Solitons in a box-shaped wavefield with noise: perturbation theory and statistics
SSolitons in a box-shaped wavefield with noise: perturbation theory and statistics
Rustam Mullyadzhanov , ∗ and Andrey Gelash , † Institute of Thermophysics SB RAS, Novosibirsk 630090, Russia Novosibirsk State University, Novosibirsk 630090, Russia Institute of Automation and Electrometry SB RAS, Novosibirsk 630090, Russia and Skolkovo Institute of Science and Technology, Moscow 121205, Russia
We investigate the fundamental problem of the nonlinear wavefield scattering data corrections inresponse to a perturbation of initial condition using inverse scattering transform theory. We presenta complete theoretical linear perturbation framework to evaluate first-order corrections of the fullset of the scattering data within the integrable one-dimensional focusing nonlinear Schr¨odinger(NLSE) equation. The general scattering data portrait reveals nonlinear coherent structures solitonsplaying the key role in the wavefield evolution. Applying the developed theory to a classic box-shaped wavefield we solve the derived equations analytically for a single Fourier mode acting as aperturbation to the initial condition, thus, leading to the sensitivity closed-form expressions for basicsoliton characteristics, i.e. the amplitude, velocity, phase and its position. With the appropriatestatistical averaging we model the soliton noise-induced effects resulting in compact relations forstandard deviations of soliton parameters. Relying on a concept of a virtual soliton eigenvaluewe derive the probability of a soliton emergence or the opposite due to noise and illustrate thesetheoretical predictions with direct numerical simulations of the NLSE evolution. The presentedframework can be generalised to other integrable systems and wavefield patterns.
The propagation of nonlinear waves is well-describedby a number of integrable models leading to the con-cept of the scattering data also known as the nonlinearFourier spectrum. Inverse scattering transform theoryuncovers a trivial evolution of this spectrum and pro-vides an elegant integration method, for example, for theone-dimensional Korteweg–de Vries (KdV) and nonlinearSchr¨odinger (NLSE) equations representing fundamentalmodels of nonlinear physics [1, 2]. The scattering dataportrait reveals nonlinear coherent structures – solitons,which are parametrized by eigenvalues and norming con-stants as well as dispersive waves described by the re-flection coefficient. Solitons represent the backbone ofthe evolution of water wave groups [3–7] or propagationof light pulses in a fiber [5, 8, 9]. These nondispersivewaves play the key role in nonlinear features such as theformation of rogue waves [3, 10, 11] and considered as themain carriers of information in nonlinear optical telecom-munication systems [12–14].In practice the wavefield typically evolves in the pres-ence of noise altering the scattering data and leading tothe important issue of sensitivity [15–19]. For KdV andNLSE models the perturbation theory has been devel-oped in the case of small continuous pumping or dissi-pation [15, 20–28] as well as for an instant perturbation[15, 17, 29], see also some recent advancements [30, 31].However, the analytical insight for perturbed scatteringdata is still missing even for simple model problems.In this work we develop the perturbation theory for abasic rectangular (box) wavefield initially perturbed bystochastic noise within the focusing NLSE. The evolu- ∗ Electronic address: [email protected] † Electronic address: [email protected] tion of a box field within the NLSE model representinga classical so-called dam-break problem [32] attracts ex-perimental attention in optics [33] and hydrodynamics[34] as well as theoretical efforts [35]. A wide box-shapedfield is unstable to long wave perturbations constitutingthe modulation instability [36, 37] typically stimulatedby adding noise [38].We provide a complete first-order perturbation ansatzfor the full scattering data including soliton parameters:amplitudes, velocities, phases and positions. The derivedequations are solved analytically for a box field perturbedby a single Fourier mode. Then using statistical averag-ing we model the effect of noise on solitons resulting incompact expressions for standard deviations. Finally, us-ing a concept of a virtual soliton eigenvalue we derive theprobability of a noise-induced soliton emergence event ordisappearance revisiting a fundamental problem using anew tool [15, 39–42].We write the focusing NLSE for a complex wavefield q ( t, x ) in a non-dimensional form: iq t + 12 q xx + | q | q = 0 , (1)where t and x are the time and spatial coordinate. Thescattering data can be found with the direct scatteringtransform (DST) based on the Zakharov–Shabat (ZS)equation [43] representing an auxiliary linear system fora vector wave function Φ = ( φ , φ ) T L Φ − ζ Φ = 0 , L = (cid:18) i∂ x − iq ( x ) − iq ∗ ( x ) − i∂ x (cid:19) , (2)where ζ = ξ + iη is the time-independent complex spectralparameter with real ξ and η , the superscripts T and thestar stand for a transposition and complex conjugation.Eq. (2) is typically solved for a fixed moment of time t with q ( x ) = q ( t , x ) playing the role of a potential. a r X i v : . [ n li n . PS ] A ug In case of potentials with compact support the wavefunction has the following asymptotics [44]:Φ | x →−∞ = ( e − iζx , T , Φ | x →∞ = ( ae − iζx , be iζx ) T . (3)The scattering coefficients a ( ζ ) and b ( ζ ) are connected tothe scattering data { ζ n , ρ n ; r } as follows: a ( ζ n ) = 0 , ρ n = b ( ζ n ) /a (cid:48) ( ζ n ); r ( ξ ) = b ( ξ ) /a ( ξ ) , (4)where { ζ n , ρ n } is a countable set of eigenvalues (discretespectrum) and associated norming constants, while r ( ξ )is the reflection coefficient defined on a real axis (contin-uous spectrum). Each eigenvalue ζ n = ξ n + iη n corre-sponds to a soliton with the amplitude 2 η n , group veloc-ity 2 ξ n while the position and phase are characterized by ρ n , see [1]. The condition a ( ζ n ) = 0 with { η n } > n = 1 , ..., N guaranties the decay of the wave functionaccording to asymptotics (3) leading to physically mean-ingful soliton eigenvalues { ζ n } [1]. At the same time, thecondition a ( ζ n ) = 0 can also be satisfied for { η n } < n = − , − , ... , see also [45], with the exponentiallygrowing wave function (3). We refer to these { ζ n } dis-tinguished by negative indexes n as nonphysical zeros of a ( ζ ) or virtual soliton eigenvalues, the number of whichcan be inifinite.With the inner product of two vectors (cid:104) Ψ , Φ (cid:105) = (cid:82) ∞−∞ Ψ ∗ T Φ dx we derive an eigensystem adjoint to (2): L † Φ † − ζ ∗ Φ † = 0 , L † = (cid:18) i∂ x iqiq ∗ − i∂ x (cid:19) , (5)where the adjoint operator satisfies the relation (cid:104) Φ † , L Φ (cid:105) = (cid:104)L † Φ † , Φ (cid:105) . Note that Φ † = ( φ ∗ , φ ∗ ) T satisfiesEq. (5).We are interested in the variation of { ζ n , ρ n } and r as-sociated with a small perturbation δq ( x ) of the potential.Let us take the variation of Eq. (2): δ (cid:0) L Φ − ζ Φ (cid:1) = ( δ L − δζ )Φ + ( L − ζ ) δ Φ = 0 . (6)To cancel out the second term in the last expression wetake the inner product of Eq. (6) with Φ † resulting in (cid:104) Φ † , ( δ L − δζ )Φ (cid:105) = 0. Extracting δζ , we end up with thefollowing expression [15]: δζ = (cid:104) Φ † , δ L Φ (cid:105)(cid:104) Φ † , Φ (cid:105) , δ L = − i (cid:18) δqδq ∗ (cid:19) , (7)where (cid:104) Φ † , δ L Φ (cid:105) = − i (cid:82) ∞−∞ ( φ δq ∗ + φ δq ) dτ and (cid:104) Φ † , Φ (cid:105) = (cid:82) ∞−∞ φ φ dτ .A deviation in ζ leads to small changes in Φ and as aconsequence to non-zero δa and δb as well as their deriva-tives with respect to ζ . To find the perturbation δρ , wetake the variation of ρ : δρ = δb/a (cid:48) − bδa (cid:48) /a (cid:48) = ρ ( δb/b − δa (cid:48) /a (cid:48) ) . (8)The values b and a (cid:48) at some { ζ n } are assumed to bealready known. According to boundary conditions, see (3), in order to obtain δb and δa (cid:48) we have to explore thevariation of the solution Φ and Φ (cid:48) = ∂ ζ Φ at x → ∞ .However, a multiplier e iζx in boundary conditions doesnot make it straightforward. We rewrite the ZS systemusing a new variable [46]: (cid:101) Φ = e iζ Λ x Φ , where Λ = diag(1 , − , (9)leading to the system ∂ x (cid:101) Φ = (cid:101) Q − (cid:101) Φ , (cid:101) Q ± = (cid:18) qe + ± q ∗ e − (cid:19) , (10)where we use the notation e ± = e ± iζx and (cid:101) Q ± . Wearrive to a modified set of boundary conditions: (cid:101) Φ | x →−∞ = (1 , T , (cid:101) Φ | x →∞ = ( a, b ) T . (11)This important simplification let us express δb and δa (cid:48) using variations δ (cid:101) Φ and δ (cid:101) Φ (cid:48) at x → ∞ : δ (cid:101) Φ | x →∞ = ( δa, δb ) T , δ (cid:101) Φ (cid:48) | x →∞ = ( δa (cid:48) , δb (cid:48) ) T . (12)Thus, to obtain δρ we have to compute δ (cid:101) Φ( x ) and δ (cid:101) Φ (cid:48) ( x ).Taking the variation of Eq. (10), we obtain: ∂ x δ (cid:101) Φ = (cid:101) Q − δ (cid:101) Φ + δ (cid:101) Q − (cid:101) Φ , (13)arriving to a nonhomogeneous equation for δ (cid:101) Φ. To de-rive the equation for δ (cid:101) Φ (cid:48) we first differentiate Eq. (10)with respect to ζ and then take the variation since theseoperations do not commute: ∂ x δ (cid:101) Φ (cid:48) = (cid:101) Q − δ (cid:101) Φ (cid:48) + δ (cid:101) Q − (cid:101) Φ (cid:48) + (cid:101) Q (cid:48)− δ (cid:101) Φ + δ (cid:101) Q (cid:48)− (cid:101) Φ . (14)According to (11), at x → −∞ zero boundary conditionshave to be imposed for δ (cid:101) Φ and δ (cid:101) Φ (cid:48) . The full expressionsfor the matrices are as follows: (cid:101) Q (cid:48)− = 2 ix (cid:101) Q + , δ (cid:101) Q − = (cid:18) δqe + − δq ∗ e − (cid:19) + 2 iδζx (cid:101) Q + ,δ (cid:101) Q (cid:48)− = 2 ix (cid:18) δqe + δq ∗ e − (cid:19) − δζx (cid:101) Q − . (15)We extend the treatment for Eqs. (13), (14) to find δb and δa (cid:48) appearing in Eq. (8). Using both independentsolutions of the ZS system [1], i.e. Φ = ( φ , φ ) T andΨ = ( ψ , ψ ) T = ( − φ ∗ , φ ∗ ) T | ζ = ζ ∗ , (16)we represent the solution of Eq. (13) as δ (cid:101) Φ = f ( x ) (cid:101) Φ + f ( x ) (cid:101) Ψ , (17)where variables with tilde are obtained in agreement withEq. (9) and f = ( f , f ) is to be determined. Substitut-ing this form of δ (cid:101) Φ to Eq. (13) and using Eq. (10) andthe notation W , we obtain: f (cid:48) (cid:101) Φ + f (cid:48) (cid:101) Ψ = W f (cid:48) = δ (cid:101) Q − (cid:101) Φ , W = ( (cid:101) Φ T , (cid:101) Ψ T ) . (18)The solution of Eq. (18) is as follows: f ( x ) = (cid:90) x −∞ W − ( y ) δ (cid:101) Q − ( y ) (cid:101) Φ( y ) dy, (19)where the integration constant is zero due to zero bound-ary conditions of δ (cid:101) Φ at x → −∞ . Using the expression(19), we recover the solution for δ (cid:101) Φ, see (17). A sim-ilar scheme can be applied to Eq. (14) with the form δ (cid:101) Φ (cid:48) = g (cid:101) Φ + g (cid:101) Ψ where for g = ( g , g ) we can obtain: g ( x ) = (cid:90) x −∞ W − ( δ (cid:101) Q − (cid:101) Φ (cid:48) + (cid:101) Q (cid:48)− δ (cid:101) Φ + δ (cid:101) Q (cid:48)− (cid:101) Φ) dy. (20)The perturbation of the reflection coefficient is ex-pressed as follows: δr = δb/a − bδa/a = r ( δb/b − δa/a ) . (21)Note that δr is defined on the real axis ξ , thus, δζ = 0 inEqs. (13), (14), see also expressions (15).As the basic unperturbed potential we consider a boxfunction q (cid:117) ( x ) = A for | x | < L/ A is a real-valued constant, while q (cid:117) = 0 otherwise. The scatteringcoefficients are as follows [47]: a ( ζ ) = e iLζ (cid:0) cos( χL ) − iζ sin( χL ) /χ (cid:1) , (22) b ( ζ ) = − A sin( χL ) /χ, χ = (cid:112) A + ζ . (23)The wave function in the region | x | < L/ e iζL ν − ν (cid:18) ν − e − iµ − νe iµ e − iµ − e iµ (cid:19) , (24) ν = i ( ζ − χ ) /A, µ = χL/ χx, (25)while for | x | > L/ a ( ζ n ) = 0 for both cases { η n } > { η n } < χ n L ) = − iχ n /ζ n , χ n = (cid:112) A + ζ n . (26)Using (22)–(26) we express norming constants (4) as: ρ n = − iχ n e − iζ n L / [ A (1 − iζ n L )] . (27)Note that the number of solitons in the box is limitedby N = Integer[1 / AL/π ] and all the eigenvalues arealinged on the imaginary axis, i.e. { ζ n } = i { η n } for n = 1 , .., N and the solitons have zero velocities, see [45,48, 49].First we study the evolution of the roots of e − iLζ a ( ζ )according to Eq. (22) for different values of A and L = 10, see Fig. 1. In addition to physically meaningfulzeros with { η n } > { η n } < A from 0 . .
778 two symmetric negative zeros approachthe imaginary axis and stick together at a slightly higher A forming a degenerate root. Further they move apart A = ξη - - A = ξη - - A = ξη - - A = ξη - - A = ξη - - A = ξη - - Figure 1: Contours of | e − iLζ a ( ζ ) | for different values of A and L = 10. Red-in-white points denote the roots of thisexpression – physical { η n } > { η n } < A . along the imaginary axis, see the case with A = 0 . A = 0 .
785 and 0 .
83. At higher A the next pair of negative roots approach the imaginaryaxis in the lower half of the ζ -plane, see A = 1 .
03, resem-bling the initial situation with A = 0 .
7. Fig. 1 illustratesa situation when a simple box-like perturbation movesa nonphysical zero to the region η > δq re = ε cos( kx + ϕ ) , δq im = iδq re , (28)where ε is a small parameter while k and ϕ are the wave-length and phase, respectively. To calculate δζ n and δρ n caused by δq as in (28) we use the explicit form ofthe wave function (24) in the relation (7) and equations(13), (14) with the conditions (26) employed for algebraicsimplifications. The following exact expressions are ob-tained: δζ re n = iεh re ( k, ζ n ) cos ϕ, δζ im n = εh im ( k, ζ n ) sin ϕ, (29) δρ re n = iε [ s re1 ( k, ζ n ) cos ϕ + s re2 ( k, ζ n ) sin ϕ ] , (30) δρ im n = ε [ s im1 ( k, ζ n ) cos ϕ + s im2 ( k, ζ n ) sin ϕ ] , (31)with the real-valued functions h re / im and s re / im1 , givenin the Appendix as well as the derivation details, ex-plicit expressions for δr re / im and verification of these re-sults using numerical DST tools [50, 51] for the potential q (cid:117) (1 + δq re / im ). Note that according to (29)-(31), δq re changes only imaginary parts of ζ n and ρ n , while δq im af-fects on their real pars. The formulas (29)-(31) are valid Re δζ × Im δζ × - - - - - - Re ( δρ ρ - ) × Im δρ ρ × - - - - - - Figure 2: Soliton scattering data deviations induced by 10 realizations of noise superimposed on the box with A = 1 and L = 12 computed numerically. The insets show numerical(green solid) and theoretical (grey dashed) PDFs for solitonparameters. The unperturbed values of ζ and ρ are com-puted using (26) and (27). for both physical and virtual soliton eigenvalues. In thelatter case they describe the migration of nonphysical ze-ros which might result in a birth of a new soliton, similarto the situation illustrated in Fig. 1.We consider a sum of modes (28) with random phasesand distributed as F ( k ) with respect to k . Integrating(29)-(31) over ϕ , we obtain the following expressions forstandard deviations:( σ re / im ζ,n ) = ε (cid:90) ∞−∞ F ( k ) | h re / im | dk, (32)( σ re / im ρ,n ) = ε (cid:90) ∞−∞ F ( k ) (cid:16) | s re / im1 | + | s re / im2 | (cid:17) dk. (33)These expressions describe the effect of noise on the dis-crete spectrum for the box potential. The convergence of(32), (33) is guaranteed by an algebraic decay of δζ n and δρ n for large k . These integrals were evaluated analyti-cally for the white noise model, i.e. F ( k ) = 1. The resultfor σ ζ,n has a compact form (see Appendix for details):( σ re ζ,n ) = πε χ n ( η n + 2 LA )2 A (1 + Lη n ) + 3 πε η n Lη n ) , (34)( σ im ζ,n ) = πε χ n ( η n + LA ) / [2 A (1 + Lη n ) ] , (35)while σ ρ,n is rather cumbersome and omitted in the text.For a direct comparison with analytical results for σ ζ,n and σ ρ,n we simulated a white noise signal as the fol-lowing normalized collection of M modes with randomphases ϕ ns ,jm : δq ns ,j ( x ) = ε √ ∆ k M (cid:88) m =1 cos( x ∆ km + ϕ ns ,jm ) , (36)where the subscript ‘ns , j ’ denotes a particular j th setof random phases. For each case of 10 realizations ofthe complex-valued noise δq ( x ) = δq ns , (1 + iδq ns , ) with ε = 0 . k = 0 . M = 200 superimposed ontop of the box potential with L = 12, A = 1, we com-puted eigenvalues and norming constant using both the Figure 3: (a) The shift of the nonphysical root to the up-per ζ -plane for a particular realization of a real-valued noisewithin a numerical (green solid) and theoretical (grey dashed)amplitude PDF. Contour plots of | q ( t, x ) | from numerical sim-ulations of NLSE. Evolution of q (cid:117) perturbed by real (b, d)and imaginary (c, e) noise when a soliton is induced and not,respectively. The theoretically predicted and numerically ob-tained parameters of the induced soliton are ζ pr = 0 . i , ζ num = 0 . i with ρ pr = − . i , ζ act = − . i . developed perturbation theory and numerical DST, seeAppendix. Fig. 2 shows statistical results for the scat-tering data for the first (largest) soliton out of N = 4.The Gaussian probability density functions (PDFs) withtheoretical standard deviations (33)-(35) accurately de-scribe the corresponding numerical data.Our theory applied to the nonphysical zeros predicts abirth of a noise-induced soliton. As an example we con-sider a box potential with L = 1 .
46 and A = 1 with nosolitons and the largest zero ζ − = − . i . We used 10 realizations of a real-valued noise (36) which affects onlyimaginary part of the virtual eigenvalue with ε = 0 . k = 0 . M = 500. Fig. 3(a) shows theoretical andnumerical PDFs for the noise-induced values of η − withthe tail η > δq ns , , see Fig.3(b), which shifts the nonphysical zero (green dot) tothe upper ζ -plane (red dot) and compute its temporalevolution numerically using NLSE (1) and a standardRunge–Kutta method (see Appendix) with the initialcondition q (0 , x ) = q (cid:117) (1 + δq ns , ). A second computa-tion is performed for the evolution of the initial condi-tion q (0 , x ) = q (cid:117) (1 + iδq ns , ), shown in Fig. 3(c). Notethat we slightly smoothed q (cid:117) on the edges for numer-ical simulations, see Appendix for details. Figs. 3(d,e)show the spatio-temporal contour plots of | q ( t, x ) | reveal-ing the presence of a strong soliton in the first case, seeFig. 3(d) for parallel contour levels, while in the secondcase the contours indicate simple decay of the continuousspectrum potential as expected [1]. Similarly one can de-scribe migration of the physical root to the nonphysicalregion, i.e. soliton disappearance.In this work we presented a complete theoretical frame-work to evaluate first-order corrections of the full setof scattering data within the NLSE model and appliedit to a classic box potential, which can be generalisedto other integrable systems and wavefields. In additionto the classical result for eigenvalues as in (7), we de-rived general expressions, see Eqs. (13), (14), leading tothe knowledge of soliton phase and position sensitivity.Starting from a single Fourier mode we obtained statisti-cal integrals (32), (33) allowing to determine the impactof a random-phase noise on soliton parameters which isimportant in the studies of the spontaneous modulationinstability [52–54] and in a number of applications. Theintroduction of a concept of a virtual soliton with non-physical zeros of a ( ζ ) allowed us to accurately predictthe noise-induced emergence of a soliton. A similar con- cept to describe soliton emergence can be further devel-oped for the NLSE model with external pumping, see[42, 55, 56]. Acknowledgments – First part of the work was sup-ported by Russian Science Foundation grant No. 19-79-10225 (RM for derivation of the perturbation frame-work). Second part was supported by Russian ScienceFoundation grant No. 20-71-00022 (AG for the workon the noise-induced effects). The authors thank DrD. Agafontsev for fruitful discussions on virtual solitoneigenvalues. Statistical simulations were performed atthe Novosibirsk Supercomputer Center (NSU). [1] S. Novikov, S. Manakov, L. Pitaevskii, and V. Za-kharov,
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