Introduction

Rogue waves are a broad multidisciplinary subject of research with areas spanning from oceanography1 to hydrodynamics2,3, optics and photonics4,5,6, acoustics7, Bose-Einstein condensation8, and artificial intelligence9,10. From a microscopic perspective, they can be referred to as a class of deterministic rational solutions of the integrable nonlinear partial differential equations, which entail a doubly-localized peak on a finite background11,12. The well-known Peregrine soliton is an example of such deterministic rational solutions13, which provides an appealing paradigm to confront the mystique surrounding realistic rogue waves14,15,16,17,18,19. So far, Peregrine soliton events have been experimentally observed in such physical settings as the water-wave tank2, deep-water flumes3, optical fibres20, plasmas21, and irregular oceanic sea states22.

The relation of rogue wave states to integrable systems can be virtually reminiscent of that of the topological edge modes to Hermitian systems23,24. For instance, rogue wave states are exact solutions of the integrable models obtained with inverse scattering transform25 or Darboux transformation12, while gapless edge modes can be analytically obtained by solving the eigenproblem of Hermitian Hamiltonian26,27. Although rogue waves and topological edge modes are two different wave localization phenomena, both of them are immune to perturbations (more precisely, the former always come with perturbations17, while the latter are topologically protected against perturbations28), and can exist for a broad range of system parameters24. Further, it has been demonstrated, both theoretically and experimentally, that rogue waves are seen in either integrable2,11,15 or non-integrable4,5,29 systems as often as the topological edge modes are in either Hermitian23,24 or non-Hermitian27,30 systems. Interestingly, a direct application of topological concept into rogue wave phenomena so as to realize topological control of extreme waves has also been recently reported31. As is well known, the basic principle behind the topological edge modes is the bulk-boundary correspondence32,33, which relates the robust edge states to the topological invariants of the bulk34,35,36. Now a natural question arises: What is the genuine mechanism underlying the rogue wave formation? Or equivalently, is there a similar correspondence principle intended for the rogue waves in integrable systems? An affirmative answer to this question will be provided in this paper.

In fact, there have been intense research activities on the mechanisms or physics behind the rogue wave events in the past decade. Numerous studies37,38,39,40,41,42,43,44,45,46 show that a possible mechanism for such rogue formation is the modulation instability (MI) (sometimes called Benjamin-Feir instability47), which traces the growth of periodic perturbations on an unstable continuous-wave (cw) background seeded by noises. In the initial stage of MI, the spectral sidebands associated with the instability undergo exponential growth, but subsequently a dramatic energy exchange may occur among multiple spectral modes, resulting in a series of high-contrast peaks of random intensity where Peregrine soliton structures can be identified12,17. In addition to MI, there are also emerging several other explanations for rogue wave generation, e.g., integrable turbulence48,49, breather or soliton collisions50,51,52, random superposition of Stokes waves53, and the spatial asymmetry or inhomogeneity54,55. These investigations provide a good perspective on the fundamental origin of rogue waves, but how to clarify such a controversial issue is still an open challenge of this emerging field.

In this work, we unveil, on an analytic level, a direct kinship between the baseband MI39 of cw backgrounds and the rogue wave solutions of integrable models, which goes beyond the prediction of the domains of existence for rogue waves12,40. Here, by baseband MI we mean the instability experienced by the cw background in a region where the perturbations feature infinitesimally small frequencies12. Such MI-rogue wave correspondence is an analog of the bulk-boundary correspondence in topological physics, with rogue waves corresponding to the boundary modes and MI corresponding to the bulk topological invariant, respectively. It applies universally to almost all integrable systems, even to those whose rogue wave solutions can not be expressed in terms of elementary functions. To support the universal scope of our results, we consider a model of N-component generalized nonlinear Schrödinger (GNLS) equation, which includes the celebrated NLS56 and Manakov57,58,59 models as well as their N-component versions60,61,62 as special cases. Within this framework, we establish the one-to-one correspondence between the rogue wave solutions and the MI baseband spectra, and numerically confirm it by demonstrating the excitation of three coexisting Peregrine solitons from a turbulent wave field63 for the specific N = 3 case. Meanwhile, the universality of this MI-rogue wave correspondence has been corroborated with various integrable models. As is expected, aside from offering a way of obtaining exact rogue wave solutions from the MI analysis, our work unequivocally substantiates the insight that MI is the most fundamental mechanism underlying the rogue wave formation.

Results and discussion

GNLS model and MI evolution equation

For our studies, we write the N-component GNLS model in dimensionless form

$${{{{{{{\rm{i}}}}}}}}\frac{\partial {u}_{j}}{\partial z}+\frac{\sigma }{2}\frac{{\partial }^{2}{u}_{j}}{\partial {t}^{2}}+\left(\mathop{\sum }\limits_{m=1}^{N}| {u}_{m}{| }^{2}\right)\left({u}_{j}+{{{{{{{\rm{i}}}}}}}}\gamma \frac{\partial {u}_{j}}{\partial t}\right)=0,$$
(1)

where j = 1,..., N and uj(z, t) is the normalized complex envelope of the jth-component wave field, with z and t the independent variables. The real parameter σ represents the type of dispersion, i.e., the anomalous dispersion for σ > 0 and the normal dispersion for σ < 0. The third term combines the self- and cross-phase modulation effect, whose coefficient has been normalized to unity, and the self-steepening effect defined by the parameter γ64,65. In quadratic nonlinear media, the self-steepening seen by ultrashort pulses is controllable in both sign and magnitude and thus provides a new degree of freedom wherever shock formation is significant66,67,68. We should emphasize that the multiple-component model (1) is of considerable physical interest not only because it includes the celebrated NLS56 and Manakov57,58,59 models as well as their integrable extensions46,69,70,71, but also because it itself (N ⩾ 3) may find applications in such fields as wavelength division multiplexing72, Bose-Einstein condensation73, and Davydov solitons in biophysics74. More fundamentally, the model (1) owns complete integrability (also known as complete solvability)75 and all complicated rogue wave signatures known so far46, and is therefore appropriate for investigating the intriguing MI-rogue wave correspondence.

Let us first derive the MI evolution equation based on the plane-wave solutions of Eqs. (1),

$${u}_{j0}={a}_{j}\exp ({{{{{{{\rm{i}}}}}}}}{\omega }_{j}t-{{{{{{{\rm{i}}}}}}}}{k}_{j}z),\quad j=1,\cdots \,,N,$$
(2)

where the amplitudes aj ∈ R, the wavenumbers kj ∈ R, and the frequencies ωj ∈ R obey the dispersion relations:

$${k}_{j}=A{\upsilon }_{j}+\frac{1}{2}\sigma {\omega }_{j}^{2},\quad {\upsilon }_{j}=\gamma {\omega }_{j}-1,$$
(3)

with \(A=\mathop{\sum }\nolimits_{j = 1}^{N}{a}_{j}^{2}\) being the sum of background intensities. Then, we add the small amplitude Fourier modes associated with rj and wj (j = 1,...,N) to the plane-wave solutions (2) and express them as

$${u}_{j}={u}_{j0}\{1+{r}_{j}\exp [{{{{{{{\rm{i}}}}}}}}\Omega (\varkappa z+t)]+{w}_{j}^{* }\exp [-{{{{{{{\rm{i}}}}}}}}\Omega ({\varkappa }^{* }z+t)]\},$$
(4)

where Ω ∈ R denotes the frequency offset from the central frequencies ωj, ϰ is a complex wavenumber induced by perturbations, and the asterisk means complex conjugate. When these perturbed plane-wave solutions are substituted into Eqs. (1), followed by linearization, a system of 2N coupled linear equations about rj and wj can be obtained, which will have a nontrivial solution only when ℘ = (ϰ + Aγ)/σ satisfies the dispersion relation (see Supplementary Note 1 for details):

$$\mathop{\sum }\limits_{j=1}^{N}\frac{{a}_{j}^{2}{\upsilon }_{j}}{{(\wp +{\omega }_{j})}^{2}-{\Omega }^{2}/4}=\sigma .$$
(5)

It is seen that only if ϰ or ℘ is a complex number does the MI occur. Therefore, we define the MI gain as γh = ∣Ωℑ(ϰ)∣ =∣σΩℑ(℘)∣, where the symbol ℑ means taking the imaginary part12. It should be noted that the MI evolution Eq. (5) depicts the instability of the cw backgrounds, not that of the solitons or other localized entities allowed.

Closed-form Peregrine soliton solutions

On the other hand, thanks to integrability, the GNLS model (1) can be solved by means of Darboux transformation46,76,77 or the Hirota bilinear method61,78. Its general fundamental rogue wave solutions are expressed as, in an elegant closed form

$${u}_{j}={u}_{j0}\left(1+\frac{{{{{{{{\rm{i}}}}}}}}{G}_{j}}{M}\right),\quad j=1,\cdots \,,N,$$
(6)

where the polynomials M and Gj are defined by

$$M={\tau }^{2}+{\sigma }^{2}{\nu }^{2}{z}^{2}+\frac{{\eta }^{2}}{4\alpha {\nu }^{2}}-\frac{{{{{{{{\rm{i}}}}}}}}\gamma }{\alpha }(\sigma \gamma {\nu }^{2}z-\eta \tau ),$$
(7)
$${G}_{j}=\, \frac{2\left[(\mu +{\omega }_{j})\tau -\sigma {\nu }^{2}z\right]}{{(\mu +{\omega }_{j})}^{2}+{\nu }^{2}}+\frac{2\gamma (\sigma \gamma {\nu }^{2}z-\eta \tau )}{\alpha }\\ -\frac{{{{{{{{\rm{i}}}}}}}}\eta {\upsilon }_{j}}{\alpha \left[{(\mu +{\omega }_{j})}^{2}+{\nu }^{2}\right]},$$
(8)

with τ = t + (σμ − γA)z, η = γμ + 1, and α = γ2ν2 + η2. Here, the parameters μ ∈ R and ν ∈ R are the real and imaginary parts of the complex root ℘ = μ + iν of the 2N-degree polynomial equation:

$$\mathop{\sum }\limits_{j=1}^{N}\frac{{a}_{j}^{2}{\upsilon }_{j}}{{(\wp +{\omega }_{j})}^{2}}=\sigma .$$
(9)

We should point out that when N = 1, such fundamental rogue wave solutions has been presented in ref. 71, and when N = 2, they have just recently been reported in ref. 46, all obtained with the Darboux transformation method.

We should point out that the closed-form solutions (6) can reveal all forms of Peregrine soliton structures that are allowed by the GNLS model (1), including those that can not be expressed in terms of elementary functions, which always occur for N ⩾ 3 (see Supplementary Note 2). Therefore, for the latter circumstances, one needs to first solve the characteristic polynomial Eq. (9) numerically for μ and ν, which can easily be done by commercial analytic tools. Then, the exactitude of the solutions (6) can be well confirmed.

As one can see, the characteristic Eq. (9) is none other than the MI evolution Eq. (5) in the baseband limit Ω = 0, although they are derived in a different way. This implies that there is a direct one-to-one relationship between the baseband MI and the rogue wave solutions, namely, for every complex root ℘ = μ + iν (ν ≠ 0) of Eq. (5) obtained with Ω = 0, the rogue wave solutions (6) are uniquely determined, and if more pairs of complex ℘ exist, more rogue wave states are allowed to occur. This is what we called MI-rogue wave correspondence, which is a universal principle hidden in integrable nonlinear equations not limited to the GNLS model (1). In our Methods or Supplementary Note 3, we also demonstrate its validity in the context of other integrable models.

MI-rogue wave correspondence and exemplification

Subsequently, to exemplify this interesting MI-rogue wave correspondence, we examine below the three-component case (N = 3) without loss of generality. In this situation, one needs to solve Eq. (5) for ℘ = μ + iν in the baseband limit Ω = 0. Generally, it is a sextic real-coefficient polynomial equation of ℘ and only a very few cases do admit the analytic solutions (see Supplementary Note 2). Let us cite here one example of such analytic solutions, where ω3 = (ω1 + ω2)/2 = κ12/2 is taken. When δ12 = ω1−ω2 = 0 (i.e., the simple decoupled situation), it is easily found that there is at most one pair of complex roots of ℘, given by

$$\mu =-{\omega }_{1},\quad \nu =\sqrt{-A{\upsilon }_{1}/\sigma },\quad {{{{{{{\rm{for}}}}}}}}\,\sigma {\upsilon }_{1} \, < \,0.$$
(10)

When δ12 ≠ 0, this sextic equation can still be solved analytically under the amplitude conditions

$${a}_{1}=\sqrt{-\frac{\sigma {\delta }_{12}^{2}}{3{\upsilon }_{1}}},\quad {a}_{2}=\sqrt{-\frac{\sigma {\delta }_{12}^{2}}{3{\upsilon }_{2}}},\quad {a}_{3}=\sqrt{-\frac{\sigma {\delta }_{12}^{2}}{3{\upsilon }_{3}}},$$
(11)

for συ1,2,3 < 0, yielding three pairs of complex roots:

$$({{{{{{{\rm{I}}}}}}}})\,\mu =-\frac{{\kappa }_{12}}{2},\quad \nu =\frac{\sqrt{3}}{6}| {\delta }_{12}| \left({\varpi }^{2}+\frac{1}{{\varpi }^{2}}\right),$$
(12)
$$({{{{{{{\rm{II}}}}}}}})\,\mu =\frac{{\delta }_{12}}{4}\left({\varpi }^{2}-\frac{1}{{\varpi }^{2}}\right)-\frac{{\kappa }_{12}}{2},\quad \nu =\frac{\sqrt{3}}{12}| {\delta }_{12}| \left({\varpi }^{2}+\frac{1}{{\varpi }^{2}}\right),$$
(13)
$$({{{{{{{\rm{III}}}}}}}})\,\mu =-\frac{{\delta }_{12}}{4}\left({\varpi }^{2}-\frac{1}{{\varpi }^{2}}\right)-\frac{{\kappa }_{12}}{2},\quad \nu =\frac{\sqrt{3}}{12}| {\delta }_{12}| \left({\varpi }^{2}+\frac{1}{{\varpi }^{2}}\right),$$
(14)

where \(\varpi ={(\sqrt{2}+1)}^{1/3}\approx 1.3415\). It should be noted that we have dropped the ± sign from the above ν formulas, as the rogue wave solutions depend on ν2 only.

Figure 1 a–c displays the landscapes of the positive imaginary parts of ℘ versus ω1 and ω2 in the baseband limit (Ω = 0), obtained numerically using a typical set of initial parameters σ = − 1 (normal dispersion), γ = 1, a1 ≃ 1.1656, a2 ≃ 0.5693, a3 ≃ 0.7234, and letting ω3 = (ω1 + ω2)/2. The cross-sectional profiles ℑ(℘j) with respect to ω1 or ω2 are illustrated in Fig. 1d–f. Since in such a baseband limit, the real and imaginary parts of ℘ give the values of μ and ν, respectively, these maps clearly reveal the rogue wave signatures in the phase space (ω1, ω2). To be specific, as every points beyond the line δ12 = 0 are completely covered by these maps (Fig. 1a–c), the solutions (6) can therefore exist in the whole domain of ω1 and ω2, apart from some particular points on that line which exclude any rogue wave solutions in both dispersion situations. It is also suggested that in the top right domain defined by συ1,2,3 < 0 and δ12 ≠ 0, there would occur three different rogue wave structures for given set of initial parameters (Fig. 1f). In other domains, at most two different structures are allowed, as indicated in Fig. 1d, e. Here we should point out that although two curves of ℑ(℘) = ν may coincide at some points in Fig. 1d–f, they still represent two different rogue wave structures since the real parts ℜ(℘) = μ are now different at these points.

Fig. 1: Landscapes of the positive imaginary parts of ℘ in the baseband limit and cross-sectional profiles.
figure 1

a–c Landscapes of ℑ(℘j) as a function of the frequencies ω1 and ω2. Here ℘j(j = 1,2,3) are the complex roots of Eq. (5) in the baseband limit Ω = 0, with ω3 = (ω1 + ω2)/2, and ℑ denotes the positive imaginary part of these roots, if any. The other parameters are specified by σ = − 1 (normal dispersion), γ = 1, a1 ≃ 1.1656, a2 ≃ 0.5693, and a3 ≃ 0.7234. The white dashed lines in a, b, c are auxiliary ones that can separate the maps into different regions and the colour bars on the top quantify the magnitude of ℑ(℘j). d–f Cross-sectional profiles of ℑ(℘j) with respect to (d) ω1 or (e, f) ω2, given the ω2 or ω1 value on the top of each panel. Here, the blue dashed line, red solid line, and cyan dash-dotted line represent ℑ(℘1), ℑ(℘2), and ℑ(℘3), respectively. The green crosses in (f) indicate the values of ν = ℑ(℘) for Cases I, II, and III calculated respectively from Eqs. (12), (13), and (14), wherein ω1 = 1.4 and ω2 ≃ 2.68.

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We also calculate numerically ℑ(℘j)( ⩾ 0) and the corresponding MI gains γhj = ∣σΩℑ(℘j)∣ from Eq. (5) under the same parameter conditions as in Fig. 1, but now allowing Ω to vary. The results corresponding to ω1 = 0 (left column) and ω1 = 1.4 (middle and right columns) are presented in Fig. 2. It is seen that when ω1 = 0, there exist only two symmetrically distributed imaginary parts ℑ(℘1,2) for almost all ω2 values (Fig. 2a, b), whereas when ω1 = 1.4, there are three imaginary parts ℑ(℘1,2,3) for ω2 > 1 (Fig. 2c–e). Each ℑ(℘j) involves a baseband part that traverses the central line Ω = 0, but the main spectrum ℑ(℘1) is usually more complicated, as it may possess several passbands as well (Fig. 2a, c). At Ω = 0, these imaginary parts will take exactly the same ω2-dependent profiles as shown in Fig. 1e, f, of course when ω1 takes 0 and 1.4, respectively. Similarly, each MI map γhj is symmetric with respect to Ω and in the Ω ⩾ 0 domain it is composed of one baseband part that has a limiting zero value at Ω = 0 and some passbands that stand away from the central line Ω = 0, as indicated in Fig. 2f, h. Generally, the appearance of passbands is a precursor of multiple rogue wave structures that could coexist on the same background46. It follows from Eqs. (4)–(6) that only the baseband of each MI spectrum determines the rogue wave domain39,40, but either its baseband or passband may contribute to the rogue wave generation (Fig. 2l). In general, due to competition, the rogue wave structure associated with larger MI gain will be more prone to be excited, and hence occur more frequently, than the one with smaller MI gain.

Fig. 2: Maps of ℑ(℘j) and γhj = ∣σΩℑ(℘j)∣ versus (ω2, Ω).
figure 2

a–e Numerical results of the positive imaginary parts ℑ(℘j) calculated from Eq. (5) with respect to the central frequency ω2 and the frequency offset Ω, for given (a, b) ω1 = 0 or (c, d, e) ω1 = 1.4. The other parameter conditions are same as in Fig. 1. f-j Maps of the corresponding modulation instability gains γhj with respect to (ω2, Ω), for (f, g) ω1 = 0 or (h, i, j) ω1 = 1.4. The BB and PB in (f, h) are short for baseband and passband, respectively. In all maps, the colour bar on the bottom quantifies the magnitude of (a–e) ℑ(℘j) or (f–j) γhj. k Cross-sectional profiles of ℑ(℘j) along the red dashed lines in c–e, for given ω1 = 1.4 and ω2 ≃ 2.68. The green crosses in k indicate the values of ν for Cases I, II, and III that are identical to those in Fig. 1f. l Cross-sectional profiles of γhj along the red dashed lines at ω2 ≃ 2.68 in h–j. In k, l, blue solid curves stand for ℑ(℘1) and γh1, red solid curves for ℑ(℘2) and γh2, and cyan dashed curves for ℑ(℘3) and γh3. For the purpose of comparison, the cross-sectional gain profiles along the green dashed lines at ω2 = 3.25 in h–j are also illustrated in (l), denoted by thin dashed curves.

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The three families of Peregrine soliton states corresponding to the green crosses in Fig. 1f or 2k, which are denoted by I, II, and III, are demonstrated in Fig. 3, using ω1 = 1.4, ω2 = ω1 + 2(ω1 − 1/γ)(ϖ − 1/ϖ + 1) ≃ 2.68, ω3 = (ω1 + ω2)/2 ≃ 2.04, and the same aj values as in Fig. 1. This set of initial parameters has been made to strictly fulfil the amplitude conditions (11) and hence the (μ, ν) values can be determined by Eqs. (12) (Case I), (13) (Case II), and (14) (Case III), respectively. It is exhibited that, in Case I (Fig. 3a), the u1 component features the black rogue wave state whose amplitude falls to zero in the dip center, while the u2 and u3 components manifest as the bright states, one of which (i.e., u2) grows beyond the 3-fold amplitude. However, for Cases II and III, these field components may take other different structures, all localized in both space and time (Fig. 3b, c). Analytically, all these different rogue wave families are obtained with the same set of initial parameters, implying that they can coexist and develop on the same continuous background63.

Fig. 3: Analytic results of three coexisting Peregrine soliton families on the continuous backgrounds.
figure 3

a–c 3D surface (top) and contour (bottom) plots of the analytic Peregrine soliton solutions (6) for three field components u1,2,3. The real parameters μ and ν in the solutions are given by Eq. (12) for a Case I, by Eq. (13) for b Case II, and by Eq. (14) for c Case III, respectively. The same set of system parameters is used, which corresponds to the green crosses in Fig. 1f or 2k. Here all the rogue wave components have been normalized by their background heights aj, denoted by \(| {u}_{j}^{{\prime} }| =| {u}_{j}/{a}_{j}|\).

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Numerical verification

We performed extensive numerical simulations to check our analytic predictions, using the exponential time differencing Crank–Nicolson (ETD-CN) and spectral method (see Methods for more detail)79,80. We first demonstrate in Fig. 4a–c the numerical solutions of the GNLS Eqs. (1) with N = 3, using the profiles of our analytic solutions (6) in the far region (namely, using uj(z = −3, t) for Case I rogue waves, and uj(z = −6, t) for the Case II and Case III rogue waves) as initial conditions, given the same set of system parameters as in Fig. 3. Note that, in Fig. 4a, a white noise of the strength that equals 0.1%aj has been put on the initial profiles, in order to excite MI waves earlier in a short distance. It is evident that these numerical solutions for all Case I, Case II, and Case III rogue waves exhibit excellent agreement with their analytic counterparts seen in Fig. 3 over a wide propagation range. Intriguingly, in our simulation results for Case III rogue waves (Fig. 4c), we also observe that there would occur Case I rogue waves on the same background (see arrows at around z = 8), which are not the outcome of input profiles, but result from the spontaneous MI. This confirms definitely the coexistence of two different types of rogue waves, for given set of system parameters. Now one may further wonder whether it is possible to see all three types of rogue waves on the same background, so as to conform to the MI-rogue wave correspondence established above.

Fig. 4: Numerical simulations of three coexisting Peregrine solitons and their numerical excitation.
figure 4

a–c Numerical simulation results of Case I, Case II, and Case III rogue wave states seen in Fig. 3. To produce them, we integrate Eqs. (1) (N = 3) using the exponential time differencing Crank--Nicolson scheme armed with (3, 3)-order Padé approximation. The initial profiles for numerics are taken to be the analytical solutions at a z = − 3 and b, c z = −6, respectively. It is exhibited that all rogue wave states can be numerically reproduced till the far trailing region where the periodic wave trains begin to appear. The arrows in (c) indicate a wave state that takes after Case I rogue wave. d Numerical excitation of three coexisting Peregrine solitons from a turbulent field. The initial condition is now supposed to be the plane-wave solutions (2) at z = 0 perturbed by strong harmonic waves, with other system parameters the same as in Fig. 3. The ellipses I, II, and III have singled out these Peregrine soliton structures. The right and left insets show the comparison of the cross-sectional profiles (solid curves) at z = 2.52 and 2.93 with the analytic solutions (red dashed curves) for Case I and Case III rogue waves, respectively. In all panels, the colour bar on the right quantifies the magnitude of the normalized amplitude \(| {u}_{j}^{{\prime} }| =| {u}_{j}/{a}_{j}|\), where aj is the background height.

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To confirm the latter issue, we then use in our numerical test the plane-wave solutions (2) at z = 0 perturbed by strong harmonic waves as initial conditions, which will soon develop into a turbulent sea of different waves (Fig. 4d). We find that, in this turbulent wave field, the three Peregrine solitons circled, respectively, by the ellipses I, II, and III at around z = 2.52, 6.85, and 2.93 can be identified as the corresponding ones shown in Fig. 3. This is more evident in the right and left insets therein, where a good agreement between the cross-sectional profiles (solid curves) at z = 2.52 and 2.93 and the corresponding analytic solutions (red dashed curves) can be seen. We should point out that the Peregrine rogue waves denoted by II have been strongly distorted by the surrounding waves, as compared to the patterns shown in Fig. 4b. Besides, one can observe that the Peregrine soliton structures like Case I can appear in many places, but those like Cases II and III are much fewer (Fig. 4d). This is not surprising because Case I has a much larger MI gain value than the latter two, as seen by the bold curves in Fig. 2l, and naturally dominates the rogue wave occurrence. When the value of ω2 is increased from 2.68 to 3.25, while keeping the background amplitudes aj unchanged, one can observe more easily the coexistence of three types of rogue wave structures on the same background (Fig. 5), since now the related MI gain profiles become closer to each other, as seen by the thin dotted lines in Fig. 2l. Noting that in Fig. 5, we used the initial profiles denoted by \({u}_{j}(z=0,t)={u}_{j0}[1+{{{{{{{\rm{i}}}}}}}}{G}_{j}^{(1)}(z-4,t)/{M}^{(1)}(z-4,t)+{{{{{{{\rm{i}}}}}}}}{G}_{j}^{(2)}(z-4,t+20)/{M}^{(2)}(z-4,t+20)+{{{{{{{\rm{i}}}}}}}}{G}_{j}^{(3)}(z-4,t-20)/{M}^{(3)}(z-4,t-20)]_{z = 0}\), where M and Gj are defined by Eqs. (7) and (8), while the superscripts over them signify that the parameters μ and ν in these functions take (1) μ ≃ − 2.3981, ν ≃ 0.7674, (2) μ ≃ −1.5515, ν ≃ 0.5484, and (3) μ ≃ −3.0254, ν ≃ 0.6778, respectively. This implies that the input profiles are a superposition of three well-separated yet low-amplitude rogue wave profiles, and thus can be thought of as some kind of strong perturbation to plane-wave solutions (2). Although the initial profiles are not the analytic solutions of the model equation, we observe that the rogue wave states numerically generated, as indicated by red arrows in Fig. 5a–c, could agree well with our analytic solutions (6). One can also notice that, for either set of parameters used in Fig. 4 or 5, there appear three different baseband MI spectra (Fig. 2h–j). Considering that three baseband MI spectra have resulted in three different rogue wave structures, our numerical results positively confirm the MI-rogue wave correspondence.

Fig. 5: Numerical excitation of three Peregrine soliton families.
figure 5

Numerical simulation results for the a u1, b u2, and c u3 field components, which develop on the backgrounds that allow the modulation instability spectra indicated by thin dashed lines in Fig. 2l. Different from Fig. 4d, we now use the initial condition defined by a superposition of three well-separated low-amplitude rogue wave profiles. The input frequencies are given by ω1 = 1.4, ω2 = 3.25, and ω3 = (ω1 + ω2)/2 = 2.325, while the other parameters are kept the same as in Fig. 2. This set of system parameters would yield: (1) μ ≃ − 2.3981, ν ≃ 0.7674; (2) μ ≃ − 1.5515, ν ≃ 0.5484; and (3) μ ≃ − 3.0254, ν ≃ 0.6778. All field components are normalized by their respective backgrounds and the red arrows mark out three Peregrine soliton structures that exhibit a good agreement with the analytic solutions (6).

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Corollary: Rogue waves on periodic backgrounds

Finally, we would like to point out that the GNLS model (1) owns an inherent SU(N) symmetry and hence any SU(N) group transformation of the solutions (6) will also give the new solutions \({u}_{j}^{p}\) of the model, which can be expressed as

$${u}_{j}^{p}=\mathop{\sum }\limits_{m=1}^{N}{\alpha }_{jm}{u}_{m},\quad j=1,\cdots \,,N,$$
(15)

where um are the old rogue wave solutions (6), αjm are the complex elements of the SU(N) matrix, \({{{{{{{\bf{A}}}}}}}}={({\alpha }_{jm})}_{N\times N}\), which should satisfy AA† = A†A = 1 and \(\det ({{{{{{{\bf{A}}}}}}}})=1\). Demonstrably, the background fields, defined now by \({u}_{j}^{b}=\mathop{\sum }\nolimits_{m = 1}^{N}{\alpha }_{jm}{u}_{m0}\), could cause interference fringes that depend not only on the temporal beat frequencies δmn = ωm − ωn, but also on the entries αjm of A, with the average background height

$${d}_{j}={\left(\mathop{\sum }\limits_{m = 1}^{N}| {\alpha }_{jm}{| }^{2}{a}_{m}^{2}\right)}^{1/2}.$$
(16)

In this regard, the solutions defined by Eq. (15) will represent the rogue wave states developed on the periodic backgrounds, as demonstrated in ref. 46 where N = 2.

For illustration, we still consider N = 3 case (see Supplementary Note 4 for detailed information) and demonstrate in Fig. 6 three families of rogue wave states sitting on the periodic backgrounds, using the same set of initial parameters as in Fig. 3, but with

$${{{{{{{\bf{A}}}}}}}}=\left[\begin{array}{ccc}\sqrt{3}/4-\sqrt{6}/8&1/4+3\sqrt{2}/8&\sqrt{6}/4\\ -3/4-\sqrt{2}/8&-\sqrt{3}/4+\sqrt{6}/8&\sqrt{2}/4\\ \sqrt{2}/4&-\sqrt{6}/4&\sqrt{2}/2\end{array}\right].$$
(17)

It is seen that all backgrounds that support the rogue waves become now periodic, owing to the constructive or destructive interferences between three continuous waves. Despite their multifarious patterns, however, the occurrence of such periodic rogue waves are still governed by the MI evolution Eq. (5), namely, by the MI-rogue wave correspondence.

Fig. 6: Peregrine solitons on the periodic backgrounds.
figure 6

a–c 3D surface (top) and contour (bottom) plots of the Peregrine soliton solutions (15) on the periodic backgrounds characterized by the coefficient matrix (17), with system parameters all the same as in Fig. 3. This set of parameters could result in : a μ ≃ −2.0384, ν ≃ 0.8682; b μ ≃ −2.4355, ν ≃ 0.4341; and (c) μ ≃ −1.6413, ν ≃ 0.4341, exactly the same as indicated by green crosses in Fig. 1f or 2k. Here all the rogue wave components are normalized as \(| {u}_{j}^{{\prime} p}| =| {u}_{j}^{p}| /{d}_{j}\), where dj is the average background height defined by Eq. (16).

Full size image

Conclusion

In conclusion, we established an explicit universal one-to-one correspondence between the baseband MI and the Peregrine rogue wave solutions, with which complex rogue wave dynamics as well as their signatures can be unveiled. Such MI-rogue wave correspondence is an analog of the bulk-boundary correspondence in topological physics, with rogue waves corresponding to the edge modes and MI corresponding to the bulk topological invariant, respectively. We exemplified this MI-rogue wave correspondence with the specific three-component GNLS model and numerically confirmed the excitation of three coexisting Peregrine solitons from a turbulent wave field, which exactly occurs as predicted by the MI analysis. In view of that MI is a particular feature of nonlinear systems, whether they are integrable or not, we expect the MI-rogue wave correspondence to be applicable to non-integrable systems as well, just as the bulk-boundary correspondence can work in non-Hermitian systems34,35,36. In that sense, our work unequivocally substantiates the insight that MI is the most fundamental mechanism underlying the rogue wave formation.

Given the MI-rogue wave correspondence established above, our work also offers an alternative way of obtaining exact rogue wave solutions from the MI analysis. In the Methods section, we provide an example to expatiate how to figure out the exact general fundamental rogue wave solutions of the three-wave resonant interaction (TWRI) equation81,82,83 from the MI analysis instead of from the complicated Darboux transformation or Hirota bilinearization84. It should be mentioned that the closed-form solutions thus obtained are much clearer in form than those reported most recently in ref. 84.

Methods

The MI equation for GNLS model

Here we outline the derivation of the MI evolution Eq. (5) within the multi-component GNLS equation framework. More derivation details along with the obtainment of rogue wave solutions can be found in Supplementary Note 1.

It is easy to check that this GNLS model admits the plane-wave solutions (2). Because of MI induced by nonlinearity and noises, such cw backgrounds are highly unstable and will eventually develop into a sea of waves, from which a series of high-contrast peaks of random intensity can be identified. Basically, the MI gain can be obtained via the linear stability analysis, which assumes the above plane-wave solutions to be perturbed according to Eqs. (4), where the terms associated with rj and wj represents the small-amplitude Fourier modes. Subsequently, substitution of Eqs. (4) into Eqs. (1) followed by a collection of the linear terms yields a system of 2N coupled linear equations about rj and wj:

$$\sigma \Omega \left(\frac{\Omega }{2}+\wp +{\omega }_{j}\right){r}_{j}+{\upsilon }_{j}\mathop{\sum }\limits_{m=1}^{N}{a}_{m}^{2}{r}_{m}+{\upsilon }_{j}\mathop{\sum }\limits_{m=1}^{N}{a}_{m}^{2}{w}_{m}=0,$$
(18)
$$\sigma \Omega \left(\frac{\Omega }{2}-\wp -{\omega }_{j}\right){w}_{j}+{\upsilon }_{j}\mathop{\sum }\limits_{m=1}^{N}{a}_{m}^{2}{r}_{m}+{\upsilon }_{j}\mathop{\sum }\limits_{m=1}^{N}{a}_{m}^{2}{w}_{m}=0,$$
(19)

where j = 1,..., N and ℘ = (Aγ + ϰ)/σ. This system of equations has a nontrivial solution only when the determinant of the coefficient matrix of the vector \({({r}_{1},\cdots ,{r}_{N},{w}_{1},\cdots ,{w}_{N})}^{T}\) equals zero. By elementary transformations of the coefficient matrix, one can easily obtain Eq. (5), which defines the evolution of MI for any complex ℘ or ϰ.

Simulations

In our simulations for Figs. 4 and 5, an efficient, second-order convergent, ETD-CN method was used79,80. Specifically, Eqs. (1) can be rewritten, in the spectral domain, as

$$\frac{\partial {\tilde{u}}_{j}}{\partial z}+L{\tilde{u}}_{j}=\tilde{F}(z,{u}_{j}),$$
(20)

where \({\tilde{u}}_{j}\) is the fast Fourier transform, which we will denote by the symbol \({{{{{{{\mathcal{F}}}}}}}}\) below, of the wave component uj with respect to time, namely, \({\tilde{u}}_{j}={{{{{{{\mathcal{F}}}}}}}}({u}_{j})\), L = iσω2/2 is the linear operator with ω being the transformed frequency variable, and \(\tilde{F}(z,{u}_{j})\) is the fast Fourier transform of the nonlinear term, i.e., \(\tilde{F}(z,{u}_{j})={{{{{{{\mathcal{F}}}}}}}}[{{{{{{{\rm{i}}}}}}}}(\mathop{\sum }\nolimits_{m = 1}^{N}| {u}_{m}{| }^{2})({u}_{j}+{{{{{{{\rm{i}}}}}}}}\gamma \frac{\partial {u}_{j}}{\partial t})]\). Then, let h = zn+1 − zn be the spatial step size, we arrive at the following recurrence formulas based on the ETD-CN scheme79:

$${\tilde{b}}_{jn}={{{{{{{{\mathcal{R}}}}}}}}}_{1}{\tilde{u}}_{jn}+{{{{{{{{\mathcal{R}}}}}}}}}_{2}\tilde{F}({z}_{n},{u}_{jn}),$$
(21)
$${\tilde{u}}_{j(n+1)}={\tilde{b}}_{jn}+{{{{{{{{\mathcal{R}}}}}}}}}_{3}[\tilde{F}({z}_{n}+h,{{{{{{{{\mathcal{F}}}}}}}}}^{-1}({\tilde{b}}_{jn}))-\tilde{F}({z}_{n},{u}_{jn})],$$
(22)

where

$${{{{{{{{\mathcal{R}}}}}}}}}_{1}={e}^{-hL},\quad {{{{{{{{\mathcal{R}}}}}}}}}_{2}=-\frac{{e}^{-hL}-I}{L},\quad {{{{{{{{\mathcal{R}}}}}}}}}_{3}=\frac{{e}^{-hL}-I+hL}{h{L}^{2}},$$
(23)

with I being the identity operator. To achieve higher spectral (or equivalently, temporal) accuracy, we now use (3, 3)-order Padé approximation for the exponential matrix e−hL, i.e.,

$${e}^{-hL}\simeq \frac{120I-60hL+12{h}^{2}{L}^{2}-{h}^{3}{L}^{3}}{120I+60hL+12{h}^{2}{L}^{2}+{h}^{3}{L}^{3}}\equiv {{{{{{{{\mathcal{R}}}}}}}}}_{1}.$$
(24)

Then \({{{{{{{{\mathcal{R}}}}}}}}}_{2}\) and \({{{{{{{{\mathcal{R}}}}}}}}}_{3}\) can also be transformed into the rational polynomials via Eq. (24). Basically, by using Eqs. (21)–(24), the coupled spectral Eqs. (20) can be numerically solved with high accuracy. Finally, an inverse fast Fourier transform of \({\tilde{u}}_{j}\) yields the numerical solutions uj of GNLS Eqs. (1).

Specifically, in our simulations, we used the simulation window of t in the range [−800, 800], with the time step size Δt = 0.006. The spatial step size h should be chosen smaller (here, h = 5 × 10−4) so that hπ/Δt < 1, which will ensure the (3, 3)-order Padé approximation (24) for e−hL to be accurate enough over the whole spectral domain. Despite the large simulation window, we only present what happens in the central portion, say, in [−18, 18] in Fig. 4d. Basically, any numerical algorithms can bring numerical errors, which will serve as intrinsic numerical noise to excite spontaneous MI. In our numerics, we also imposed strong external perturbations (e.g., white noise or harmonic waves) in order to excite MI as early as possible, so that we can observe the occurrence of rogue waves within shorter propagation distance.

MI–rogue wave correspondence in other integrable systems

We now demonstrate the MI-rogue wave correspondence in other different integrable systems, i.e., the vector Kaup–Newell equation19 and the (2+1)-component long-wave short-wave resonance equation63. Although the fundamental rogue wave solutions have been reported for these models, their intimate relation to MI was not revealed therein.

(i) Kaup–Newell model. We first consider the vector Kaup–Newell model which can be written as19

$${{{{{{{\rm{i}}}}}}}}\frac{\partial {u}_{j}}{\partial z}+\frac{\sigma }{2}\frac{{\partial }^{2}{u}_{j}}{\partial {t}^{2}}+\left(1+{{{{{{{\rm{i}}}}}}}}\gamma \frac{\partial }{\partial t}\right)[(| {u}_{1}{| }^{2}+| {u}_{2}{| }^{2}){u}_{j}]=0,$$
(25)

where j = 1, 2, and the parameter γ accounts for the self-steepening effect often seen by pulses with spectral widths comparable to the optical frequency. This two-component model possesses the plane-wave solutions:

$${u}_{j0}={a}_{j}\exp ({{{{{{{\rm{i}}}}}}}}{\omega }_{j}t-{{{{{{{\rm{i}}}}}}}}{k}_{j}z),\quad j=1,2,$$
(26)

where

$${k}_{j}=A(\gamma {\omega }_{j}-1)+\frac{1}{2}\sigma {\omega }_{j}^{2},\quad A=\mathop{\sum }\limits_{j=1}^{2}{a}_{j}^{2}={a}_{1}^{2}+{a}_{2}^{2}.$$

We assume that these plane-wave solutions are perturbed by the same Fourier modes as in Eqs. (4). A direct MI analysis of them results in a system of four linear equations for r1,2 and w1,2. It is easy to show that this system of linear equations has a nontrivial solution only when the dispersion relation (see Supplementary Note 3 for detailed derivation)

$$\frac{{a}_{1}^{2}(2\gamma \wp +\gamma {\omega }_{1}+1)}{{(\wp +{\omega }_{1})}^{2}-{\Omega }^{2}/4}+\frac{{a}_{2}^{2}(2\gamma \wp +\gamma {\omega }_{2}+1)}{{(\wp +{\omega }_{2})}^{2}-{\Omega }^{2}/4}+\sigma =0,$$
(27)

is satisfied. It is seen that the dispersion relation Eq. (27) has a more complicated form than Eq. (5). This implies that the rogue wave solutions of the vector Eqs. (25) will be quite different from those of the vector GNLS Eqs. (1).

In fact, it has been reported in ref. 19 that the general fundamental rogue wave solutions can take the form

$${u}_{j}={u}_{j0}\left(1+\frac{{R}_{j}}{M-{{{{{{{\rm{i}}}}}}}}K}\right)\left(\frac{M+{{{{{{{\rm{i}}}}}}}}K}{M-{{{{{{{\rm{i}}}}}}}}K}\right),$$
(28)

where the polynomials M, K, R1,2 are defined by

$$M={\tau }^{2}+{\sigma }^{2}{\nu }^{2}{z}^{2}+\frac{{\eta }^{2}}{4\alpha {\nu }^{2}},\quad K=\frac{\gamma }{\alpha }(\sigma \gamma {\nu }^{2}z-\eta \tau ),$$
(29)
$${R}_{j}=\frac{2{{{{{{{\rm{i}}}}}}}}[(\mu +{\omega }_{j})\tau -\sigma {\nu }^{2}z]}{{(\mu +{\omega }_{j})}^{2}+{\nu }^{2}}-\frac{2\alpha +\eta (\gamma {\omega }_{j}-1)}{\alpha [{(\mu +{\omega }_{j})}^{2}+{\nu }^{2}]},$$
(30)

with τ = t + (σμ − γA)z, η = γμ + 1, and α = ν2γ2 + η2. The real parameters μ and ν in Eqs. (29) and (30) are determined by the complex root ℘ = μ + iν of the quartic equation:

$$\frac{{a}_{1}^{2}(2\gamma \wp +\gamma {\omega }_{1}+1)}{{(\wp +{\omega }_{1})}^{2}}+\frac{{a}_{2}^{2}(2\gamma \wp +\gamma {\omega }_{2}+1)}{{(\wp +{\omega }_{2})}^{2}}+\sigma =0.$$
(31)

Once again, the characteristic Eq. (31) is the same as the dispersion relation Eq. (27) in the baseband limit Ω = 0, although they are derived in a different way.

(ii) Long-wave short-wave model. We then demonstrate the MI-rogue wave correspondence in the (2+1)-component long-wave short-wave model, which describes the nonlinear interaction between two complex short-wave field envelopes, u1,2, and the real long-wave field, ϕ. In dimensionless form, this multi-component model reads63

$${{{{{{{\rm{i}}}}}}}}\frac{\partial {u}_{j}}{\partial z} +\frac{1}{2}\frac{{\partial }^{2}{u}_{j}}{\partial {t}^{2}}+{u}_{j}\phi =0,\quad (\,j=1,2),\\ \frac{\partial \phi }{\partial z} -\frac{\partial }{\partial t}(| {u}_{1}{| }^{2}+| {u}_{2}{| }^{2})=0,$$
(32)

where t and z are two independent variables. The plane-wave solutions of this model can be easily found to be

$${u}_{j0}={a}_{j}\exp ({{{{{{{\rm{i}}}}}}}}{\omega }_{j}t-{{{{{{{\rm{i}}}}}}}}{k}_{j}z),$$
(33)

where \({k}_{j}=\frac{1}{2}{\omega }_{j}^{2}-b\) and b ⩾ 0 defines the background of the real long-wave field.

In a similar fashion, substituting the following perturbed plane-wave solutions

$${u}_{j}={u}_{j0}\{1+{r}_{j}\exp [{{{{{{{\rm{i}}}}}}}}\Omega (\varkappa z+t)]+{w}_{j}^{* }\exp [-{{{{{{{\rm{i}}}}}}}}\Omega ({\varkappa }^{* }z+t)]\},$$
(34)
$$\phi =b+p\exp [{{{{{{{\rm{i}}}}}}}}\Omega (\varkappa z+t)]+{p}^{* }\exp [-{{{{{{{\rm{i}}}}}}}}\Omega ({\varkappa }^{* }z+t)],$$
(35)

into Eqs. (32) and collecting the linear terms, one can obtain a system of five coupled equations. As noted in linear algebra, only when its coefficient matrix has a zero determinant does the vector \({({r}_{1},{r}_{2},{w}_{1},{w}_{2},p)}^{T}\) have a nontrivial solution. This zero determinant condition can give rise to the following dispersion relation (see Supplementary Note 3):

$$\frac{{a}_{1}^{2}}{{(\varkappa +{\omega }_{1})}^{2}-{\Omega }^{2}/4}+\frac{{a}_{2}^{2}}{{(\varkappa +{\omega }_{2})}^{2}-{\Omega }^{2}/4}+\varkappa =0,$$
(36)

which is now a quintic equation of ϰ.

On the other hand, the general fundamental rogue wave solutions of Eqs. (32) can be written as63

$${u}_{j}={u}_{j0}\left\{1-\frac{2{{{{{{{\rm{i}}}}}}}}{\nu }^{2}z-2{{{{{{{\rm{i}}}}}}}}(\mu +{\omega }_{j})(t+\mu z)+1}{[{(\mu +{\omega }_{j})}^{2}+{\nu }^{2}][{(t+\mu z)}^{2}+{\nu }^{2}{z}^{2}+1/(4{\nu }^{2})]}\right\},$$
(37)
$$\phi =b+\frac{2[{\nu }^{2}{z}^{2}-{(t+\mu z)}^{2}+1/(4{\nu }^{2})]}{{[{(t+\mu z)}^{2}+{\nu }^{2}{z}^{2}+1/(4{\nu }^{2})]}^{2}}.$$
(38)

As demonstrated in the preceding example, the real parameters μ and ν in Eqs. (37) and (38) are again determined by the dispersion relation Eq. (36) in the baseband limit Ω = 0, that is, by the following quintic equation of ϰ = μ + iν:

$$\frac{{a}_{1}^{2}}{{(\varkappa +{\omega }_{1})}^{2}}+\frac{{a}_{2}^{2}}{{(\varkappa +{\omega }_{2})}^{2}}+\varkappa =0.$$
(39)

Another route to exact rogue wave solutions

Finally, let us demonstrate how to figure out the exact rogue wave solutions from the MI analysis, based on the MI-rogue wave correspondence established (see Supplementary Note 5 for more detail). We limit our attention to the popular TWRI equation, which enjoys a prominent status in nonlinear optics81. In dimensionless form, the TWRI model reads81,82,83

$$ \frac{\partial {u}_{1}}{\partial t}+{V}_{1}\frac{\partial {u}_{1}}{\partial z}={u}_{2}^{* }{u}_{3}^{* },\quad \frac{\partial {u}_{2}}{\partial t}+{V}_{2}\frac{\partial {u}_{2}}{\partial z}=-{u}_{1}^{* }{u}_{3}^{* },\\ \frac{\partial {u}_{3}}{\partial t}+{V}_{3}\frac{\partial {u}_{3}}{\partial z}={u}_{1}^{* }{u}_{2}^{* },$$
(40)

where u1,2,3(z, t) are the slowly varying complex envelopes of three wave fields. The coefficients Vj correspond to the relative group velocities of three waves and we suppose V1,2 > V3. Without loss of generality, we set V3 = 0, which implies that Eqs. (40) are written in a reference frame comoving with u3.

Hence, the initial plane-wave solutions that satisfy Eqs. (40) can be found to be

$${u}_{10}(z,t) =\, {a}_{1}\exp \left[-{{{{{{{\rm{i}}}}}}}}({k}_{1}z-{\omega }_{1}t)\right],\\ {u}_{20}(z,t) =\, {a}_{2}\exp \left[{{{{{{{\rm{i}}}}}}}}({k}_{2}z-{\omega }_{2}t)\right],\\ {u}_{30}(z,t) =\, {{{{{{{\rm{i}}}}}}}}{a}_{3}\exp \left[{{{{{{{\rm{i}}}}}}}}({k}_{1}-{k}_{2})z-{{{{{{{\rm{i}}}}}}}}({\omega }_{1}-{\omega }_{2})t\right],$$
(41)

where

$${k}_{1}=\frac{{\omega }_{1}}{{V}_{1}}+\frac{{a}_{2}^{2}}{\delta {V}_{1}},\quad {k}_{2}=\frac{{\omega }_{2}}{{V}_{2}}+\frac{{a}_{1}^{2}}{\delta {V}_{2}},\quad {a}_{3}=\frac{{a}_{1}{a}_{2}}{\delta },$$
(42)

with a1,2,3 being the background heights and δ = ω1 − ω2. Subsequently, substituting the perturbed plane-wave solutions

$${u}_{j}={u}_{j0}\{1+{r}_{j}\exp [{{{{{{{\rm{i}}}}}}}}\Omega (\varkappa z+t)]+{w}_{j}^{* }\exp [-{{{{{{{\rm{i}}}}}}}}\Omega ({\varkappa }^{* }z+t)]\},$$
(43)

where j = 1, 2, 3, into the TWRI equation followed by linearization, one can obtain a system of six coupled equations. Under the condition that the system of equations admits only nontrivial solutions, one can find the dispersion relation:

$${\left(\frac{{a}_{1}^{2}}{{V}_{2}\varkappa +1}+\frac{{a}_{2}^{2}}{{V}_{1}\varkappa +1}\right)}^{2}+2{\delta }^{2}\left(\frac{{a}_{1}^{2}}{{V}_{2}\varkappa +1}-\frac{{a}_{2}^{2}}{{V}_{1}\varkappa +1}\right)+{\delta }^{4}={\Omega }^{2}{\delta }^{2}.$$
(44)

Now, to find the rogue wave solutions, we first assume Ω = 0 in Eq. (44) and rewrite it as

$${\left(\frac{{a}_{1}^{2}}{{V}_{2}\varkappa +1}+\frac{{a}_{2}^{2}}{{V}_{1}\varkappa +1}\right)}^{2}+2{\delta }^{2}\left(\frac{{a}_{1}^{2}}{{V}_{2}\varkappa +1}-\frac{{a}_{2}^{2}}{{V}_{1}\varkappa +1}\right)+{\delta }^{4}=0,$$
(45)

which can give the values of ϰ = μ + iν for given set of system parameters. Then we suppose the rogue wave solutions to take

$${u}_{1}=\, {u}_{10}\left[1+\frac{{{{{{{{\rm{i}}}}}}}}{f}_{1}(t+\mu z)+{{{{{{{\rm{i}}}}}}}}{g}_{1}z+{e}_{1}}{{(t+\mu z)}^{2}+{\nu }^{2}{z}^{2}+1/\alpha }\right],\\ {u}_{2}=\, {u}_{20}\left[1+\frac{{{{{{{{\rm{i}}}}}}}}{f}_{2}(t+\mu z)+{{{{{{{\rm{i}}}}}}}}{g}_{2}z+{e}_{2}}{{(t+\mu z)}^{2}+{\nu }^{2}{z}^{2}+1/\alpha }\right],\\ {u}_{3}=\, {u}_{30}\left[1-\frac{{{{{{{{\rm{i}}}}}}}}{f}_{3}(t+\mu z)+{{{{{{{\rm{i}}}}}}}}{g}_{3}z+{e}_{3}}{{(t+\mu z)}^{2}+{\nu }^{2}{z}^{2}+1/\alpha }\right],$$
(46)

where the real parameters e1,2,3, f1,2,3, g1,2,3, and α are to be determined. Substituting Eqs. (46) into the TWRI Eqs. (40) and equating the coefficients of zltm (l, m are powers) to zero, one can readily find that

$${f}_{1}=\frac{\sqrt{2({\ell }_{2}-{V}_{2}\mu -1)}}{{a}_{1}},\quad {g}_{1}=-\frac{\nu \sqrt{2({\ell }_{2}+{V}_{2}\mu +1)}}{{a}_{1}},$$
(47)
$${f}_{2}=\frac{\sqrt{2({\ell }_{1}+{V}_{1}\mu +1)}}{{a}_{2}},\quad {g}_{2}=\frac{\nu \sqrt{2({\ell }_{1}-{V}_{1}\mu -1)}}{{a}_{2}},$$
(48)
$${\ell }_{j}=\sqrt{{({V}_{j}\mu +1)}^{2}+{V}_{j}^{2}{\nu }^{2}},\,\,(\,j=1,2),$$
(49)
$$\alpha =\frac{16{\nu }^{2}{g}_{1}^{2}}{{(\,{f}_{1}^{2}{\nu }^{2}+{g}_{1}^{2})}^{2}}=\frac{16{\nu }^{2}{g}_{2}^{2}}{{(\,{f}_{2}^{2}{\nu }^{2}+{g}_{2}^{2})}^{2}},$$
(50)
$${e}_{1}=-\frac{{f}_{1}^{2}}{4}-\frac{{g}_{1}^{2}}{4{\nu }^{2}},\quad {e}_{2}=-\frac{{f}_{2}^{2}}{4}-\frac{{g}_{2}^{2}}{4{\nu }^{2}},$$
(51)
$${f}_{3}={f}_{1}+\,{f}_{2},\quad {g}_{3}={g}_{1}+{g}_{2},\quad {e}_{3}={f}_{1}{f}_{2}-({e}_{1}+{e}_{2})-\frac{{f}_{1}+{f}_{2}}{\delta }.$$
(52)

It is seen that all these coefficients have been uniquely determined by the real parameters μ and ν, and hence, according to Eqs. (46), the resultant fundamental rogue wave solutions of the TWRI equation are obtained.