼ e ± i ( γ + δ + κ x ) e iκ ξ ± U SS ( ξ ± ) ,ε → ±∞ (3.35)where ˜ f is the function (2.6) and U SS is the function (2.8). Finally, the phase factor e ± i ( γ + δ + κ x ) can be combined with e iφ to get a shifted phase angle φ ± = φ ± η , η = γ + δ + κ x . (3.36)In a similar way, the expansion (3.30) gives e − k | ε | e ∓ k ξ e iκ ( ξ + ε ) V ( ξ , ξ + ε ) ∼ , ε → ±∞ (3.37)whence e iκ ( ξ + ε ) V ( ξ , ξ + ε ) /W SS ( ξ , ξ + ε ) = ˜ f ( ξ , ξ + ε ) ∼ , ε → ±∞ . (3.38) ombining equations (3.35) and (3.38) with equation (2.15), we see that the asymptoticexpansion of the Sasa-Satsuma oscillatory 2-soliton solution with respect to its leftmostcoordinate is given by u ( t, x ) ∼ e iφ ± e iν t e iκ ξ ± U SS ( ξ ± ) = u ± ( t, x ) , ξ = const ., ε = ξ − ξ → ±∞ . (3.39)Next using the expansion (3.17) and neglecting subdominant terms for ε → ±∞ , we find e iκ ( ξ − ε ) V ( ξ − ε, ξ ) ∼ O ( e k | ε | ) (3.40) e iκ ξ V ( ξ − ε, ξ ) ∼ e k | ε | (cid:18) k ( k + κ ) / | κ / | / | κ | e iκ ξ e ± k ξ × (cid:16) √ ∆Γ e − i ( α + γ ) e k ξ + 1 √ ∆Γ e − k ξ e i ( υ − γ ) (cid:17)(cid:19) + O ( e k | ε | )(3.41) W SS ( ξ − ε, ξ ) ∼ e k | ε | (cid:18) | κ | e ± k ξ (cid:16) | κ | (cid:16) ∆Γ e ∓ k ξ + 1∆Γ e ± k ξ (cid:17) + ( k + κ ) / (cid:17)(cid:19) + O ( e k | ε | ) (3.42)(with k > e − k | ε | e ∓ k ξ W SS ( ξ − ε, ξ ) ∼ | κ | (cid:0) cosh(2 k ξ ± ) + ( k + κ ) / (cid:1) , ε → ±∞ (3.43)where ξ ± = ξ ∓ x , x = ln(∆Γ)2 k < e − k | ε | e ∓ k ξ e iκ ξ V ( ξ − ε, ξ ) ∼ k ( k + κ ) / | κ | / | κ | e iκ ξ e ∓ i ( γ + δ − κ x ) e iκ ξ ± cosh( k ξ ± + iλ ) , ε → ±∞ . (3.45)Hence we have e iκ ξ V ( ξ − ε, ξ ) /W SS ( ξ − ε, ξ ) = ˜ f ( ξ − ε, ξ ) ∼ e ∓ i ( γ + δ − κ x ) e iκ ξ ± U SS ( ξ ± ) ,ε → ±∞ . (3.46)The phase factor e ∓ i ( γ + δ − κ x ) can be combined with e iφ to get a shifted phase angle φ ± = φ ± η , η = − γ − δ + κ x . (3.47)Similarly, the expansion (3.40) gives e − k | ε | e ∓ k ξ e iκ ( ξ − ε ) V ( ξ − ε, ξ ) ∼ , ε → ±∞ (3.48)whence e iκ ( ξ − ε ) V ( ξ − ε, ξ ) /W SS ( ξ − ε, ξ ) = ˜ f ( ξ − ε, ξ ) ∼ , ε → ±∞ . (3.49) ombining equations (3.46) and (3.49) with equation (2.15), we see that the asymptoticexpansion of the Sasa-Satsuma oscillatory 2-soliton solution with respect to its rightmostcoordinate is given by u ( t, x ) ∼ e iφ ± e iν t e iκ ξ ± U SS ( ξ ± ) = u ± ( t, x ) , ξ = const ., ε = ξ − ξ → ±∞ . (3.50)3.3. Asymptotic expansion for large time.
The precise correspondence between themoving coordinate expansion given by ε → ±∞ and an asymptotic expansion t → ±∞ will now be explained. In particular, through equations (3.3) and (3.4), we will determinehow large | t | must so that the expansions (3.16) and (3.28) derived for the Hirota oscillatory2-soliton and the expansions (3.39) and (3.50) derived for the Sasa-Satsuma oscillatory 2-soliton are approximately valid over some interval in x at a finite time −∞ < t < ∞ .From equations (3.6)–(3.8) and equations (3.29)–(3.31), we see that the expansions (3.16)and (3.39) remain approximately valid if ± k ξ (cid:29) k | ξ ± | = O (1). These twoconditions can be expressed explicitly as conditions on t, x after we use equations (3.3), (3.4),(3.10) and (3.33) to get ξ = ξ ± ± x + ( c − c ) t . Then the condition ± k ξ (cid:29) ± ( ξ ± + ( c − c ) t ) (cid:29) k − x (3.51)while the other condition k | ξ ± | = O (1) implies ± ξ ± (cid:38) − k . (3.52)We now combine these two inequalities (3.51) and (3.52), yielding ± t (cid:29) k + 1 k − x c − c (3.53)which determines the minimum size of t . Finally, from inequality (3.52), we have c t ± x − k (cid:46) x (cid:46) c t ± x + 1 k (3.54)which determines the interval in which x lies. These are the conditions on t, x under whichthe expansions (3.16) and (3.39) approximately hold, giving u ( t, x ) (cid:39) e iφ ± e iν t ˜ f ( x − c t ∓ x ) = u ± ( t, x ) . (3.55)There are similar conditions for the expansions (3.16) and (3.39) to hold approximatelyfrom equations (3.18)–(3.20) and equations (3.40)–(3.42), so thus u ( t, x ) (cid:39) e iφ ± e iν t ˜ f ( x − c t ∓ x ) = u ± ( t, x ) . (3.56)We see that these equations remain approximately valid if ∓ k ξ (cid:29) k | ξ ± | = O (1). By using equations (3.3) and (3.4), we get ξ = ξ ± ± x − ( c − c ) t . The condition ∓ k ξ (cid:29) ∓ ( ξ ± − ( c − c ) t ) (cid:29) k + x (3.57)while the other condition k | ξ ± | = O (1) implies ∓ ξ ± (cid:38) − k . (3.58) hen, combining these two inequalities (3.57) and (3.58), we obtain ± t (cid:29) k + 1 k + x c − c (3.59)which determines the minimum size of t . Then from inequality (3.58), we have c t ± x − k (cid:46) x (cid:46) c t ± x + 1 k (3.60)which determines the interval in which x lies.An important observation now is that the approximate expansions (3.55) and (3.56) willhold simultaneously if t satisfies both conditions (3.53) and (3.59). Since equations (3.10),(3.23), (3.33), (3.44) show that x > x < t is given by | t | (cid:29) k + k k k ( c − c ) . (3.61)Another useful observation is that the previous analysis holds independently of the signsof c and c , including cases when one of c or c is zero. Hence, we have established thefollowing results. Lemma 1.
For t satisfying the condition (3.61) , the Hirota oscillatory -soliton solution (2.14) , (2.16) – (2.18) with parameters φ , φ , ν , ν , c > c has the form of an asymptoticsuperposition u (cid:39) u ± + u ± in which u ± and u ± are distinct oscillatory waves having respectivespeeds c and c , temporal frequencies ν and ν , phase angles φ ± and φ ± given by expres-sions (3.13) and (3.25) , and having positions that are determined by the respective movingcoordinates (3.10) and (3.23) . Lemma 2.
For t satisfying the condition (3.61) , the Sasa-Satsuma oscillatory -solitonsolution (2.14) , (2.19) – (2.21) with parameters φ , φ , ν , ν , c > c has the form of anasymptotic superposition u (cid:39) u ± + u ± in which u ± and u ± are distinct oscillatory waveshaving respective speeds c and c , temporal frequencies ν and ν , phase angles φ ± and φ ± given by expressions (3.36) and (3.47) , and having positions that are determined by therespective moving coordinates (3.33) and (3.44) . When these oscillatory 2-soliton solutions for the Hirota equation and the Sasa-Satsumaequation have either c = 0 or c = 0, then the respective asymptotic wave u ± or u ± as t → ±∞ is a standing wave. 4. Constants of Motion
For the Hirota equation (2.1) and the Sasa-Satsuma equation (2.2), we recall that theconserved integrals defining momentum, energy, and Galilean energy are given by [9] (up toarbitrary normalization factors) P = (cid:90) + ∞−∞ | u | dx (4.1) E = (cid:90) + ∞−∞ | u x | − | u | ) dx (4.2) = (cid:90) + ∞−∞ t ( | u x | − | u | ) − x | u | dx (4.3)which yield constants of motion for all smooth solutions u ( t, x ) with sufficiently rapid decay u → x → ±∞ . These integrals are related to the center of momentum defined by X ( t ) = 1 P (cid:90) + ∞−∞ x | u | dx = X (0) + EP t (4.4)where C = t E − PX ( t ) = C (0) = −PX (0) . (4.5)This is the same relation that holds for the corresponding constants of motion of the mKdVequation [7].The Hirota equation admits an additional conserved integral given by the angular twist[9] (up to an arbitrary normalization factor) W = (cid:90) + ∞−∞ Re ( iu ¯ u x ) dx = − i (cid:90) + ∞−∞ | u | arg( u ) x dx (4.6)holding for all smooth solutions u ( t, x ) with sufficiently rapid decay u → x → ±∞ .This integral is not conserved for the Sasa-Satsuma equation.It is straightforward to evaluate these constants of motion explicitly for the oscillatory 1-soliton solutions from Proposition 1 for the Hirota equation and the Sasa-Satsuma equation.For notional convenience we will denote β ± = (cid:113)(cid:112) ( c/ + ( ν/ ± ν/ . (4.7) Theorem 1.
The Hirota oscillatory -soliton (2.5) , (2.7) has angular twist, momentum,energy, and Galilean energy given by W = κk = √ β − − α ) / P = k = √ β − + α + ) / E = k ( k − κ ) = √ β − + α + ) c/ C = 0 (4.11) The Sasa-Satsuma oscillatory -soliton (2.5) , (2.8) has momentum, energy, and Galileanenergy given by P = k = √ β − + α + ) / E = k ( k − κ ) = √ β − + α + ) c/ C = 0 (4.14) In both cases, the center of momentum is X ( t ) = ct with c = E / P . We note that the center of momentum of these oscillatory 1-solitons can be shifted ar-bitrarily by means of a space translation x → x − x applied to the moving coordinate ξ = x − ct in equation (2.5), which leads to X ( t ) = x + ct. (4.15) his changes the Galilean energy C = − x P (4.16)while the momentum and energy are unchanged.From the previous expressions, we can evaluate the momentum, energy, and Galileanenergy of the oscillatory 2-soliton solutions from Proposition 2 for the Hirota equation andthe Sasa-Satsuma equation. In particular, we know from Lemmas 1 and 2 that each solution u ∼ u ± + u ± is asymptotically a superposition of two waves u ± and u ± as t → ±∞ . Hencethe conserved integrals (4.1), (4.2), (4.3) are respectively given by a sum of the momenta P , P , the energies E , E , and the Galilean energies C , C associated with each individualwave. This yields the following result, using the notation (2.11). Theorem 2.
The Hirota oscillatory -soliton (2.14) – (2.15) , (2.16) – (2.18) has angular twist,momentum, energy, and Galilean energy given by W = W + W = κ k + κ k = √ β − − β + β − − β ) / P = P + P = k + k = √ β − + β + β − + β ) / E = E + E = k ( k − κ ) + k ( k − κ ) = √ β − + β ) c + ( β − + β ) c ) / C = C + C = ∓ ( k x + k x ) = 0 (4.20) where x and x are given by equations (3.10) and (3.23) . The Sasa-Satsuma oscillatory -soliton (2.14) – (2.15) , (2.19) – (2.21) has momentum, energy, and Galilean energy given by P = P + P = k + k = √ β − + β + β − + β ) / E = E + E = k ( k − κ ) + k ( k − κ ) = √ β − + β ) c + ( β − + β ) c ) / C = C + C = ∓ ( k x + k x ) = 0 (4.23) where x and x are given by equations (3.33) and (3.44) . In both cases, X ( t ) = EP t (4.24) is the center of momentum, which moves at constant speed c = EP = P c + P c P + P = ( β − + β ) c + ( β − + β ) c β − + β + β − + β . (4.25)5. Position shifts and phase shifts
As shown by Lemmas 1 and 2, in the asymptotic past t → −∞ and future t → ∞ , theHirota and Sasa-Satsuma oscillatory 2-soliton solutions given in Proposition 2 reduce to asuperposition u ∼ u ± + u ± of oscillatory 1-solitons u ± and u ± having speeds c , c , temporalfrequencies ν , ν , phase angles φ ± , φ ± , and having centers of momentum χ ± ( t ) = c t ± x , χ ± ( t ) = c t ± x , with c (cid:54) = c . Without loss of generality, we will assume c > c hereafter,since u is symmetric under simultaneously interchanging c ←→ c , ν ←→ ν , φ ←→ φ .We begin by examining some properties of the asymptotic oscillatory 1-solitons u ± = exp( iφ ± ) exp( iν t ) exp( iκ ξ ± ) U ( ξ ± ) u ± = exp( iφ ± ) exp( iν t ) exp( iκ ξ ± ) U ( ξ ± ) (5.1) here ξ ± = x − c t ∓ x , ξ ± = x − c t ∓ x (5.2)are shifted moving coordinates, and where both U and U are given by the envelope function(2.7) in the Hirota case and (2.8) in the Sasa-Satsuma case. First, the functions U and U are symmetric around ξ ± = 0 and ξ ± = 0, coinciding with the positions of the centers ofmomentum x = χ ± ( t ) = c t ± x , x = χ ± ( t ) = c t ± x (5.3)for the two asymptotic oscillatory waves. Second, at these positions, the phase of bothfunctions U and U vanishes,arg( U ) | ξ ± =0 = arg( U ) | ξ ± =0 = 0 . (5.4)Third, away from the positions (5.3), the amplitude of the two asymptotic oscillatory waveshas exponential decay | u ± | = | U | ∼ O (exp( − k | ξ ± | )) , | ξ ± | (cid:29) /k | u ± | = | U | ∼ O (exp( − k | ξ ± | )) , | ξ ± | (cid:29) /k (5.5)while their phase has linear behaviourarg( u ± ) = φ ± + ν t + κ ξ ± + arg( U ) ∼ ψ + φ ± + ν t + κ ξ ± , | ξ ± | (cid:29) /k arg( u ± ) = φ ± + ν t + κ ξ ± + arg( U ) ∼ ψ + φ ± + ν t + κ ξ ± , | ξ ± | (cid:29) /k (5.6)where ψ = ψ = 0 in the Hirota case, and ψ = sgn( ξ ± ) arg( κ ( κ + ik )), ψ =sgn( ξ ± ) arg( κ ( κ + ik )) in the Sasa-Satsuma case. For graphical and analytical purposes,it will be more useful to work with the envelope phase of the two waves (5.1).We recall that the envelope phase of an oscillatory wave u = exp( iφ ) exp( iνt ) exp( iκξ ) U ( ξ )expressed in terms of a moving coordinate ξ = x − ct − χ , with phase angle φ , temporalfrequency ν , speed c , and center of momentum χ ( t ) = ct + χ , is defined by [8] ϕ ( u ) = arg( u ) − κx − ( ν − κc ) t = φ − κχ + arg( U ) . (5.7)Note arg( u ) = φ + νt + κ ( x − ct − χ ) + arg( U ) = κx + ( ν − κc ) t + ϕ ( u ) is the total phase of u , so thus ϕ ( u ) represents the contribution to the phase of u after the linear contributions κx − ( ν − κc ) t from the harmonic modulation are removed, which corresponds to writing u = exp( iϕ ) exp( i ( κx − ( ν − κc ) t )) | U ( ξ ) | . (5.8)Applied to the asymptotic oscillatory waves (5.1), this yields the envelope phases ϕ ( u ± ) =arg( U ) + φ ± ∓ κ x , ϕ ( u ± ) = arg( U ) + φ ± ∓ κ x . The phase property (5.4) shows that ϕ ( u ± ) | ξ ± =0 = φ ± ∓ κ x = ϕ ± , ϕ ( u ± ) | ξ ± =0 = φ ± ∓ κ x = ϕ ± , (5.9)where ϕ ± = φ ± γ , ϕ ± = φ ∓ γ , in the Hirota case, and ϕ ± = φ ± ( γ + δ ), ϕ ± = φ ∓ ( γ + δ ) in the Sasa-Satsuma case. Away from the center of momentum positions(5.3), the envelope phases approach constant values ϕ ( u ± ) ∼ ϕ ± + ψ , | ξ ± | (cid:29) /k ϕ ( u ± ) ∼ ϕ ± + ψ , | ξ ± | (cid:29) /k . (5.10) s t → ±∞ , the asymptotic positions (5.3) of the two oscillatory waves (5.1) lie on straightlines in the ( t, x )-plane, with the lines x = χ +1 ( t ) and x = χ +2 ( t ) being each shifted relativeto the lines x = χ − ( t ) and x = χ − ( t ) by a constant value∆ x = χ +1 ( t ) − χ − ( t ) = 2 x , ∆ x = χ +2 ( t ) − χ − ( t ) = 2 x . (5.11)Likewise, the asymptotic phase angles of the two oscillatory waves are each shifted by aconstant value ∆ φ = φ +1 − φ − , ∆ φ = φ +2 − φ − . (5.12)Hence the envelope phases also undergo shifts∆ ϕ = ϕ ( u +1 ) − ϕ ( u − ) = ∆ φ − κ ∆ x = ϕ +1 − ϕ − ∆ ϕ = ϕ ( u +2 ) − ϕ ( u − ) = ∆ φ − κ ∆ x = ϕ +2 − ϕ − (5.13)which are determined entirely by the asymptotic shifts (5.11) and (5.12).From Lemmas 1 and 2, we have the following expressions for the shifts (5.11) and (5.13). Theorem 3.
For t → ±∞ in the Hirota oscillatory -soliton (2.14) , (2.16) – (2.18) , with c > c , the asymptotic soliton with speed c and temporal frequency ν undergoes a shift inposition and envelope phase given by ∆ x = 1 k ln (cid:18) ( k + k ) + ( κ − κ ) ( k − k ) + ( κ − κ ) (cid:19) > ϕ = − (cid:0) ( k + k )( k − k ) + ( κ − κ ) + i k ( κ − κ ) (cid:1) (5.15) while the asymptotic soliton with speed c and temporal frequency ν undergoes a shift inposition and envelope phase given by ∆ x = − k ln (cid:18) ( k + k ) + ( κ − κ ) ( k − k ) + ( κ − κ ) (cid:19) < ϕ = − (cid:0) ( k + k )( k − k ) − ( κ − κ ) + i k ( κ − κ ) (cid:1) (5.17) where k , k , κ , κ are given in terms of c , c , ν , ν by equations (2.9) – (2.11) . Theorem 4.
For t → ±∞ in the Sasa-Satsuma oscillatory -soliton (2.14) , (2.19) – (2.21) ,with c > c , ν (cid:54) = 0 and ν (cid:54) = 0 , the asymptotic soliton with speed c and temporal frequency ν undergoes a shift in position and envelope phase given by ∆ x = 1 k ln (cid:18) ( k + k ) + ( κ − κ ) ( k − k ) + ( κ − κ ) (cid:115) ( k + k ) + ( κ + κ ) ( k − k ) + ( κ + κ ) (cid:19) > ϕ = − (cid:0) ( k + k )( k − k ) + ( κ − κ ) + i k ( κ − κ ) (cid:1) + arg (cid:0) ( k + k )( k − k ) + ( κ + κ ) + i k ( κ + κ ) (cid:1) (5.19) while the asymptotic soliton with speed c and temporal frequency ν undergoes a shift inposition and envelope phase given by ∆ x = − k ln (cid:18) ( k + k ) + ( κ − κ ) ( k − k ) + ( κ − κ ) (cid:115) ( k + k ) + ( κ + κ ) ( k − k ) + ( κ + κ ) (cid:19) < ϕ = − (cid:0) ( k + k )( k − k ) − ( κ − κ ) + i k ( κ − κ ) (cid:1) + arg (cid:0) ( k + k )( k − k ) − ( κ + κ ) − i k ( κ + κ ) (cid:1) (5.21) here k , k , κ , κ are given in terms of c , c , ν , ν by equations (2.9) – (2.11) . For both the Hirota and Sasa-Satsuma oscillatory 2-solitons, as t → ±∞ the centers ofmomentum and the phase angles of the two asymptotic oscillatory waves (5.1) are given by χ ± ( t ) = c t ± ∆ x , χ ± ( t ) = c t ± ∆ x (5.22)and φ ± = φ ± ∆ φ , φ ± = φ ± ∆ φ . (5.23)5.1. Oscillatory wave collisions.
The Hirota and Sasa-Satsuma oscillatory 2-soliton solu-tions (2.14)–(2.15) describe a collision between two asymptotic oscillatory waves with speeds c > c (or c < c ). The collision is a right-overtake if c > c ≥ c > c ≥ ≥ c > c (or 0 ≥ c > c ), and a head-on if c > ≥ c (or c > ≥ c ).As will be now illustrated, in all cases the net effect of the collision is only to shift theasymptotic position and asymptotic phase angle of each wave, where these shifts are givenin Theorems 3 and 4.The positions shifts are seen graphically in the asymptotic amplitude of the 2-solitonsolution, since for large | t | we have | u | ∼ (cid:40) | u ± | = | U ( ξ ± ) | , x (cid:39) χ ± ( t ) | u ± | = | U ( ξ ± ) | , x (cid:39) χ ± ( t ) (5.24)from Lemmas 1 and 2, where ξ ± and ξ ± are the shifted moving coordinates (5.2) whichdetermine the positions (5.3) of the two asymptotic oscillatory waves, and where U and U are the envelope functions for these waves, given in Proposition 1. Similarly, we havearg( u ) ∼ (cid:40) arg( u ± ) = ϕ ( u ± ) + κ x + ( ν − c κ ) t, x (cid:39) χ ± ( t )arg( u ± ) = ϕ ( u ± ) + κ x + ( ν − c κ ) t, x (cid:39) χ ± ( t ) (5.25)yielding the asymptotic phase of the 2-soliton solution.To see the phase shifts graphically, it is useful to remove the asymptotic linear part ofarg( u ) by defining an envelope phase for the 2-soliton solution similarly to the definition(5.8) for oscillatory waves [8]. Consider, for a 2-soliton solution, the factorization u = exp( iϕ ) (cid:16) exp( i ( ν t + κ ξ )) | V ( ξ , ξ ) | + exp( i ( ν t + κ ξ )) | V ( ξ , ξ ) | (cid:17) A ( ξ , ξ , t ) W ( ξ , ξ ) (5.26)where V , V , W are the envelope functions in Proposition 2, and where A is an amplitudenormalization factor. Equating this form for u to the oscillatory form (2.14)–(2.15), weobtain the envelope phase ϕ ( u ) = arctan (cid:18) | V | Im (exp( iφ )( V + exp( − i Φ) V )) + | V | Im (exp( iφ )( V + exp( i Φ) V )) | V | Re (exp( iφ )( V + exp( − i Φ) V )) + | V | Re (exp( iφ )( V + exp( i Φ) V )) (cid:19) (5.27)withΦ = φ + ν t + κ ξ − φ − ν t − κ ξ = φ − φ + ( κ − κ ) x + ( ν − ν + c κ − c κ ) t. (5.28)The envelope phase (5.27) essentially represents the contribution to the phase of u after thelinear contributions from the harmonic modulation of the two asymptotic oscillatory waves re removed. In particular, let θ = | V | / ( | V | + | V | ) , θ = | V | / ( | V | + | V | ) (5.29)denote normalized envelope functions satisfying the properties θ + θ = 1 (5.30) θ ∼ (cid:40) , x (cid:39) χ ± ( t )0 , x (cid:39) χ ± ( t ) (5.31) θ ∼ (cid:40) , x (cid:39) χ ± ( t )0 , x (cid:39) χ ± ( t ) (5.32)Then we can write the envelope phase as ϕ ( u ) = arctan (cid:18) θ sin( σ ) + θ sin( σ ) + θ θ (sin( σ − Φ) + sin( σ + Φ)) θ cos( σ ) + θ cos( σ ) + θ θ (cos( σ − Φ) + cos( σ + Φ)) (cid:19) (5.33)with σ = arg( V ) + φ , σ = arg( V ) + φ . (5.34)The properties (5.30)–(5.32) combined with the asymptotic phase (5.25) show that ϕ ( u ) ∼ (cid:40) ϕ ( u ± ) , x (cid:39) χ ± ( t ) ϕ ( u ± ) , x (cid:39) χ ± ( t ) (5.35)whereby the envelope phase of u asymptotically matches the envelope phase (5.9) of eachasymptotic oscillatory wave.The amplitude and envelope phase of the oscillatory 2-soliton solutions (2.14)–(2.15) areillustrated in Fig. 1– Fig. 3 for the Hirota case, and in Fig. 4– Fig. 6 for the Sasa-Satsumacase. (a) amplitude in asymptotic future (b) envelope phase in asymptotic future Figure 1.
Hirota oscillatory 2-soliton right-overtake (in solid) and oscillatory1-solitons (in dots and dot-dash) with c = 4, c = 2, ν = 2, ν = 5, ϕ − = 0, ϕ − = π/ a) amplitude in asymptotic future (b) envelope phase in asymptotic future Figure 2.
Hirota oscillatory 2-soliton left-overtake (in solid) and oscillatory1-solitons (in dots and dot-dash) with c = − c = − ν = 2, ν = 5, ϕ − = 0, ϕ − = π/ (a) amplitude in asymptotic future (b) envelope phase in asymptotic future Figure 3.
Hirota oscillatory 2-soliton head-on (in solid) and oscillatory 1-solitons (in dots and dot-dash) with c = 4, c = − ν = 2, ν = 5, ϕ − = 0, ϕ − = π/ (a) amplitude in asymptotic future (b) envelope phase in asymptotic future Figure 4.
Sasa-Satsuma oscillatory 2-soliton right-overtake (in solid) andoscillatory 1-solitons (in dots and dot-dash) with c = 4, c = 2, ν = 2, ν = 5, ϕ − = 0, ϕ − = π/ a) amplitude in asymptotic future (b) envelope phase in asymptotic future Figure 5.
Sasa-Satsuma oscillatory 2-soliton left-overtake (in solid) and os-cillatory 1-solitons (in dots and dot-dash) with c = − c = − ν = 2, ν = 5, ϕ − = 0, ϕ − = π/ (a) amplitude in asymptotic future (b) envelope phase in asymptotic future Figure 6.
Sasa-Satsuma oscillatory 2-soliton head-on (in solid) and oscilla-tory 1-solitons (in dots and dot-dash) with c = 4, c = − ν = 2, ν = 5, ϕ − = 0, ϕ − = π/ Position shifts in oscillatory wave collisions.
In a right-overtake collision with c > c >
0, the asymptotic solitons u ± and u ± are oscillatory waves that each move to theright, where u ± is the faster wave and u ± is the slower wave. The effect of the collision onthe asymptotic positions of these waves is to shift the fast wave forward (i.e. to the right,since ∆ x >
0) and the slow wave backward (i.e. to the left, since ∆ x < > c > c , the asymptotic solitons u ± and u ± are oscillatorywaves that each move to the left, where now u ± is the slower wave and u ± is the fasterwave. The collision affects the asymptotic positions of the two waves by shifting the fastwave forward (i.e. to the left, since ∆ x <
0) and the slow wave backward (i.e. to the right,since ∆ x > c > > c , the asymptotic soliton u ± is a right-moving oscillatory wave while the other asymptotic soliton u ± is a left-movingoscillatory wave. The collision has the effect that the asymptotic positions of both waves areshifted forward relative to their directions of motion, since the right-moving wave undergoesa shift to the right (due to ∆ x >
0) and the left-moving wave undergoes a shift to the left(due to ∆ x < x and ∆ x satisfy the algebraic relation k ∆ x + k ∆ x = 0 (5.36) hich holds as a direct consequence of the oscillatory 2-soliton solution having a center ofmomentum that moves at a constant speed, as shown by equation (4.24).5.3. Standing waves.
An oscillatory 1-soliton (1.5) with c = 0 and ν (cid:54) = 0 is a time-periodic standing wave. The standing wave solutions for the Hirota equation (2.1) and theSasa-Satsuma equation (2.2) are presented in oscillatory form in Proposition 1.Collisions of an oscillatory wave with a standing wave are described by the oscillatory2-soliton (2.14) when c = 0, c (cid:54) = 0, ν (cid:54) = 0, or when c = 0, c (cid:54) = 0, ν (cid:54) = 0. Thesecollision solutions for the Hirota and Sasa-Satsuma equations are special cases of the solutionspresented in Proposition 2. They have not previously appeared in the literature.We remark that Theorems 3 and 4 hold for collisions of an oscillatory wave and a standingwave. Thus, in a right-overtake collision with c > c = 0, the effect of the collision on theasymptotic positions of the waves is to shift the right-moving asymptotic oscillatory wave u ± in a forward direction (i.e. to the right, since ∆ x >
0) while the asymptotic standing wave u ± is displaced in the opposite direction (i.e. to the left, since ∆ x < c > c , the collision affects the asymptotic positions of the twowaves by shifting the left-moving asymptotic oscillatory wave u ± in a forward direction (i.e.to the left, since ∆ x <
0) while the asymptotic standing wave u ± is displayed in the oppositedirection (i.e. to the right, since ∆ x > (a) amplitude in asymptotic future (b) envelope phase in asymptotic future Figure 7.
Hirota 2-soliton collision (in solid) of a left-moving oscillatory wave(in dot-dash) and a standing wave (in dots) with c = 0, c = − ν = 2, ν = 5, ϕ − = 0, ϕ − = π/ a) amplitude in asymptotic future (b) envelope phase in asymptotic future Figure 8.
Hirota 2-soliton collision (in solid) of a right-moving oscillatorysoliton (in dots) and a standing wave (in dot-dash) with c = 4, c = 0, ν = 2, ν = 5, ϕ − = 0, ϕ − = π/ (a) amplitude in asymptotic future (b) envelope phase in asymptotic future Figure 9.
Sasa-Satsuma collision (in solid) of a left-moving oscillatory soliton(in dot-dash) and a standing-wave soliton (in dots) with c = 0, c = − ν = 2, ν = 5, ϕ − = 0, ϕ − = π/ (a) amplitude in asymptotic future (b) envelope phase in asymptotic future Figure 10.
Sasa-Satsuma collision (in solid) of a right-moving oscillatorysoliton (in dots) and a standing-wave soliton (in dot-dash) with c = 4, c = 0, ν = 2, ν = 5, ϕ − = 0, ϕ − = π/ Interaction features and Concluding remarks
In previous work [7], collisions of ordinary solitary waves (1.4) (i.e. with no temporal har-monic modulation) have been studied for the Hirota equation (2.1) and the Sasa-Satsuma quation (2.2). A collision in this case consists of a right-moving faster solitary wave withspeed c and phase φ overtaking a right-moving slower solitary wave with speed c andphase φ . The corresponding ordinary 2-soliton solutions exhibit several distinguishing prop-erties. First, at a particular time t = t the amplitude displays invariance | u ( t , x − χ ( t ) | = | u ( t , χ ( t ) − x | under spatial reflection around the center of momentum x = χ ( t ) of thetwo solitary waves. This time t = t can be understood to represent the moment of greatestnonlinear interaction of the waves during the collision. Second, the amplitude is always non-zero, | u ( t, x ) | (cid:54) = 0, throughout the collision. As a consequence of these two properties, theinteraction of the two waves can be characterized primarily by the convexity of | u ( t , x ) | atthe center of momentum x = χ ( t ) at time t = t . The case of negative convexity describes acollision such that the waves undergo a merge-split interaction in which | u ( t , x ) | has a singlepeak with an exponentially decreasing tail, while the case of positive convexity describes acollision such that the waves exhibit either a bounce-exchange interaction in which | u ( t , x ) | has a double peak with an exponentially decreasing tail, or an absorb-emit interaction inwhich | u ( t , x ) | has a pair of side peaks around a central peak and an exponentially decreas-ing tail, depending on the speed ratio and relative phase angle of the two waves, as explainedin Ref. [7].In contrast, collisions of oscillatory waves (1.5) described by the 2-soliton solu-tions from Proposition 2 have very different features (animations can be seen athttp://lie.math.brocku.ca/ ~ sanco/solitons/oscillatory.php):(1) the amplitude | u | exhibits invariance under spatial reflections only in special cases;(2) the amplitude | u | vanishes at certain positions x and times t (i.e. u has nodes);(3) the phase arg( u ) exhibits rapid spatial change at certain positions x and times t (i.e. u has phase coils with large spatial winding);(4) the phase gradient arg( u ) x changes sign at certain positions x and times t (i.e. u hasspatial reversals of phase winding).A detailed study of the interactions of oscillatory waves for these equations will be pre-sented in a sequel paper. Our work in the present paper has two immediate extensions.First, the Hirota equation (2.1) and the Sasa-Satsuma equation (2.2) are known to begauge-equivalent to third-order NLS equations [13] q ˜ t ± i (cid:112) v/ q ˜ x ˜ x + α | q | q ) + α | q | q ˜ x + β ( | q | ) ˜ x q + q ˜ x ˜ x ˜ x = 0 (6.1)through the Galilean-phase transformation˜ t = t, ˜ x = x + vt, u ( t, x ) = q (˜ t, ˜ x ) exp (cid:0) ± i (cid:112) v/ x − (2 v/ t ) (cid:1) (6.2)where v > α = 24, β = 0 in the Hirota case and α = 12, β = 6 in the Sasa-Satsuma case. Under this transformation, the oscillatory 1-solitons (2.5)–(2.8) shown in Proposition 1 for the Hirota and Sasa-Satsuma equations correspond to NLSsolitons of the same form q (˜ t, ˜ x ) = exp( iφ ) exp( i ˜ ν ˜ t )˜ q ( ˜ ξ ) , ˜ ξ = ˜ x − ˜ c ˜ t (6.3)parameterized by a speed ˜ c , a temporal frequency ˜ ν , and a phase φ , where˜ c = c + v, ˜ ν = ν ± (cid:112) v/ c + 4 v/ , ˜ q ( ˜ ξ ) = exp( ∓ i (cid:112) v/ ξ ) ˜ f ( ˜ ξ ) . (6.4)Consequently, we obtain NLS oscillatory 2-solitons q (˜ t, ˜ x ) = exp( iφ ) exp( i ˜ ν ˜ t )˜ q ( ˜ ξ , ˜ ξ ) + exp( iφ ) exp( i ˜ ν ˜ t )˜ q ( ˜ ξ , ˜ ξ ) (6.5) ith ˜ ξ = ˜ x − ˜ c ˜ t, ˜ ξ = ˜ x − ˜ c ˜ t, (6.6)˜ c = c + v, ˜ c = c + v, (6.7)˜ ν = ν ± (cid:112) v/ c + 4 v/ , ˜ ν = ν ± (cid:112) v/ c + 4 v/ , (6.8)˜ q ( ˜ ξ , ˜ ξ ) = exp( ∓ i (cid:112) v/ ξ ) ˜ f ( ˜ ξ , ˜ ξ ) , ˜ q ( ˜ ξ , ˜ ξ ) = exp( ∓ i (cid:112) v/ ξ ) ˜ f ( ˜ ξ , ˜ ξ ) , (6.9)where the functions ˜ f ( ˜ ξ , ˜ ξ ) and ˜ f ( ˜ ξ , ˜ ξ ) are given in Proposition 2 for the oscillatory2-solitons (2.14)–(2.21) of the Hirota and Sasa-Satsuma equations. In addition, we obtainNLS oscillatory breathers in the special case c = c (cid:54) = 0, discussed in Ref. [7].The main results stated in Theorems 3 and 4 on the properties of collisions described byoscillatory 2-solitons, carry over directly to the third-order NLS equation (6.1). In particular,the net effect of a collision is to shift the asymptotic positions and phases of the individual os-cillatory waves while the speed and the temporal frequency of each wave remains unchanged,such that the center of momentum of the waves is preserved in the collision.Second, the Hirota equation (2.1) has two natural multi-component generalizations givenby U ( N )-invariant integrable mKdV equations [14] (cid:126)u t + 12( | (cid:126)u | (cid:126)u x + ( (cid:126)u x · (cid:126)u ) (cid:126)u ) + (cid:126)u xxx = 0 (6.10)and (cid:126)u t + 24( | (cid:126)u | (cid:126)u x + ( (cid:126)u x · (cid:126)u ) (cid:126)u − ( (cid:126)u x · (cid:126)u ) (cid:126)u ) + (cid:126)u xxx = 0 (6.11)where (cid:126)u ( t, x ) is a N -component complex vector variable. For all N ≥
2, these two vectorequations admit vector oscillatory wave solutions of the form (cid:126)u ( t, x ) = exp( iνt ) ˜ f H ( x − ct ) ˆ ψ (6.12)with wave speed c and temporal frequency ν , satisfying the kinematic relation (3.1), whereˆ ψ is an arbitrary constant complex unit vector and ˜ f H is the complex envelope function (2.7)for the oscillatory 1-soliton solution of the scalar Hirota equation (2.1). In forthcoming work,we plan to generalize the results in the present paper to study the vector oscillatory 2-solitonsolutions and vector oscillatory breather solutions of both equations (6.10) and (6.11). Acknowledgement
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