Soliton trapping, transmission and wake in modulationally unstable media
Interactions between solitons and the coherent oscillation structures generated by localized disturbances via modulational instability are studied within the framework of the focusing nonlinear Schrödinger equation. Two main interaction regimes are identified based on the relative value of the velocity of the incident soliton compared to the amplitude of the background: soliton transmission and soliton trapping. Specifically, when the incident soliton velocity exceeds a certain threshold, the soliton passes through the coherent structure and emerges on the other side with its velocity unchanged. Conversely, when the incident soliton velocity is below the threshold, once the soliton enters the coherent structure, it remains confined there forever. It is demonstrated that the soliton is not destroyed, but its velocity inside the coherent structure is different from its initial one. Moreover, it is also shown that, depending on the location of the discrete eigenvalue associated to the soliton, these phenomena can also be accompanied by the generation of additional, localized propagating waves in the coherent structure, akin to a soliton-generated wake.
pacs:05.45.-a, 02.30.Ik, 05.45.Yv, 42.65.Sf, 47.20.-k
The dynamics of nonlinear media affected by modulational instability displays a number of interesting nonlinear phenomena ZO2009 . In particular, modulational instability (i.e., the instability of the background with respect to long wavelength perturbations) is the main mechanism for supercontinuum generation DudleyTaylor , integrable turbulence zakharov2009 ; az2014 ; randoux2014 , and the formation of rogue waves onoratoosborne ; solli ; naturephys . Modulational instability is described quantitatively by the one-dimensional focusing nonlinear Schrödinger (NLS) equation, which is a universal model for the evolution of weakly nonlinear dispersive wave packets, and as such arises in such diverse fields as water waves, plasmas, optics and Bose-Einstein condensates AS1981 ; Agrawal2007 ; IR2000 ; NMPZ1984 ; PS2003 . Indeed, by linearizing the focusing NLS equation around the background, one can find the range of unstable Fourier modes as well as their growth rate benjaminfeir . But the linearization cannot capture the dynamics once the perturbations become comparable with the background, which is referred to as the nonlinear stage of MI.
A qualitative explanation proposed for the nonlinear stage of modulational instability is the formation of solitons (in particular the so-called “superregular breathers” zakharovgelash ; gelashzakharov ; kibler ). As shown in biondinifagerstrom , however, there exist broad classes of perturbations of the constant background that generate no solitons. Therefore, solitons cannot be the main vehicle for the instability (because all generic perturbations are linearly unstable, whereas for some perturbations no solitons are present in the solution). Instead, in biondinifagerstrom we showed that the signature of the instability lies in the nonlinear analogue of the unstable Fourier modes. In biondinimantzavinos ; CPAM2017 we then characterized the nonlinear stage of modulational instability for localized perturbations of the constant background. We showed that, generically, the nonlinear stage of modulational instability displays universal behavior, with the -plane decomposing into two quiescent, or “plane wave” regions, where the solution asymptotically equals the background up to a phase, separated by a central region in which the leading order behavior is described by a slow modulation of the periodic, traveling wave solutions of the focusing NLS equation. In biondinilimantzavinos we further characterized the details of the asymptotic state and showed that, for large times, the solution in the modulated oscillation region becomes a coherent collection of classical (i.e., sech-shaped) solitons of the NLS equation. Moreover, in SIREV we demonstrated that this kind of behavior is not limited to the NLS equation, but is instead a more general feature of nonlinear dynamical systems subject to modulational instability.
A key requirement in biondinimantzavinos ; CPAM2017 ; biondinilimantzavinos , however, was that no solitons be present in the initial conditions. A natural and important question is thus what happens when both solitons and a localized disturbance are simultaneously present. The purpose of this work is to answer this question. We do so by studying the interactions between solitons and the coherent oscillation structures generated by localized disturbances due to modulational instability, which for brevity hereafter we refer to as the “wedge”. We identify three interaction regimes based on the location of the discrete eigenvalue that gives rise to the soliton: soliton transmission, soliton trapping, and a mixed regime in which the soliton transmission or trapping is accompanied by the generation of additional, localized contributions in the wedge, which can be considered a soliton-generated wake.
NLS, solitons and modulational instability.
The starting point for our study is the focusing NLS equation,
where is the complex envelope of a quasi-monochromatic, weakly nonlinear dispersive wave packet, and the physical meaning of the variables and depends on the physical context. (E.g., in optics, represents propagation distance while is a retarded time.) Here [with ] is the amplitude of the nonzero background (NZBG). The constant background solution of Eq. (1) is simply . Equation (1) is the compatibility condition of the Lax pair ZS1972 and , with and , where is the third Pauli matrix, and
The first half of the Lax pair and are referred to as the scattering problem and the potential, respectively.
The inverse scattering transform allows one to solve the initial-value problem for Eq. (1) by associating to time-independent scattering data via the solutions of the scattering problem. Once the scattering data are obtained from the initial condition, is reconstructed in terms of them by inverting the scattering transform. Specifically, the nonlinearization of the Fourier modes are the Jost eigenfunctions , which are the solutions of the Lax pair that reduce to plane waves as . In the case of NZBG,
where , and
The Jost solutions are defined over the continuous spectrum , which is the image of the Fourier wavenumbers. In particular, the range is the image of the modulationally unstable Fourier modes biondinifagerstrom . The scattering matrix is defined by the scattering relation
and the reflection coefficient is the nonlinearization of the Fourier transform (see Appendix for details).
As usual, each discrete eigenvalue, if present, generates a soliton. In the case of NZBG, the velocity of a soliton generated by a discrete eigenvalue at in the absence of localized disturbances is biondinikovacic , with and , implying and .
When no solitons are present, it was shown in biondinimantzavinos ; CPAM2017 that an initial disturbance localized near generates a coherent oscillation structure confined to the wedge-shaped region , whereas outside this region the solution remains equal to the background value up to a phase. The same approach as in CPAM2017 can be used to show that these features remain the same when solitons are present.
Transmission, trapping and soliton wake.
To study the interactions between solitons and localized disturbances, we performed careful numerical simulations of the focusing NLS equation (1) with a variety of initial conditions consisting of a localized disturbance of the constant background initially placed at together with a soliton with initial velocity (generated by a corresponding discrete eigenvalue ) and initially placed at a location . The kinds of possible outcomes are shown in Fig. 2 and Fig. 2(b) in three representative cases corresponding to different choices for . Those in Fig. 2 result in a soliton transmission and a soliton trapping, respectively, whereas that in Fig. 2(a) results in a mixed outcome in which the soliton transmission is accompanied by the generation of additional contributions inside the wedge. As we show next, all of these phenomena can be accurately described quantitatively by computing the long-time asymptotics of solutions.
Solitons and long-time asymptotics.
where is the solution of a matrix Riemann-Hilbert problem Gakhov ; TrogdonOlver defined in terms of the reflection coefficient, and, when discrete eigenvalues are present, the corresponding poles and associated norming constants biondinikovacic ; CPAM2017 (see Appendix for details).
As usual whitham ; AS1981 , we compute the long-time asymptotics along lines with const. The difference between the present scenario and the one in biondinimantzavinos ; CPAM2017 is the additional presence of poles in the Riemann-Hilbert problem coming from the discrete spectrum, i.e., the discrete eigenvalue at that produces the soliton. Next we briefly discuss how solitons arise in the calculation of the long-time asymptotics.
Both the jump condition and the residue conditions contain the phase , with biondinikovacic ; CPAM2017 (see Appendix for details). As in dkkz1996 , one can show that when the reflection coefficient is zero the poles give a vanishingly small contribution to as for all such that . Conversely, when is such that , the poles give an contribution to the solution. As a result, the soliton velocity is simply the value of such that . When (i.e., with zero background), the phase reduces to . In this case the above criterion recovers the familiar expression , with . When instead (i.e., with NZBG), the same criterion yields , with , which coincides with the expression given earlier. Thus, one can identify the soliton velocity without computing the solution of the NLS equation.
Figure 2(b) shows the contour lines of the soliton velocity in the spectral plane with NZBG. Note that the curves of constant soliton velocity are simply given by for different values of . These curves touch the real -axis twice for , once for and never for . This feature affects the calculation of the long-time asymptotics. Also, Fig. 2(b) shows that the contour line divides the spectral plane into three domains: (gray region), where extending to infinity; (dark and light blue regions, respectively and ), where ; and (orange region), where touching the segment . Below we show that these regions correspond to the locations of discrete eigenvalues that result in soliton transmission, trapping, and the mixed regimes.
Consider a discrete eigenvalue . Recall that the boundary of the wedge is given by the lines , and the range is the plane wave region. The value lies in this range. Even in the presence of a discrete eigenvalue, for the calculations proceed as in CPAM2017 , and the discrete eigenvalue yields no contribution to the solution in the long-time asymptotics. Conversely, for it yields a leading-order contribution that results in the soliton, as before. In other words, the long-time asymptotics predicts that the soliton appears as a localized traveling object outside the wedge, and that the soliton velocity after the interaction coincides with , in perfect agreement with the numerical results. Moreover, the long-time asymptotics also recovers the asymptotic phase difference of the solution as , consistently with the constraint coming from the discrete eigenvalue biondinikovacic .
Soliton trapping and velocity change.
The outcome changes when the soliton is generated by . In this case , hence now the value occurs inside the wedge. Thus, the long-time asymptotics now predicts that no soliton is present in the plane wave region, consistently with the numerics. A natural question is then whether the soliton is destroyed by the interaction or whether it persists inside the wedge. To this end, Fig. 3(a) shows the difference between the amplitude of a solution with a soliton present and that of a solution in which the soliton is absent. The permanent change across the wedge in Fig. 3(a) clearly demonstrates the persistent presence of the soliton trapped inside the structure.
At the same time, Fig. 3 also clearly shows that the soliton velocity inside the wedge differs from . To understand this phenomenon, recall that the calculation of the long-time asymptotics changes for biondinimantzavinos ; CPAM2017 . Specifically, in that range the controlling phase in the Riemann-Hilbert problem must be modified in order to regularize the problem, and one must replace with a new phase function defined in terms of Abelian integrals CPAM2017 (see Appendix for details). It is thus , not , that controls the soliton velocity inside the wedge. More precisely, setting yields an implicit equation that determines (see Appendix for details). As shown in Fig. 3(a), the value of predicted by the long-time asymptotics is in excellent agreement with the numerical results. Figure 4(a) shows a plot of as a function of . Note that is always less than , meaning that the soliton always slows down as a result of the trapping. The change in the soliton velocity can be interpreted as the result of the interaction between the soliton and the infinite number of sech-like peaks inside the wedge biondinilimantzavinos . At , the solution is locally a nonlinear superposition of a soliton and a periodic solution of the NLS equation, similarly to trogdondeconinck . Note that does not depend just on , as demonstrated by Fig. 4, which shows solitons with the same but generated by different eigenvalues yield different values of in general.
Mixed regimes and soliton wake.
Yet a different scenario arises when or . When , we again have , implying occurs in the plane wave region, so again we expect a soliton transmission with an unchanged soliton velocity after the interaction. The twist, however, is that when the implicit equation also has a solution for . Thus, the long-time asymptotics predicts that a single discrete eigenvalue generates two distinct contributions to the long-time behavior, propagating at different velocities: and . We are unaware of any previous instances in which a similar phenomenon was reported in any physical context.
The above predictions are borne out by the numerical simulations, as shown in Fig. 2(a) and Fig. 3(b). Both figures clearly show that the soliton is transmitted through the wedge, but at the same time a soliton-generated wake is clearly visible inside the wedge. Also, Fig. 3(b) demonstrates once more an excellent agreement with the prediction of the velocity of the soliton-generated wake coming from the long-time asymptotics. A plot of the velocity of the soliton-generated wake as a function of is shown in Fig. 4(b). Physically, the difference between and in is that, for the same amplitude and velocity, eigenvalues in always lead to much broader solitons than eigenvalues in (cf. Appendix).
A similar situation arises when , except that no soliton arises outside the wedge in the long-time asymptotics, and two contributions are generated inside the wedge. That is, twice for : once for , corresponding to the trapped soliton, and again for , corresponding to the soliton-generated wake (cf. Appendix).
In summary, we presented a study of nonlinear interactions between solitons and localized disturbances in focusing media with nonzero background, we classified the possible outcomes, which include soliton transmission, soliton trapping and the generation of a soliton wake, and we identified precise conditions that determine the velocity of the trapped solitons and the soliton wake.
We emphasize that in the pure transmission regime, the interaction is “clean”. That is, no permanent residuals are left in the wedge, and (apart from a constant phase shift) the solution for is virtually undistinguishable from one without the soliton (as confirmed by the analogue of Fig. 3 for a soliton transmission supplement ). The same is true in the trapping regime except for the presence of the soliton inside the structure (cf. Fig. 2). Conversely, in the mixed regimes the wedge acquires additional permanent localized traveling objects, and interacts nonlinearly with them. The analytical predictions for the velocity of the soliton-generated wake remain valid in the limit when the discrete eigenvalue touches the segment , in which case the soliton is replaced by an Akhmediev breather and supplement .
Soliton interactions with NZBG in the absence of localized disturbances have recently been studied in gelash ; LiBiondini . Interactions between solitons and localized disturbances had been previously studied in the case of zero background AS1981 ; NMPZ1984 ; gordon ; kuznetsovmikhailov , but in that case they only lead to small effects, even in modulationally unstable media. It is only in focusing media with NZBG that dramatic effects such as a soliton velocity change and a soliton-generated wake arise.
The phenomena discussed here are related to interactions between solitons and dispersive shock waves (DSWs) elhoefer . Soliton tunneling was recently studied in the context of DSWs in defocusing media in sprenger . Also, soliton trapping by an initial discontinuity was recently studied in maiden ; ablowitzluocole . Note, however, that phenomena governed by the Korteweg-deVries (KdV) equation or the defocusing NLS equation are very different from those described by the focusing NLS equation. In both the KdV equation and the defocusing NLS equation, the soliton velocity is proportional to its amplitude. Thus, the soliton velocity and amplitude decrease as the soliton travels up the ramp generated by the discontinuity, with the soliton eventually disappearing altogether. In the focusing NLS equation, in contrast, the soliton velocity and the soliton amplitude are entirely decoupled (i.e., independent of each other), the mechanism which drives the change in the soliton velocity is completely different, and the soliton amplitude and velocity inside the wedge are constant.
We also emphasize that the applicability of our results is expected to be very broad, since, similarly to biondinimantzavinos ; CPAM2017 ; biondinilimantzavinos ; SIREV , the results are essentially independent of the specific details of the initial localized disturbance. Moreover, since the NLS equation arises in many physical contexts, including nonlinear optics, deep water waves, acoustics, plasmas and Bose-Einstein condensates, the results of this work apply to all of these areas. (Recall that suitable scalings to observe NLS dynamics in each of these domains are well known, e.g., see DudleyTaylor ; AS1981 ; Agrawal2007 ; IR2000 ; PS2003 .) Finally, since the results of biondinimantzavinos were shown in SIREV to extend to a broader class of modulationally unstable systems, we expect that the same will apply in this case. In particular, nonlinear optical fibers and gravity waves in one-dimensional deep water channels are especially promising candidates for the experimental verification of the phenomena described here.
We thank M. J. Ablowitz, M. A. Hoefer, M. Onorato and S. Trillo for many interesting discussions. This work was partially supported by the National Science Foundation under grant numbers DMS-1614623 and DMS-1615524.
Here we give a few details of the inverse scattering transform (IST) for the focusing NLS equation with NZBG, the wedge structure, the calculation of the long-time asymptotics, the numerical methods used and some further numerical results.
IST with NZBG: Direct problem.
The direct problem in the IST consists in computing the scattering data (i.e., reflection coefficient, discrete eigenvalues and norming constants) from the potential . This is done through the Jost eigenfunctions , which are the simultaneous matrix solutions of both parts of the Lax pair that reduce to plane waves as . In the case of NZBG, biondinikovacic , they are given by Eq. (3), where and are respectively the eigenvalues and corresponding eigenvector matrices of . The value of is uniquely determined for all by requiring that the branch cut is on , is continuous from the right on the cut, and that for . These Jost eigenfunctions, which are the nonlinearization of the Fourier modes, are defined for all values of such that , namely the continuous spectrum . The reflection and transmission coefficients are and , respectively. The zeros of and define the discrete spectrum of the problem, which leads to solitons. As usual, the time evolution in the IST is trivial. In particular, with the above normalization of the Jost eigenfunctions, all the scattering data are independent of time biondinikovacic .
IST with NZBG: Inverse problem.
The inverse problem in the IST consists in reconstructing the solution of the NLS equation from the scattering data, and is formulated in terms of a Riemann-Hilbert problem, namely the problem of finding the sectionally meromorphic matrix which in terms of the direct problem is given by biondinimantzavinos ; CPAM2017
where are the upper half and lower half of the complex -plane, respectively, for denote the columns of and . More precisely, the RHP consists in computing from the knowledge of the jump condition,
where superscripts denote projection from the left/right of the contour (oriented rightward along the real -axis and upward along the segment ), together with the normalization as , residue conditions at the discrete eigenvalues and suitable growth conditions at the branch points BilmanMiller . Note that for . The jump matrix, obtained using the scattering relation and symmetries, is CPAM2017 \cref@addtoresetequationparentequation
with and , where the asterisk denotes complex conjugation. The residue condition at the poles induced by the discrete eigenvalues are biondinikovacic \cref@addtoresetequationparentequation
being an arbitrary, complex-valued norming constant.
As shown in biondinifagerstrom , the signature of MI in the inverse problem is the exponentially growing entries of for through the time dependence of .
The coherent oscillation region.
As shown in biondinilimantzavinos ; CPAM2017 , when no solitons are present the leading-order solution in the coherent oscillation region is expressed in terms of Jacobi elliptic functions, and represents a slow modulation of the traveling wave (periodic) solutions of the focusing NLS equation. In particular,
where the elliptic parameter and the constants are \cref@addtoresetequationparentequation
and the slowly varying offset is explicitly determined by the reflection coefficient. The four points , and are the branch points of the elliptic solutions of the focusing NLS equation. In particular, is a slowly varying function of determined implicitly via the system of modulation equations el ; kamchatnov \cref@addtoresetequationparentequation
where and are the complete elliptic integrals of the first and second kind NIST , respectively. The detailed properties of the asymptotic state in the coherent oscillation region were characterized in biondinilimantzavinos . When , Eqs. (13) yield and ; when , one has and . The trajectory described by in the complex -plane for is given by the red curve in Fig. 2 in the main text, which delimits the upper and lower portions of , respectively denoted .
Long-time asymptotics with a discrete spectrum.
Consider the situation in which a discrete eigenvalue at is present in the spectrum. Owing to the invariance of the NLS equation under spatial reflections, it is enough to consider the case in which is in the first quadrant of the spectral plane, implying . Note that the asymptotic results are independent of whether the sign of the initial soliton position is positive or negative, the only difference being whether the interaction between the soliton and the localized disturbance occurs at positive versus negative times.
Let and , with . As in CPAM2017 , the calculation of the long-time asymptotics differs depending on whether . When , one can deform the RHP to remove the exponentially growing jumps without introducing additional branch cuts or modifying the phase function. Conversely, when , in order to remove the exponential growth one must introduce an additional branch cut and replace with a new phase function defined by the Abelian integral CPAM2017 \cref@addtoresetequationparentequation
with . Without loss of generality, we take the two branch cuts of respectively along the segment of the imaginary axis from to and along the curve connecting to CPAM2017 .
The only difference between the long-time asymptotics in our case and that in CPAM2017 arises when in the plane wave region or in the wedge. Those are the only values of at which the poles in the RHP give an contribution to the solution. Conversely, as in the case of zero background, for all values of in the plane wave region such that , and all values of in the wedge such that , the poles give an exponentially small contribution to the solution as .
As discussed in the main text, the condition is satisfied for when the discrete eigenvalue lies in (transmission region) or (wake region), leading to the appearance of a soliton in the plane wave region in those cases, but not when (trapping region), leading to the absence of a soliton in the plane wave region in that case. It therefore remains to examine whether the condition is satisfied for in each of these three cases. We discuss this issue next.
Implicit equation for the soliton velocity in the wedge.
The integrals in Eq. (14) can be carried out and expressed in terms of incomplete elliptic functions. On the other hand, we found it more convenient to just evaluate numerically the imaginary part of , which can be shown to be simply
The value of is then computed numerically with standard root finding algorithms.
Figure 5 shows the value of as a function of in the wedge (i.e., in the range ) for a few representative values of in the transmission, trapping and mixed regimes. From these plots we see that the equation has no solution when the discrete eigenvalue is in , i.e., in the pure transmission regime. Conversely, exactly once when , corresponding to the trapped soliton, and when , corresponding to the soliton-generated wake. Finally, twice when , with one zero corresponding to the trapped soliton and the other to the wake. Note that is always less than in absolute value, and whenever .
All numerical simulations of the NLS equation were performed using an eighth-order Fourier split-step method yoshida with periodic boundary conditions and grid points. The spatial domain used was much larger than the spatial window shown in the figures, so that the phase discontinuity at the edge of the domain generated by the soliton did not affect the solution in the plot window. Specifically, we took with , implying a spatial grid size of . The initial disturbance was realized by taking near . The time integration was performed with an integration step size of . This setup allowed us to obtain accurate results until about , at which point roundoff errors become .
Further numerical results.
Figure 8(a) shows the amplitude difference between the solution in Fig. 2(a) of the main text and a solution without the corresponding soliton, clearly demonstrating that no permanent effects remain in the wedge. The right panel of Fig. 8 shows a comparison between the width of two solitons with the same amplitude and velocity, one generated by a discrete eigenvalue and the other by , demonstrating that eigenvalues in always generate broader solitons than those in .
For completeness, Fig. 8 shows an interaction with a soliton generated by a discrete eigenvalue in , in which case the interaction results in a soliton trapping and a soliton-generated wake, and Fig. 8 shows the amplitude of the solutions in Fig. 4 of the main text. Finally, we note that the long-time asymptotics results also apply if the soliton is replaced by an Akhmediev breather, in which case one simply obtains .
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