Darboux Transformation: New Identities
11 Darboux Transformation: New Identities
Vishal Vaibhav
Abstract —This letter reports some new identities for multi-soliton potentials that are based on the explicit representationprovided by the Darboux matrix. These identities can be usedto compute the complex gradient of the energy content of thetail of the profile with respect to the discrete eigenvalues and thenorming constants. The associated derivatives are well definedin the framework of the so-called Wirtinger calculus which canaid a complex variable based optimization procedure in orderto generate multisolitonic signals with desired effective temporaland spectral width.
Index Terms —Multisolitons, Darboux Transformation
I. I
NTRODUCTION
This letter deals with the Darboux representation of multi-soliton solutions of the nonlinear Schrödinger equation. Ascarriers of information, these multisolitonic signals offer apromising solution to the problem of nonlinear signal dis-tortions in fiber optic channels [1]. In any nonlinear Fouriertransform (NFT) based transmission methodology seekingto modulate the discrete spectrum of the multisolitons, theunbounded support of such signals (as well as its Fourierspectrum) presents some challenges in achieving the bestpossible spectral efficiency [2], [3] which forms part of themotivation for this work.The Darboux transformation has proven to be an extremelypowerful tool in handling the discrete part of the nonlinearFourier spectrum. The rational structure of the associatedDarboux matrix was recently exploited to obtain fast inverseNFT algorithms in [4], [5]. In the particular case of K -solitonsolutions, the rational structure of the Darboux matrix facil-itates the exact solution of the Zakharov-Shabat problem [6]for (doubly-) truncated version of the signal via the solutionof an associated Riemann-Hilbert problem [7]. In [4], [7], anexact method for computing the energy content of the “tails”of K -soliton solutions was reported which was again basedon the Darboux transformation. This method was further usedin [8] to establish the sufficient conditions for either one-sidedor compact support of the signals resulting from “addition”of boundstates. Given that the complexity of computing theDarboux matrix coefficients is O (cid:0) K (cid:1) , these methods turn outto be extremely efficient eliminating any need for heuristicapproaches based on the asymptotic expansions (with respectto the windowing parameter, say, τ ). The present work,therefore, tries to further reinforce the idea that the Darbouxrepresentation can potentially facilitate a number of designand signal processing aspects of K -soliton solutions. For thegeneral case, when the reflection coefficient is bandlimited,the work presented in [9] may allow us to compute the Jostsolutions of the seed potential with extremely high accuracy. Email: [email protected]
The present work is also motivated by the fact that therecent attempts [2], [3], [10] towards optimizing the generatedmultisolitonic signals are either based on brute-force methodsor asymptotic expansions with respect to the windowing pa-rameter. These methods have serious drawbacks either becausethey do not scale well in complexity when the number ofboundstates are only moderately high or because they are notreliable in the absence of a prior knowledge of the goodness ofthe approximations made. In this letter, we present some newidentities that can potentially make the optimization problemamenable to some of the powerful optimization proceduresavailable in the literature (see [12] and the references therein)at the same time completely circumventing the need for anyheuristics. Given that the independent variables (i.e. discreteeigenvalues and the norming constants) in the optimizationprocedure are complex in nature, the framework based onWirtinger calculus presented in [12] appears to be moreconvenient.The main contributions of this work are presented in Sec. IIwhich deals with the temporal width which is followed by abrief discussion of estimation of spectral width in Sec. III.The letter concludes with some examples in Sec. IV wherecalculation can be carried out in a simple manner.II. T EMPORAL W IDTH
The temporal width a K -soliton solution can be defined viathe L -norm of the profile which is also related to the energyof the pulse. Let the energy content of the “tails” of the profile,denoted by E (±) ( t ) , be defined by E (−) ( t ) = ∫ t −∞ | q ( s )| ds , E ( + ) ( t ) = ∫ ∞ t | q ( s )| ds , (1)so that E( t ) = (cid:104) E (−) (− t ) + E ( + ) ( t ) (cid:105) , (2)characterizes the total energy in the tails which is a fractionof the total energy of a K -soliton solution. The total energyis given by (cid:107) q (cid:107) = (cid:205) Kk = Im ζ k . Now, if the tolerance forthe fraction of total energy in the tails is (cid:15) tails , then effectivethe support, [− τ, τ ] , of the profile must be chosen such that E( τ ) ≤ (cid:15) tails (cid:107) q (cid:107) . Therefore, the effective temporal width ofthe K -soliton solution can then be defined as τ which is afunction of (cid:15) tails . The readers are warned that the representation of the norming constantsused in these papers do not follow the standard convention and are incorrectin many cases. The relationship b k = b ( ζ k ) , where ( ζ k , b k ) in the tuplecomprising the discrete eigenvalue and the corresponding norming constant,does not hold when b does not have an analytic continuation in C + . In fact,from [11] it is known that the b remains invariant when boundstates are addedto an arbitrary profile so that b k (cid:44) b ( ζ k ) even if b ( ζ k ) exists.c (cid:13) a r X i v : . [ n li n . S I] J a n For the specific case of K -soliton solutions, an exact recipefor computing E( τ ) was reported in [4], [7] which we sum-marize briefly as follows: Let v ( t ; ζ ) = ( φ , ψ ) be the matrixform of the Jost solutions. Then, from the standard theory ofscattering transforms [6], it is known that, for ζ ∈ C + , v K e i σ ζ t = (cid:32) + i ζ E (−) i ζ q ( t )− i ζ r ( t ) + i ζ E ( + ) (cid:33) + O (cid:18) ζ (cid:19) . (3)Now, the K -soliton potentials along with their Jost solutionscan be computed quite easily using the Darboux transforma-tion (DT) [4], [7]. Let S K be the discrete spectrum of a K -soliton potential. The seed solution here corresponds to the null potential; therefore, v ( t ; ζ ) = e − i σ ζ t . The augmentedmatrix Jost solution v K ( t ; ζ ) can be obtained from the seedsolution v ( t ; ζ ) using the Darboux matrix as v K ( t ; ζ ) = µ K ( ζ ) D K ( t ; ζ, S K ) v ( t ; ζ ) for ζ ∈ C + . The Darboux transfor-mation can be implemented as a recursive scheme. Let usdefine the successive discrete spectra ∅ = S ⊂ S ⊂ S ⊂ . . . ⊂ S K such that S j = {( ζ j , b j )} ∪ S j − for j = , , . . . , K where ( ζ j , b j ) are distinct elements of S K . The Darboux matrixof degree K > can be factorized into Darboux matrices ofdegree one as D K ( t ; ζ, S K | S ) = D ( t ; ζ, S K | S K − )× D ( t ; ζ, S K − | S K − ) × . . . × D ( t ; ζ, S | S ) , (4)where D ( t ; ζ, S j | S j − ) , j = , . . . , K are the successiveDarboux matrices of degree one with the convention that ( ζ j , b j ) = S j ∩ S j − is the bound state being added to theseed potential whose discrete spectra is S j − . Note that theDarboux matrices of degree one can be stated as D ( t ; ζ, S j | S j − ) = ζ σ − (cid:169)(cid:173)(cid:173)(cid:171) | β j − | ζ j + ζ ∗ j + | β j − | ( ζ j − ζ ∗ j ) β j − + | β j − | ( ζ j − ζ ∗ j ) β ∗ j − + | β j − | ζ j + ζ ∗ j | β j − | + | β j − | (cid:170)(cid:174)(cid:174)(cid:172) , (5)and β j − ( t ; ζ j , b j ) = φ ( j − ) ( t ; ζ j ) − b j ψ ( j − ) ( t ; ζ j ) φ ( j − ) ( t ; ζ j ) − b j ψ ( j − ) ( t ; ζ j ) , (6)for ( ζ j , b j ) ∈ S K and the successive Jost solutions, v j = ( φ j , ψ j ) , needed in this ratio are computed as v j ( t ; ζ ) = ( ζ − ζ ∗ j ) − D ( t ; ζ, S j | S j − ) v j − ( t ; ζ ) . (7)The potential is given by q j = q j − − i ( ζ j − ζ ∗ j ) β j − + | β j − | . (8)and E (−) j = E (−) j − + ( ζ j ) + | β j − | − , E ( + ) j = E ( + ) j − + ( ζ j ) + | β j − | , (9)so that E (∓) ( t ) = E (∓) K ( t ) with E (∓) ( t ) ≡ . Next, our objectiveis to compute the derivatives of E( τ ) with respect to thediscrete spectra of K -soliton in order to facilitate optimizationprocedures that are based on gradients. In the following, weuse the notation ∂ Z j = ∂∂ Z j , ∂ Z j = ∂∂ ( Z ∗ j ) . (10) For real-valued function f , it is known that (cid:16) ∂ Z j f (cid:17) ∗ = ∂ Z j f .In Wirtinger calculus, the complex gradient is defined as ( ∂ Z , . . . ∂ Z K , ∂ Z , . . . , ∂ Z K ) (cid:124) Remark II.1.
Here, we have described the Darboux trans-formation only for the case of K-soliton potentials; however,the recipe can be easily adapted to the case of arbitrary seedpotentials. Note that this would require explicit knowledge of v ( t ; ζ ) and E (∓) ( t ) .A. Derivatives with respect to norming constants Note that E (−) K − (− τ ) and E ( + ) K − ( τ ) are independent of b K ;therefore, ∂ b K E (∓) (∓ τ ) = ∓ ( ζ K ) ∂ b K (cid:2) + | β K − (∓ τ )| (cid:3) − . (11)By direct calculation, we have ∂ b K β K − ( t ; ζ K , b K ) = a K − ( ζ K ) (cid:104) φ ( K − ) − b K ψ ( K − ) (cid:105) ( t ; ζ K ) , (12)where we have used to the Wronskian relation a K − ( ζ ) = W ( φ K − , ψ K − ) = K − (cid:214) k = (cid:18) ζ − ζ k ζ − ζ ∗ k (cid:19) . (13)Using the identity (12), it is straightforward to work out: ∂ b K E (∓) (∓ τ ) = ± ( ζ K ) | β K − | (cid:2) + | β K − | (cid:3) × a K − ( ζ K ) (cid:104) φ ( K − ) − b K ψ ( K − ) (cid:105) (cid:104) φ ( K − ) − b K ψ ( K − ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = ∓ τ . (14)Note that ζ K is the last eigenvalue to be added using the DTiterations. Given that there is no restriction on the order inwhich the eigenvalues can be added, we can always choose ζ k to be added last. This would determine ∂ b k E (±) using DTiterations for arbitrary k . Thus, the complexity of computing K derivatives works out to be O (cid:0) K (cid:1) .Before we conclude this discussion, let us examine the caseof multisoliton solutions when τ is large. In this limit, we have β K − (− τ ) ∼ b − K e i ζ K τ a K − ( ζ K ) , β − K − ( + τ ) ∼ b K e i ζ K τ a K − ( ζ K ) , (15)so that ∂ b K E (∓) (∓ τ ) ∼ ∓ ( ζ K ) b K | β K − (∓ τ )| ± . (16)Thus, the stationary condition ∂ b K E( τ ) = translates into | b K | = . Therefore, asymptotically, | b j | = minimizes theenergy in the tails. This result can be easily verified from (9)which in the limit of large τ gives E( τ ) ∼ K (cid:213) j = ( ζ j )| a j − ( ζ j )| (cid:18) | b j | + | b j | (cid:19) e − η j τ ≤ K (cid:213) j = ( ζ j )| a j − ( ζ j )| e − η j τ . (17) B. Derivatives with respect to discrete eigenvalues
Let V j ( t ; ζ ) = ( ζ − ζ ∗ j ) v j ( t ; ζ ) and define V j = ( Φ j , Ψ j ) sothat V j ( t ; ζ ) = D ( t ; ζ, S j | S j − ) V j − ( t ; ζ ) . (18)The ratio β j can also be computed in terms of the modifiedJost solutions on account of the fact that ( ζ − ζ j ) falls out ofthe equation while taking the ratio: β j − ( t ; ζ j , b j ) = Φ ( j − ) ( t ; ζ j ) − b j Ψ ( j − ) ( t ; ζ j ) Φ ( j − ) ( t ; ζ j ) − b j Ψ ( j − ) ( t ; ζ j ) , (19)This gives us the opportunity to compute the derivatives withrespect to ζ recursively: ∂ ζ V j ( t ; ζ ) = V j − ( t ; ζ ) + D ( t ; ζ, S j | S j − ) ∂ ζ V j − ( t ; ζ ) . (20)Using the notation W ζ ( u , v ) = (cid:0) u ∂ ζ v − v ∂ ζ u (cid:1) for the Wron-skian of scalar functions, let us introduce W ( K − ) ( t ; ζ K ) = W ζ (cid:16) Φ ( K − ) , Φ ( K − ) (cid:17) ( t ; ζ K ) , W ( K − ) ( t ; ζ K ) = W ζ (cid:16) Ψ ( K − ) , Ψ ( K − ) (cid:17) ( t ; ζ K ) . (21)By direct calculation, we have ∂ ζ K β K − ( t ; ζ K , b K ) = (cid:104) W ( K − ) + b K W ( K − ) (cid:105)(cid:104) Φ ( K − ) − b K Ψ ( K − ) (cid:105) ( t ; ζ K )− b K ∂ ζ a K − ( ζ K ) (cid:104) Φ ( K − ) − b K Ψ ( K − ) (cid:105) ( t ; ζ K ) . (22)Note that E (−) K − (− τ ) and E ( + ) K − ( τ ) are independent of ζ K ;therefore, using the above identity, it is straightforward toobtain ∂ ζ K E (∓) (∓ τ ) = + | β K − | ∓ + ( ζ K )| β K − | (cid:2) + | β K − | (cid:3) × W ( K − ) − b K ∂ ζ a K − ( ζ K ) + b K W ( K − ) (cid:16) Φ ( K − ) − b K Ψ ( K − ) (cid:17) (cid:16) Φ ( K − ) − b K Ψ ( K − ) (cid:17) t = ∓ τ . (23)Following as in the case of norming constants, we can alwayschoose ζ k to be added last so that ∂ ζ k E (±) can be determinedusing DT iterations for arbitrary k . Thus, the complexity ofcomputing K derivatives again works out to be O (cid:0) K (cid:1) .III. S PECTRAL W IDTH
Consider the Fourier spectrum of the multisoliton potentialdenoted by Q ( ξ ) = ∫ q ( t ) e − i ξ t dt , ξ ∈ R . Let us observe thatthe following quantities can be expressed entirely in terms ofthe discrete eigenvalues: C = − ∫ q ∗ ( ∂ t q ) dt = i (cid:213) k Im ζ k , C = ∫ (cid:16) | q | − | ∂ t q | (cid:17) dt = − (cid:213) k Im ζ k . (24) with C = (cid:107) q (cid:107) . These quantities do not evolve as the pulsepropagates along the fiber. From [7], the variance (cid:104) ∆ ξ (cid:105) isgiven by (cid:104) ∆ ξ (cid:105) = ∫ | q | dtC + C C − C C ≤ (cid:107) q (cid:107) ∞ + C C − C C . (25)This quantity characterizes the width of the Fourier spectrum.The biquadratic integral above cannot be computed exactly ingeneral, however, (cid:107) q (cid:107) ∞ can be computed in a straightforwardmanner: From (8), we have (cid:107) q j (cid:107) ∞ ≤ (cid:107) q j − (cid:107) ∞ + ( ζ j ) , wehave (cid:107) q K (cid:107) ∞ ≤ (cid:205) Kk = Im ( ζ j ) which yields (cid:104) ∆ ξ (cid:105) ≤ S where(correcting a typographical error in [7]) S = (cid:32) C + C C − C C (cid:33) . (26)Note that this inequality holds irrespective of how the pulseevolves as it propagates along the fiber.IV. E XAMPLES
A. One-sided effective support
Let us consider the case where we want to introduce aboundstate with eigenvalue ζ to any arbitrary profile suchthat the energy content of the tail [ τ, ∞) ( τ > ) is E ( + ) ( τ ) .The problem is to determine the norming constant b whichminimizes E ( + ) = E ( + ) ( τ ) . To this end, setting ∂ E ( + ) ( τ ) = ,we have (cid:104) φ ( ) − b ψ ( ) (cid:105) (cid:104) φ ( ) − b ψ ( ) (cid:105)(cid:12)(cid:12)(cid:12) t = τ, ζ = ζ = , which yields b ∈ (cid:40) φ ( ) ( τ ; ζ ) ψ ( ) ( τ ; ζ ) , φ ( ) ( τ ; ζ ) ψ ( ) ( τ ; ζ ) (cid:41) . It is easy to verify that the first choice corresponds to maxi-mum E ( + ) ( τ ) which leaves us with b = φ ( ) ( τ ; ζ )/ ψ ( ) ( τ ; ζ ) so that E ( + ) ( τ ) = E ( + ) ( τ ) , i.e., no part of the soliton’senergy goes into the tail [ τ, ∞) . By a recursive argument,the conclusion holds for any number of boundstates provided b j = φ ( ) ( τ ; ζ j )/ ψ ( ) ( τ ; ζ j ) . B. Adding a boundstate to a symmetric profile
Let us consider the case where we want to introduce aboundstate with eigenvalue ζ to any arbitrary seed profile.The energy content of the tail R \ (− τ, τ ) of the seed profileis E ( τ ) . The problem is to determine the norming constant b which minimizes E( τ ) . To this end, setting ∂ E( τ ) = , wehave (cid:104) φ ( ) − b ψ ( ) (cid:105) (cid:104) φ ( ) − b ψ ( ) (cid:105)(cid:104) | φ ( ) − b ψ ( ) | + | φ ( ) − b ψ ( ) | (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = − τ = (cid:104) φ ( ) − b ψ ( ) (cid:105) (cid:104) φ ( ) − b ψ ( ) (cid:105)(cid:104) | φ ( ) − b ψ ( ) | + | φ ( ) − b ψ ( ) | (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = τ . (27) For the sake of simplicity, we assume that the seed profile issymmetric. Further, we also assume that ζ = i η so that (cid:40) φ ( ) (− t ; i η ) = ψ ( )∗ ( t ; i η ) ,φ ( ) (− t ; i η ) = ψ ( )∗ ( t ; i η ) , (28)with a ( i η ) = a ∗ ( i η ) . In the following, we set t = τ . Then,using the symmetry relations, we obtain (cid:104) φ ( )∗ − ( / b ) ψ ( )∗ (cid:105) (cid:104) φ ( )∗ − ( / b ) ψ ( )∗ (cid:105)(cid:104) φ ( ) − b ψ ( ) (cid:105) (cid:104) φ ( ) − b ψ ( ) (cid:105) b | b | = (cid:104) | φ ( ) − ( / b ∗ ) ψ ( ) | + | φ ( ) − ( / b ∗ ) ψ ( ) | (cid:105) (cid:104) | φ ( ) − b ψ ( ) | + | φ ( ) − b ψ ( ) | (cid:105) . (29)Physically, log | b | is related to the translation of the profile;therefore, it is easy to conclude, for a symmetrical profile, thatthe extrema is obtained for | b | = . Putting A = i (cid:16) ψ ( ) ψ ( ) − φ ( )∗ φ ( )∗ (cid:17) , B = i (cid:16) ψ ( ) φ ( ) − ψ ( )∗ φ ( )∗ (cid:17) = i (cid:16) φ ( ) ψ ( ) − φ ( )∗ ψ ( )∗ (cid:17) , (30)in (29) and using | b | = , we have Ab − Bb + A ∗ = . Thesolution of this equation works out to be b = B ± i (cid:112) | A | − B A = B ± i √ ∆ A . (31)From the relations B = | ψ ( ) | | φ ( ) | − [( ψ ( ) φ ( ) ) ] = | φ ( ) | | ψ ( ) | − [( φ ( ) ψ ( ) ) ] , | A | = | ψ ( ) | | ψ ( ) | + | φ ( ) | | φ ( ) | − [ ψ ( ) ψ ( ) φ ( ) φ ( ) ] , we have ∆ = [ a ( i η )] + | ψ ( ) | | ψ ( ) | + | φ ( ) | | φ ( ) | − | φ ( ) | | ψ ( ) | − | ψ ( ) | | φ ( ) | . (32)In order to show that ∆ ≥ , consider ∆ = (| ψ ( ) ψ ( ) | − | φ ( ) φ ( ) |) + (cid:12)(cid:12)(cid:12) φ ( ) ψ ( ) − ψ ( ) φ ( ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) | φ ( ) ψ ( ) | − | ψ ( ) φ ( ) | (cid:12)(cid:12)(cid:12) = (| ψ ( ) ψ ( ) | − | φ ( ) φ ( ) |) + (cid:16)(cid:12)(cid:12)(cid:12) φ ( ) ψ ( ) − ψ ( ) φ ( ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) | φ ( ) ψ ( ) | − | ψ ( ) φ ( ) | (cid:12)(cid:12)(cid:12)(cid:17) × (cid:16)(cid:12)(cid:12)(cid:12) φ ( ) ψ ( ) − ψ ( ) φ ( ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) | φ ( ) ψ ( ) | − | ψ ( ) φ ( ) | (cid:12)(cid:12)(cid:12)(cid:17) , which shows that ∆ ≥ . Therefore, the extremal points for b are given by arg b = ± arg (cid:34) B + i √ ∆ | A | (cid:35) − arg A . (33)
1) Symmetric -soliton: The general result derived abovecan be applied to a symmetric -soliton potential. Let usassume that the seed potential is a symmetric -soliton po-tential with the discrete spectrum given by {( i η , e i θ )} . Theboundstate being introduced is characterized by ( i η , e i θ ) .Expression for the Jost solutions can be obtained from (4)which leads to B = so that b = ± e i θ . It can be directlyverified that b = e i θ corresponds to the minima of E ( τ ) for all τ > as follows: Given the symmetric nature of theprofile, it suffices to find the minima of E ( + ) ( τ ) which readsas E ( + ) ( τ ) = E ( + ) ( τ ) + η Y − X − + Y − (cid:18) HG (cid:19) (cid:18) − G cos θ − H cos θ (cid:19) , (34)where θ = θ − θ and X − = e − η τ + a ( i η ) e η τ = η cosh ( η τ ) η + η (cid:20) − η η tanh ( η τ ) (cid:21) , Y − = a ( i η ) e − η τ + e η τ = η cosh ( η τ ) η + η (cid:20) + η η tanh ( η τ ) (cid:21) , G = Y e η τ + Y − e − η τ , H = ( X − + Y − ) (cid:2) ( + X − ) e η τ + ( + Y − ) e − η τ (cid:3) , It is straightforward to show that < G , H ≤ , Y − > and X − + Y − > . Now, from − G cos θ − H cos θ = − G / H − H cos θ + GH , and − GH = − ( η − η ) η ( η + η ) [ Y − e − η τ + e η τ ] [ + ( η τ )] cosh ( η τ ) × (cid:20) sinh [ ( η + η ) τ ] η + η + sinh [ ( η − η ) τ ] η − η (cid:21) , it follows that ( − G / H ) ≤ ; therefore, the minima of E ( + ) ( τ ) occurs at θ = n π or b = e i θ .R EFERENCES[1] S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin,M. Kamalian, and S. A. Derevyanko, “Nonlinear Fourier transform foroptical data processing and transmission: advances and perspectives,”
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