A new nonlocal nonlinear Schroedinger equation and its soliton solutions
aa r X i v : . [ n li n . S I] J u l A new nonlocal nonlinear Schr¨odinger equation and its soliton solutions
Jianke Yang
Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05401, USA
A new integrable nonlocal nonlinear Schr¨odinger (NLS) equation with clear physical motivationsis proposed. This equation is obtained from a special reduction of the Manakov system, and itdescribes Manakov solutions whose two components are related by a parity symmetry. Since theManakov system governs wave propagation in a wide variety of physical systems, this new nonlocalequation has clear physical meanings. Solitons and multi-solitons in this nonlocal equation are alsoinvestigated in the framework of Riemann-Hilbert formulations. Surprisingly, symmetry relationsof discrete scattering data for this equation are found to be very complicated, where constraintsbetween eigenvectors in the scattering data depend on the number and locations of the underlyingdiscrete eigenvalues in a very complex manner. As a consequence, general N -solitons are difficultto obtain in the Riemann-Hilbert framework. However, one- and two-solitons are derived, and theirdynamics investigated. It is found that two-solitons are generally not a nonlinear superpositionof one-solitons, and they exhibit interesting dynamics such as meandering and sudden positionshifts. As a generalization, other integrable and physically meaningful nonlocal equations are alsoproposed, which include NLS equations of reverse-time and reverse-space-time types as well asnonlocal Manakov equations of reverse-space, reverse-time and reverse-space-time types. PACS numbers: 05.45.Yv, 02.30.Ik
I. INTRODUCTION
Integrable systems have been studied for over fiftyyears [1–5]. The most familiar integrable systems are lo-cal equations, i.e., the solution’s evolution depends onlyon the local solution value and its local space and timederivatives. The Korteweg-de Vries equation and thenonlinear Schr¨odinger (NLS) equation are such examples.In the past few years, nonlocal integrable equationsstarted to attract a lot of attention. The first suchequation, as proposed by Ablowitz and Musslimani [6]as a new reduction of the Ablowitz-Kaup-Newell-Segur(AKNS) hierarchy [7], is the NLS equation of reverse-space type, iq t ( x, t ) + q xx ( x, t ) + 2 σq ( x, t ) q ∗ ( − x, t ) = 0 , (1)where σ = ± x and − x are directly coupled, reminiscent of quantumentanglement between pairs of particles.Following the introduction of this equation, its prop-erties have been extensively investigated [6, 8–19]. Inaddition, other nonlocal integrable equations have beenreported [20–38]. A transformation between many non-local and local equations has been discovered as well [34].From a mathematical point of view, studies of thesenonlocal equations is interesting because these equationsoften feature new types of solution behaviors, such asfinite-time solution blowup [6, 17], the simultaneous ex-istence of solitons and kinks [31], the simultaneous exis-tence of bright and dark solitons [6, 39], and distinctivemulti-soliton patterns [18]. However, the physical mo-tivations of these existing nonlocal equations are ratherweak. Indeed, none of these equations was derived for aconcrete physical system [even though the nonlocal equa- tion (1) above was linked to an unconventional system ofmagnetics [40], it is not clear whether such an unconven-tional magnetics system is physically realizable]. Thislack of physical motivation dampens the interest in thesenonlocal equations from the broader scientific commu-nity.In this article, we propose a new integrable nonlocalNLS equation which has clear physical meanings. Thisequation is iu t ( x, t )+ u xx ( x, t )+2 σ (cid:2) | u ( x, t ) | + | u ( − x, t ) | (cid:3) u ( x, t ) = 0 , (2)where σ = ±
1. Here, the nonlocality is also of reverse-space type, where solutions at locations x and − x aredirectly coupled, similar to Eq. (1). The difference fromEq. (1) is that the nonlinear terms are different. Here, thenonlinearity-induced potential 2 σ [ | u ( x, t ) | + | u ( − x, t ) | ]is real and symmetric in x , which contrasts the previousequation (1), where the nonlinearity-induced potential2 σq ( x, t ) q ∗ ( − x, t ) is generally complex and parity-time-symmetric [41].The new equation (2) will be derived from a specialreduction of the Manakov system [42]. It is well knownthat the Manakov system governs nonlinear wave prop-agation in a great variety of physical situations, such asthe interaction of two incoherent light beams [43], thetransmission of light in a randomly birefringent opticalfiber [44, 45], and the evolution of two-component Bose-Einstein condensates [46]. Thus, this new nonlocal equa-tion governs nonlinear wave propagation in such physicalsystems under a certain constraint of the initial condi-tions, where the two components of the Manakov systemare related by a parity symmetry. This physical interpre-tation can help us understand the solution behaviors inthis nonlocal equation.For this new integrable nonlocal equation, we will fur-ther study its bright solitons and multi-solitons in theframework of Riemann-Hilbert formulation (which is amodern treatment of the inverse scattering transform)[2, 3, 5]. In this Riemann-Hilbert framework, the key toderiving general soliton solutions is to determine sym-metry relations of the discrete scattering data. For theprevious nonlocal NLS equation (1) and two others ofreverse-time and reverse-space-time types [6, 28], it wasfound in [18] that those symmetry relations were verysimple, and thus the general N -solitons in those equa-tions were very easy to write down. However, for thisnew nonlocal equation (2), we will show that its sym-metry relations of discrete scattering data are very com-plicated, because the constraints between eigenvectors inthe scattering data depend on the number and locationsof the underlying discrete eigenvalues in a very intricateway. Even though we do succeed in deriving these sym-metry relations for one- and two-solitons, derivation ofsuch relations for the general N -solitons is apparentlyvery difficult, at least in the Riemann-Hilbert and inversescattering framework. We are not aware of other inte-grable equations whose symmetry relations of the scat-tering data are so complicated, which makes this newnonlocal equation mathematically interesting and chal-lenging. From the derived one- and two-soliton solutions,we find that two-solitons are generally not a nonlinearsuperposition of one-solitons, and they exhibit interest-ing dynamical patterns such as meandering and suddenposition shifts. As a generalization of these results, wealso propose other new integrable and physically mean-ingful nonlocal equations, such as the NLS equations ofreverse-time and reverse-space-time types as well as non-local Manakov equations of reverse-space, reverse-timeand reverse-space-time types. II. A NEW INTEGRABLE NONLOCAL NLSEQUATION
The Manakov system iu t + u xx + 2 σ ( | u | + | v | ) u = 0 , (3) iv t + v xx + 2 σ ( | u | + | v | ) v = 0 , (4)where σ = ±
1, is a ubiquitous nonlinear wave systemwhich governs a wide variety of physical phenomena rang-ing from the interaction of two incoherent light beams[43], the transmission of light in a randomly birefringentoptical fiber [44, 45], and the evolution of two-componentBose-Einstein condensates [46]. This system was shownby Manakov to be integrable [42] (see also [5, 47]).Now, we impose the solution constraint v ( x, t ) = u ( − x, t ) . (5)Under this constraint, it is easy to see that the two equa-tions in the Manakov system are consistent, and this sys-tem reduces to a single but nonlocal equation for u ( x, t )as iu t ( x, t )+ u xx ( x, t )+2 σ (cid:2) | u ( x, t ) | + | u ( − x, t ) | (cid:3) u ( x, t ) = 0 , (6) which is the new nonlocal NLS equation (2) in the pre-vious section.The above derivation of this new nonlocal equation alsoreveals the physical interpretation of its solutions. Specif-ically, this equation describes solutions of the Manakovsystem under special initial conditions where v ( x,
0) = u ( − x, u ( x, t ) solution is governedby the nonlocal equation (6), while the v ( x, t ) solution isgiven in terms of u ( x, t ) as v ( x, t ) = u ( − x, t ). We em-phasize that even though the Manakov system has beenextensively studied before [5, 42, 47], its solutions withspecial initial conditions v ( x,
0) = u ( − x,
0) have not re-ceived much attention. We just showed that these specialsolutions are governed by a single nonlocal equation (6),which opens the door for studies of these solutions in theframework of Eq. (6).This new nonlocal equation is also integrable. To getits Lax pair, we recall that the Lax pair of the Manakovsystem (3)-(4) are Y x = ( − iζJ + Q ) Y, (7) Y t = (cid:2) − iζ J + 2 ζQ + iJ ( Q x − Q ) (cid:3) Y, (8)where J = − , Q = u v − σu ∗ − σv ∗ . The Lax pair for the nonlocal equation (6) are simplythe above ones with v ( x, t ) replaced by u ( − x, t ) in viewof the reduction (5). III. SOLITONS AND MULTI-SOLITONS IN THENEW NONLOCAL EQUATION
Since the new nonlocal equation (6) is integrable, it isnatural to seek its general soliton and multi-soliton solu-tions. Recall that this nonlocal equation is a reductionof the Manakov system. Thus, its solitons are a partof Manakov solitons. But what Manakov solitons sat-isfy this nonlocal equation? This is actually a nontrivialquestion. The present situation is similar to the previousnonlocal NLS equation (1). Even though that equationwas a reduction of the well-known coupled q - r systemin the AKNS hierarchy [5–7], its solutions were still notobvious, which prompted a lot of studies on that equa-tion in the past few years [6, 8–19]. In this article, weonly consider bright-soliton solutions, which exist underfocusing nonlinearity; thus we set σ = 1 below.In a previous article [18], we derived general N -solitonsin the previous nonlocal NLS equation (1) and two oth-ers of reverse-time and reverse-space-time types, whichwere reduced from the q - r system in the AKNS hierar-chy [6, 7, 28]. That derivation was set in the Riemann-Hilbert framework. Starting from the general N -solitonsolutions of the q - r system and deriving symmetry re-lations of the discrete scattering data for those nonlo-cal equations, general N -solitons were then obtained. Inthat approach, derivation of symmetry relations of thescattering data was the key. It turns out that those sym-metry relations were simple (as in all common integrablesystems we are aware of). Thus, general N -solitons inthose nonlocal equations were easy to write down.In this article, we follow a similar approach. Sincethe new nonlocal equation (6) is a reduction from theManakov system, we will start from the general solitonsolutions of the Manakov system in the Riemann-Hilbertformulation. As before, the key to obtaining solitons inthis new nonlocal equation is to derive symmetry rela-tions of its discrete scattering data, which we will dobelow. It turns out that these symmetry relations aresurprisingly complicated for this new nonlocal equation,which makes derivations of its general N -solitons moredifficult.General N -solitons of the Manakov system (3)-(4) arewell known [5, 42, 47]. From the Riemann-Hilbert ap-proach, such solitons can be expressed as [5] (cid:18) u ( x, t ) v ( x, t ) (cid:19) = 2 i N X j,k =1 (cid:18) α j β j (cid:19) e θ j − θ ∗ k (cid:0) M − (cid:1) jk , (9)where M is a N × N matrix whose elements are given by M jk = 1 ζ ∗ j − ζ k h e − ( θ ∗ j + θ k ) + ( α ∗ j α k + β ∗ j β k ) e θ ∗ j + θ k i ,θ k = − iζ k x − iζ k t, (10) ζ k are complex numbers in the upper half plane C + , and α k , β k are arbitrary complex constants. In the languageof inverse scattering, the discrete scattering data of thesesolitons is { ζ k , w k , ≤ k ≤ N } , where ζ k are zeros of theunderlying Riemann-Hilbert problem (which are assumedto be simple), and w k = ( α k , β k , T is the associated eigenvector at the Riemann-Hilbert zero ζ k . Here, the superscript ‘ T ’ represents transpose of avector, and the eigenvector w k has been scaled so thatits last element is unity.The nonlocal equation (6) is obtained from the Man-akov equations under the solution reduction (5). Thissolution reduction becomes a potential constraint in thescattering problem (7), which induce symmetry condi-tions on the scattering data. Imposing these symmetryconditions of the scattering data in the above Manakovsolitons, the resulting solutions would be solitons of thenonlocal equation (6).Note that the Manakov N -solitons (9) above alreadyincorporated symmetry conditions of the scattering databetween eigenvalues in the upper and lower complexplanes C + and C − , which appear as complex-conjugatepairs [2, 5]. Such symmetry conditions are valid for allManakov solitons. For the present nonlocal equation (6), we only need to determine symmetry conditions of scat-tering data for eigenvalues in the upper complex plane C + , which are induced by the new potential reduction(5). These symmetry conditions are presented in the fol-lowing theorem. Theorem 1.
For the nonlocal NLS equation (6), if ζ ∈ C + is a discrete eigenvalue, so is ˆ ζ ≡ − ζ ∗ ∈ C + .Thus, eigenvalues in the upper complex plane are eitherpurely imaginary, or appear as ( ζ, − ζ ∗ ) pairs. Symme-try relations on their eigenvectors depend on the num-ber and locations of these eigenvalues. For the one- andtwo-solitons (with a single and double eigenvalues in C + respectively), these symmetry relations are given below.1. For a single purely imaginary eigenvalue ζ = iη ,with η > , its eigenvector is of the form w = h − / e iγ , − / e iγ , i T , (11) where γ is an arbitrary real constant.2. For two purely-imaginary eigenvalues ζ = iη and ζ = iη , with η , η > , their eigenvectors w =( α , β , T and w = ( α , β , T are related as | α | + | β | = | α | + | β | , (12) g (cid:2) ( | α | + | β | ) − (cid:3) = (1 − g ) (cid:0) − | α ∗ α + β ∗ β | (cid:1) , (13) β = (1 + g )( α α ∗ + β β ∗ ) α − g ( | α | + | β | ) α , (14) β = g ( | α | + | β | ) α +(1 − g )( α ∗ α + β ∗ β ) α , (15) where g ≡ ( η + η ) / ( η − η ) . (16) These equations admit solutions for w and w ifand only if | α ∗ α + β ∗ β | ≤ , and the admitted so-lutions have four free real parameters (not countingthe eigenvalue parameters η and η ).3. For two non-purely-imaginary eigenvalues ( ζ , ζ ) ∈ C + , where ζ = − ζ ∗ , their eigen-vectors w = ( α , β , T and w = ( α , β , T are related as (cid:18) α β (cid:19) = S (cid:18) α β (cid:19) , (17) where S = (cid:18) ζ ∗ − ζ − ζ ∗ (cid:19) × − α ∗ β ζ ∗ | α | + | β | ζ ∗ − ζ − | β | ζ ∗ | α | + | β | ζ ∗ − ζ − | α | ζ ∗ − α β ∗ ζ ∗ − , and α , β are free complex constants. Proof.
The nonlocal NLS equation (6) is reducedfrom the Manakov system under the solution reduction v ( x, t ) = u ( − x, t ). In this case, the initial potential ma-trix Q ( x,
0) = u ( x, u ( − x, − u ∗ ( x, − u ∗ ( − x,
0) 0 admits the following two symmetries Q † ( x,
0) = − Q ( x, , Q ( − x,
0) = − P − Q ( x, P, where P = − , and the dagger † represents Hermitian (i.e., conjugatetranspose). The first potential symmetry Q † = − Q isvalid for the general Manakov system, and we only needto consider the second potential symmetry and its con-sequences.Switching x → − x in the scattering equation (7) andutilizing the second potential symmetry, we get[ P Y ( − x )] x = [ ζJ + Q ( x )] [ P Y ( − x )] . (18)This means that, if ζ is a discrete eigenvalue of the scat-tering problem (7), so is − ζ . But it is known for thegeneral Manakov system that eigenvalues to the scatter-ing problem (7) come in conjugate pairs ( ζ, ζ ∗ ). Thus, if − ζ is an eigenvalue, so is − ζ ∗ . This proves the eigenvaluesymmetry in Theorem 1.It is important to notice that, although we can show − ζ ∗ would be an eigenvalue so long as ζ is, there is nosimple relation between their eigenfunctions, and thusone cannot obtain a simple symmetry relation betweentheir eigenvectors in the scattering data. Eigenfunctionsfor ζ and − ζ are directly related in view of Eq. (18).But − ζ is in the opposite half plane of ζ , and the adjointeigenfunction at − ζ , which we need [2, 5], is not available.To prove symmetry relations of eigenvectors for one-and two-solitons in Theorem 1, we utilize the connectionbetween these eigenvectors and Riemann-Hilbert-based N -soliton solutions (9) of the Manakov system. By im-posing the condition v ( x, t ) = u ( − x, t ) on the Manakovsolitons, we will be able to derive symmetry conditionsof eigenvectors for the nonlocal NLS equation (6).First, we consider one-solitons, where there is a singlepurely imaginary eigenvalue ζ = iη ∈ C + , with η > (cid:18) u ( x, t ) v ( x, t ) (cid:19) = (cid:18) α β (cid:19) ηe iη t e − ηx + ( | α | + | β | ) e ηx . By requiring v ( x, t ) = u ( − x, t ), we get the conditions β = α ( | α | + | β | ) , α = β ( | α | + | β | ) . Hence, | α | + | β | = 1 , α = β , and | α | = 1 /
2. Writing α = 2 − / e iγ , where γ is a realconstant, the resulting eigenvector w is then as givenin Eq. (11).Next, we consider two-solitons, where there are twocomplex eigenvalues ζ , ζ ∈ C + . In this case, the generaltwo-Manakov-solitons from Eq. (9) can be rewritten as u ( x, t ) = 2 i det( M ) h A e θ − θ ∗ − ( θ + θ ∗ ) + A e θ − θ ∗ + θ + θ ∗ + A e θ + θ ∗ + θ − θ ∗ + A e − ( θ + θ ∗ )+ θ − θ ∗ i , (19) v ( x, t ) = 2 i det( M ) h B e θ − θ ∗ − ( θ + θ ∗ ) + B e θ − θ ∗ + θ + θ ∗ + B e θ + θ ∗ + θ − θ ∗ + B e − ( θ + θ ∗ )+ θ − θ ∗ i , (20)wheredet( M ) = C e − ( θ + θ ∗ + θ + θ ∗ ) + C e θ + θ ∗ + θ + θ ∗ + C e θ + θ ∗ − ( θ + θ ∗ ) + C e − ( θ + θ ∗ )+ θ + θ ∗ + C e θ − θ ∗ − ( θ − θ ∗ ) + C ∗ e − ( θ − θ ∗ )+ θ − θ ∗ ,θ k is given in Eq. (10), and coefficients A k , B k , C k arecertain functions of ζ , ζ , α , α , β , β whose expressionsare given below: A = (cid:18) ζ ∗ − ζ − ζ ∗ − ζ (cid:19) α ,A = α ( | α | + | β | ) ζ ∗ − ζ − α ( α α ∗ + β β ∗ ) ζ ∗ − ζ ,A = α ( | α | + | β | ) ζ ∗ − ζ − α ( α ∗ α + β ∗ β ) ζ ∗ − ζ ,A = (cid:18) ζ ∗ − ζ − ζ ∗ − ζ (cid:19) α ,B = (cid:18) ζ ∗ − ζ − ζ ∗ − ζ (cid:19) β ,B = β ( | α | + | β | ) ζ ∗ − ζ − β ( α α ∗ + β β ∗ ) ζ ∗ − ζ ,B = β ( | α | + | β | ) ζ ∗ − ζ − β ( α ∗ α + β ∗ β ) ζ ∗ − ζ ,B = (cid:18) ζ ∗ − ζ − ζ ∗ − ζ (cid:19) β , C = 1( ζ ∗ − ζ )( ζ ∗ − ζ ) + 1 | ζ ∗ − ζ | ,C = (cid:0) | α | + | β | (cid:1) (cid:0) | α | + | β | (cid:1) ( ζ ∗ − ζ )( ζ ∗ − ζ ) + | α ∗ α + β ∗ β | | ζ ∗ − ζ | ,C = | α | + | β | ( ζ ∗ − ζ )( ζ ∗ − ζ ) ,C = | α | + | β | ( ζ ∗ − ζ )( ζ ∗ − ζ ) ,C = α α ∗ + β β ∗ | ζ ∗ − ζ | . For two-solitons, there are two cases to consider.(1) If the two eigenvalues ζ and ζ are purely imaginary,i.e., ζ = iη , ζ = iη , with η , η >
0, then θ k = η k x + 2 iη k t, θ k + θ ∗ k = 2 η k x, θ k − θ ∗ k = 4 iη k t. In this case, when x → − x , θ k + θ ∗ k → − ( θ k + θ ∗ k ) , θ k − θ ∗ k → θ k − θ ∗ k . Thus, by cross multiplication of the ratio expressions for u ( − x, t ) and v ( x, t ) from (19)-(20) and requiring expo-nentials of the same power to match, we find that thenecessary and sufficient conditions for v ( x, t ) = u ( − x, t )are A = B , A = B , A = B , A = B , (21) C = C , C = C . (22)The requirement of C = C directly leads to Eq. (12)in Theorem 1, and the requirement of C = C leadsto Eq. (13). Under these two requirements on C k ’s, wefind that only two of the four conditions for A k ’s and B k ’s in Eq. (21) are independent, i.e., if two of them aresatisfied, then the other two would be satisfied automat-ically. When we choose the two conditions as A = B and A = B , these conditions would lead to equations(14)-(15).Later in Sec. IV B, we will explicitly solve the fourequations (12)-(15), and show that they admit solutionsfor w and w if and only if | α ∗ α + β ∗ β | ≤
1. Inaddition, the admitted solutions have four free real pa-rameters (not counting the eigenvalue parameters η and η ).(2) If the two eigenvalues ζ and ζ are not purely imag-inary, then ζ = − ζ ∗ . In this case, θ = − iζ x − iζ t, θ = iζ ∗ x − iζ ∗ t ; thus, θ + θ ∗ = − iζ x, θ − θ ∗ = − iζ t. Then, as x → − x , θ + θ ∗ → − ( θ + θ ∗ ) , θ − θ ∗ → θ − θ ∗ . Recalling the expressions of u ( x, t ) and v ( x, t ) in Eqs.(19)-(20), we find that in order for v ( x, t ) = u ( − x, t ), thenecessary and sufficient conditions now are A = B , A = B , A = B , A = B , (23)and C = C , C = C ∗ . (24)The A = B and A = B conditions are β ( | α | + | β | ) ζ ∗ − ζ − β ( α ∗ α + β ∗ β )2 ζ ∗ = (cid:18) ζ ∗ − ζ − ζ ∗ (cid:19) α and α ( | α | + | β | ) ζ ∗ − ζ − α ( α ∗ α + β ∗ β )2 ζ ∗ = (cid:18) ζ ∗ − ζ − ζ ∗ (cid:19) β , which can be rewritten as equations (17) in Theorem 1.Remarkably, we find that when ( α , β ) are related to( α , β ) by Eq. (17), all the other conditions in (23)-(24)are automatically satisfied. This completes the proof ofTheorem 1. (cid:3) Remark 1.
Theorem 1 shows that for the nonlocalNLS equation (6), symmetry relations of eigenvectors inthe scattering data are very complicated, because suchrelations depend on the number and locations of eigen-values in a highly nontrivial way. Given the complexity ofthese symmetry relations for two-solitons, such relationsfor three and higher solitons are expected to be even morecomplicated. This poses a challenge for deriving general N -solitons in Eq. (6), at least in the Riemann-Hilbertframework. IV. SOLITON DYNAMICS IN THE NONLOCALNLS EQUATION
In this section, we examine dynamics of one- and two-solitons of Eq. (6) as presented in Theorem 1.
A. Single solitons
Single solitons in the nonlocal NLS equation (6) can beobtained from the single Manakov-soliton (9) with onepurely imaginary eigenvalue ζ = iη ( η >
0) and with itseigenvector w given by Eq. (11) in Theorem 1. Thissoliton is u ( x, t ) = √ η e iη t + iγ sech (2 ηx ) , (25)where γ is a free real parameter. Since the nonlocal NLSequation (6) is phase-invariant, the above soliton is equiv-alent to u ( x, t ) = √ η e iη t sech (2 ηx ) , (26)which is shown in Fig. 1. This soliton is stationary withconstant amplitude, and is symmetric in x . −1 0 1−202 Re( ζ ) I m ( ζ ) x t −5 0 5−101 FIG. 1: The single soliton (26) in the nonlocal NLS equation(6) with η = 1. Left panel: positions of eigenvalues. Rightpanel: graph of solution | u ( x, t ) | . B. Two-solitons with purely imaginary eigenvalues
These solitons are obtained from the two-Manakov-solitons (9) with two purely imaginary eigenvalues in C + ,and with eigenvectors w , w satisfying the equations(12)-(15) in Theorem 1. Below, we solve these four equa-tions explicitly.First, we introduce the notations p ≡ | α | + | β | , q ≡ α ∗ α + β ∗ β , and q ≡ r e iγ , α ≡ r e iγ , α ≡ r e iγ , where r , r , r ( ≥
0) are amplitudes of complex numbers q, α , α , and γ , γ , γ their phases.Before solving equations (12)-(15), we notice that theyadmit two invariances, i.e., if α → α e i b γ , β → β e i b γ , α → α e i b γ , β → β e i b γ , where b γ and b γ are arbitrary real constants, then theseequations remain invariant. Thus, the phases γ , γ ofparameters α and α are free real constants.To solve equations (12)-(15), it is convenient to param-eterize their solutions in terms of q , i.e., r and γ , whichare two additional free real constants. We will show thatsolutions exist if and only if | q | ≤
1. For given q , we can get p from Eq. (13) as p = s g − | q | g . Recall from the definition of g in Eq. (16) that g is realand | g | >
1. Thus, the quantity under the square root inthe above expression is always positive. After the p and q values are available, we see from Eqs. (12) and (14)-(15)that β and β depend on α and α only linearly, whichis a big advantage.Now, we substitute equations (14)-(15) into (12). Af-ter simplification, we obtain a quadratic equation for theratio h ≡ r /r as ah + bh + c = 0 , (27)where the coefficients are a = 1 − (1 + g ) r , b = 2 gpr cos( γ + γ − γ ) ,c = − (cid:2) g − r (cid:3) . After this h value is obtained, we insert (14) into theequation p = | α | + | β | and use it to obtain r as r = r p Ω , whereΩ = 1+ g p +(1+ g ) r h − g (1+ g ) pr h cos( γ + γ − γ ) , and the r value is then r = r h. By now, the α and α values have been obtained, withtheir phases γ , γ being free constants, and their am-plitudes r , r related to their phases and q through theabove equations. The β , β values are determined sub-sequently from α , α , p and q through equations (14)-(15). We have verified that the α , β , α , β values thusobtained satisfy the condition α ∗ α + β ∗ β = q ; thus thecalculations are consistent.The existence and number of solutions to equations(12)-(15) depend on the existence and number of non-negative solutions to the quadratic equation (27) for h .The discriminant ∆ = b − ac of this quadratic equationcan be found to be∆ = 4 g p r (cid:20) cos ( γ + γ − γ ) − − r − r [1 + ( g − r ] (cid:21) . Without loss of generality, we let 0 < η < η ; hence g >
1. Then, utilizing this discriminant and the coef-ficient expressions of ( a, b, c ) above, we can easily reachthe following conclusions.1. If r >
1, then ∆ <
0. In this case, thequadratic equation (27) for h does not admit anynon-negative solution.2. If r = 1, then p = 1, a = c = − g , and b =2 g cos( γ + γ − γ ). In this case, the quadraticequation (27) admits a single (repeated) positiveroot h = 1 when cos( γ + γ − γ ) = 1, and thecorresponding w and w solutions are w = [2 − / e iγ , − / e iγ , T , w = [2 − / e iγ , − / e iγ , T , where γ and γ are free real constants.3. If 1 / √ g < r <
1, then this quadratic h -equation admits two positive solutions whencos( γ + γ − γ ) > s r − r [1 + ( g − r ] , and thus there are two ( w , w ) solutions. Whenthe left and right sides of the above inequality be-come equal, there is a single ( w , w ) solution.4. If r < / √ g , then c/a <
0. In this case, thequadratic equation (27) admits a single positiveroot h for arbitrary γ , γ and γ values. Thus,there is a single ( w , w ) solution for arbitraryfree parameters γ , γ , γ .To summarize, the above results reveal that equations(14)-(15) admit solutions for w and w if and onlyif | α ∗ α + β ∗ β | ≤
1, and the admitted solutions havefour free real parameters, which can be chosen as theamplitude and phase of parameter q = α ∗ α + β ∗ β , andthe phases of complex numbers α , α .Next, we illustrate the dynamics of these two-solitonswith imaginary eigenvalues. We will fix η = i and η =2 i and vary the free parameters q and phases γ , γ of α , α . For these η and η values, g = 3.First, we choose q = 0 , γ = 1 , γ = 2 . (28)For this q value, r < / √ g . Thus, it belongs tothe case (4) above, and there is a single solution for( α , β , α , β ), which is found to be α = 1 √ e i , α = 1 √ e i , β = − α , β = α . The corresponding u ( x, t ) solution from Eq. (9) is dis-played in Fig. 2(b). It is seen that this two-soliton me-anders periodically, which is an interesting and distinc-tive pattern. Physically, this meandering can be under-stood through the connection of the nonlocal NLS equa-tion (6) with the Manakov system (3)-(4). Specifically,the evolution in Fig. 2(b) corresponds to an interactionbetween this u ( x, t ) component and its opposite-paritywave u ( − x, t ) in the v -component in the Manakov sys-tem. Thus, this interesting meandering of the u ( x, t ) so-lution is caused by the interference of its opposite-paritywave u ( − x, t ). Note that this meandering in Fig. 2(b) resembles internal oscillations of vector solitons in thecoupled NLS equations [48]. However, in contrast withthe internal oscillations reported in [48], the present me-andering does not emit any radiation and thus lasts for-ever. In addition, the present meandering is described byexact analytical formulae.Next, we choose q = 0 . e i , γ = 1 . , γ = 5 . (29)For this q value, 1 / √ g < r <
1. Thus, it belongsto case (3) above. It is easy to check that the inequalitycondition in case (3) is met. Hence, there are two setsof ( α , β , α , β ) values. The corresponding two u ( x, t )solutions from Eq. (9) are displayed in Fig. 2(c,d) re-spectively. The solution in panel (c) looks like a periodicwave drifting and recovering, while the solution in panel(d) looks like asymmetric meandering. −1 0 1−303 Re( ζ ) I m ( ζ ) (a) x t (b) −3 0 3−101 x t (c) −3 0 3−101 x t (d) −3 0 3−101 FIG. 2: Three examples of two-solitons in the nonlocal NLSequation (6) with purely imaginary eigenvalues η = i and η = 2 i . (a) Positions of eigenvalues; (b) the two-soliton withparameters in Eq. (28); (c, d) the two solutions of two-solitonswith parameters in Eq. (29). C. Two-solitons with non-imaginary eigenvalues
Now we consider two-solitons with non-imaginaryeigenvalues, which are obtained from the two-Manakov-solitons (9) with a pair of non-imaginary eigenvalues( ζ , − ζ ∗ ) in C + , and with eigenvectors w , w satisfy-ing the equations (17) in Theorem 1. In these solutions, α and β are free complex parameters. To illustrate, wetake ζ = 0 . . i, β = − . . Then, for three choices of the α values of 0 . − . i ,0 .
04 and 0, the corresponding u ( x, t ) solutions are dis-played in Fig. 3. The solution in the upper right panellooks like a refection of two moving waves of different am-plitudes. The solution in the lower left panel looks likethe annihilation of the left-moving wave by the right-moving one upon collision. The solution in the lowerright panel looks like a single right-moving wave, withits position abruptly shifted near x = 0. Again, these in-teresting behaviors can be understood physically throughthe connection of the nonlocal NLS equation (6) with theManakov system (3)-(4). For instance, the abrupt posi-tion shift of the single right-moving wave in the lowerright panel is caused by a collision of this right-movingwave u ( x, t ) with its opposite-parity wave u ( − x, t ) in the v -component, which occurs near x = 0. It is interest-ing to note that for the original nonlocal defocusing NLSequation proposed in [6], single moving dark solitons withabrupt position shifts were reported in [8]. Althoughsuch dark solitons with abrupt position shifts were de-rived mathematically, they were difficult to understandphysically. In view of the moving bright solitons withabrupt position shifts in Fig. 3, those dark solitons withabrupt position shifts are now a little easier to under-stand.Recall from Sec. IV A that one-solitons in the under-lying nonlocal equation (6) are stationary. Thus, thesetwo-solitons in Fig. 3 definitely are not nonlinear super-positions of those stationary one-solitons. This behaviorresembles that in the previous nonlocal NLS equation (1)as we revealed in [18]. −0.3 0 0.3−101 Re( ζ ) I m ( ζ ) x t −10 0 10−20020 x t −10 0 10−20020 x t −10 0 10−20020 FIG. 3: Three examples of two-solitons in the nonlocal NLSequation (6) with complex eigenvalues ζ = − ζ ∗ = 0 . . i and β = − .
43. Upper left: positions of eigenvalues; upperright: α = 0 . − . i ; lower left: α = 0 .
04; lower right: α = 0. V. OTHER NEW INTEGRABLE NONLOALEQUATIONS
Extending the idea of previous sections, we can deriveother new nonlocal equations of physical relevance.Starting from the Manakov system (3)-(4), when weimpose the solution constraint v ( x, t ) = u ∗ ( x, − t ) , (30)we get iu t ( x, t )+ u xx ( x, t )+2 σ (cid:2) | u ( x, t ) | + | u ( x, − t ) | (cid:3) u ( x, t ) = 0 , (31)which is a new nonlocal NLS equation of reverse-timetype. When we impose the solution constraint v ( x, t ) = u ∗ ( − x, − t ) , (32)the Manakov system reduces to iu t ( x, t )+ u xx ( x, t )+2 σ (cid:2) | u ( x, t ) | + | u ( − x, − t ) | (cid:3) u ( x, t ) = 0 , (33)which is a new nonlocal NLS equation of reverse-space-time type. These two equations differ from the previ-ous nonlocal NLS equations of reverse-time and reverse-space-time types in [28] in the nonlinear terms. Bothequations are also integrable, and their Lax pairs are(7)-(8) with v ( x, t ) replaced by u ∗ ( x, − t ) and u ∗ ( − x, − t )respectively.Physically, the reverse-time NLS equation (31) de-scribes the solutions of the Manakov system under specialinitial conditions where v ( x,
0) = u ∗ ( x, u ( x, t ) of the reverse-time equation (31) fornegative time gives the v ( x, t ) solution of the Manakovsystem for positive time through v ( x, t ) = u ∗ ( x, − t ). Thereverse-space-time NLS equation (33) describes the solu-tions of the Manakov system under special initial condi-tions where v ( x,
0) = u ∗ ( − x, u ( x, t ) of the reverse-space-time equation (33) for nega-tive time gives the v ( x, t ) solution of the Manakov systemfor positive time through v ( x, t ) = u ∗ ( − x, − t ).The above ideas can be generalized further. For in-stance, let we consider the four-component coupled NLSequations iU t + U xx + 2 σ ( U † U ) U = 0 , (34)where U = [ u, v, w, s ] T , and σ = ±
1. These coupledequations govern the nonlinear interaction of four inco-herent light beams [43] as well as the evolution of four-component Bose-Einstein condensates [46]. These equa-tions are also integrable [5, 47]. If we impose the solutionconstraints w ( x, t ) = u ( − x, t ) , s ( x, t ) = v ( − x, t ) , (35)these equations reduce to iu t ( x, t ) + u xx ( x, t ) + 2 σ (cid:2) | u ( x, t ) | + | u ( − x, t ) | + | v ( x, t ) | + | v ( − x, t ) | (cid:3) u ( x, t ) = 0 , (36) iv t ( x, t ) + v xx ( x, t ) + 2 σ (cid:2) | u ( x, t ) | + | u ( − x, t ) | + | v ( x, t ) | + | v ( − x, t ) | (cid:3) v ( x, t ) = 0 , (37)which are a system of nonlocal Manakov equations ofreverse-space type. If we impose the solution constraints w ( x, t ) = u ∗ ( x, − t ) , s ( x, t ) = v ∗ ( x, − t ) , (38)we get iu t ( x, t ) + u xx ( x, t ) + 2 σ (cid:2) | u ( x, t ) | + | u ( x, − t ) | + | v ( x, t ) | + | v ( x, − t ) | (cid:3) u ( x, t ) = 0 , (39) iv t ( x, t ) + v xx ( x, t ) + 2 σ (cid:2) | u ( x, t ) | + | u ( x, − t ) | + | v ( x, t ) | + | v ( x, − t ) | (cid:3) v ( x, t ) = 0 , (40)which are a system of nonlocal Manakov equations ofreverse-time type. If we impose the solution constraints w ( x, t ) = u ∗ ( − x, − t ) , s ( x, t ) = v ∗ ( − x, − t ) , (41)we get iu t ( x, t ) + u xx ( x, t ) + 2 σ (cid:2) | u ( x, t ) | + | u ( − x, − t ) | + | v ( x, t ) | + | v ( − x, − t ) | (cid:3) u ( x, t ) = 0 , (42) iv t ( x, t ) + v xx ( x, t ) + 2 σ (cid:2) | u ( x, t ) | + | u ( − x, − t ) | + | v ( x, t ) | + | v ( − x, − t ) | (cid:3) v ( x, t ) = 0 , (43)which are a system of nonlocal Manakov equations ofreverse-space-time type. These three nonlocal Manakovsystems are also integrable, and they describe the solu-tion behaviors of the physical model (34) under specialinitial conditions of (35), (38) and (41) with t = 0. VI. SUMMARY AND DISCUSSION
In this paper, we proposed a new integrable nonlo-cal NLS equation (2) which has concrete physical mean-ings. This equation was derived from a reduction ofthe Manakov system, and it describes physical situa-tions governed by the Manakov system under special ini-tial conditions. Solitons and multi-solitons in this non-local equation were also investigated in the frameworkof Riemann-Hilbert formulation. We found that sym-metry relations of discrete scattering data for this non-local equation are very complicated, which makes thederivation of its general N -solitons challenging. Fromthe one- and two-solitons we obtained. it was observedthat the two-solitons are not a nonlinear superposition of one-solitons, and the two-solitons exhibit interesting dy-namical patterns such as meandering and abrupt positionshifts. As a generalization of these results, we also pro-posed other integrable and physically meaningful nonlo-cal equations, such as new NLS equations of reverse-timeand reverse-space-time types, as well as nonlocal Man-akov equations of reverse-space, reverse-time and reverse-space-time types.The results in this paper are significant in two differentways. From a mathematical point of view, we presenteda new integrable nonlocal equation which has clear physi-cal meanings. In addition, we showed that this integrableequation exhibits some unusual mathematical propertiessuch as intricate symmetry relations of its discrete scat-tering data. From a physical point of view, we derivedone- and two-solitons in this nonlocal equation, whichcorrespond to Manakov solutions under the initial paritysymmetry between the two components, and these soli-tons feature interesting physical patterns such as sym-metric and asymmetric meandering.The highly complex symmetry relations of discretescattering data for the nonlocal NLS equation (2) arevery surprising. This fact implies that general N -solitons in this equation will be very difficult to derive inthe inverse scattering and Riemann-Hilbert framework.Whether they can be derived more easily in other frame-works such as the Darboux transformation and bilinearmethods remains to be seen.In this paper, we only studied bright solitons in thenonlocal NLS equation (2). Other types of solutions suchas rogue waves and dark solitons in this equation are de-sirable too, which merit studies in the future. In addi-tion, we proposed a number of other new nonlocal equa-tions of physical relevance, such as new NLS equationsof reverse-time and reverse-space-time types, and nonlo-cal Manakov equations of reverse-space, reverse-time andreverse-space-time types. Bright solitons, dark solitonsand rogue waves in those systems are also open questionsfor further studies. Acknowledgment
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