In the last decade, the study of rogue waves has attracted increasing interest in such fields as oceanography [1], fluid dynamics [2, 3], optics and photonics [4–6], acoustics [7], magnetism [8], Bose-Einstein condensation [9, 10], artificial intelligence [11], and topological control [12]. Mathematically, rogue waves can be represented by rational functions localized on the space-time domain [13, 14]. One typical example is the fundamental rational solution of the nonlinear Schrödinger (NLS) equation, termed Peregrine soliton, which was first discovered by Peregrine in 1983 [15]. The most distinctive feature of Peregrine soliton is the bilocalized peak on a finite background. Due to its peculiar spatiotemporal structure, Peregrine soliton is generally recognized as the prototype of rogue waves in reality [16]. Nowadays, Peregrine solitons have been observed in many physical environments, such as water wave tanks [2], optical fibers [17, 18], plasmas [19], and irregular ocean states [20].

In many practical situations, multi-component nonlinear systems with more complex spatiotemporal dynamics rather than scalar integrable models need to be considered [21, 22]. Unlike scalar NLS, the vector Chen-Lee-Liu (CLL) NLS model incorporates some indispensable features. Multi-component coupling between optical fields u 1 and u 2 , essential for modeling polarization-dependent effects in birefringent media [23], and self-steepening, which induces spatiotemporal reshaping critical for ultrashort pulses [24, 25]. These enable phenomena unattainable in scalar models, such as energy-transfer-driven amplitude anomalies and higher-order rogue states. Examples include the formation of anomalous Peregrine solitons involving self-steepening effects in the multi-component systems [26, 27], as well as the formation of Peregrine rogue waves on a background of periodic waves induced by interference [28]. Meanwhile, the dynamics of higher-order Peregrine solitons have also attracted a great deal of research activities [29–31], especially the multiple rogue wave states consisting of multiple Peregrine solitons [32, 33] and the super rogue wave states superimposed by three or six Peregrine solitons [34, 35]. It should be noted that the multiple rogue wave states are different from the super rogue wave states, with the former being composed of a number of well-separated Peregrine solitons. In addition, rogue wave patterns have also been studied in recent years, and these studies are mainly related to the root structure of special polynomials (e.g., Okamoto polynomials or Adler–Moser polynomials), where the distribution of the roots determines the geometry of the patterns [36–39]. These studies have greatly enriched our understanding of the nature of Peregrine solitons.

In our previous research works [40, 41], we explored the omnipresent coexistence as well as Peregrine solitons on periodic backgrounds, using the vector CLL-NLS equation. It is also found that when having the double-root case, the n th-order multiple rogue wave states will consist of n ( n +1)/2 fundamental Peregrine solitons, as what was found in scalar systems [35, 42, 43]. As for the triple-root case, however, such n th-order rogue wave states will exhibit more complex patterns, consisting of at most n ( n +1) Peregrine solitons, with at most two and 6 fundamental Peregrine solitons seen for their first- and second-order rogue wave states, which were confirmed in most other vector system [44–46]. One fundamental question naturally arises: as for the third-order rogue wave states, does it satisfy that it consists of at most 12 fundamental Peregrine solitons? If so, what kind of pattern will its fundamental Peregrine solitons show? Will it be similar to the second-order solutions case, with an even number of multiple patterns of rogue wave states?

In this work, we conduct a systematic study of this problem. We obtain up to third-order rogue wave solutions for the vector CLL-NLS system by the nonrecursive Darboux transform (DT) method and analyze the complex patterns of rogue wave states in the triple-root case. In addition, based on the results for the first-, second-, and third-order rogue wave states, the number patterns of the fundamental Peregrine solitons of the n th-order rogue wave states are discussed. Our findings may contribute to the possibility of future experimental observations. This paper is structured as follows: first, Section 2 gives an overview of the vector CLL-NLS model and the nonrecursive DT form by which higher-order rogue wave solutions are derived. Then, Section 3 gives a detailed list of the first-, second-, and third-order rogue wave solutions, and discusses the number rule of fundamental Peregrine solitons for rogue wave states. Finally, Section 4 concludes the paper.

We consider the vector CLL-NLS model, which controls the mixing of two fundamental-frequency pulses in a second-order nonlinear crystal through the generation process of type II highly phase mismatched second-harmonic [24, 25, 47]. The dimensionless form of the CLL-NLS model is written as i u ı z + σ 2 u ı t t + ( | u 1 | 2 + | u 2 | 2 ) ( u ı + i γ u ı t ) = 0 , ( ı = 1,2 ) , (1) where u ı ( z , t ) are the normalized complex envelopes, with the subscripts denoting the partial derivatives, and z and t being the propagation distance and delay time, respectively. All variables ( z , t , u ı ) are dimensionless in this study. Physical units can be restored via scaling transformations [23, 48]. The parameter σ can be normalized to + 1 and − 1 in the case of anomalous and normal dispersion, respectively. The third term of Equation 1 includes the Kerr nonlinear effect, which arises from the intensity-dependent refractive index [23], and the self-steepening effect modulated by the parameter γ [48]. It is easy to see that when γ = 0 , the above model will regress to the well-known Manakov system, which is widely used in various nonlinear vector systems to model, for example, the propagation of optical pulses in randomly birefringent optical fibers [27], the formation of spatial solitons in planar waveguides [49], and the cross-wave states occurring in the open ocean [50]. As an important generalization of the Manakov system, the vector CLL-NLS model can more accurately describe the dynamical evolution of rogue waves in these different nonlinear media.

Due to complete integrability [51], Equation 1 can be transformed into a 3 × 3 linear eigenvalue problem R t = U R , R z = V R , (2) where R = [ r , s , w ] T is the characteristic function ( T represents transposition, r , s , and w are functions of z , t , and the complex spectral parameter λ ). Matrices U and V are given by the following equations U = − i ( λ − σ ) σ 3 2 σ γ + λ σ Q + i γ σ K Q 2 , V = − i ( λ − σ ) 2 σ 3 4 σ γ 2 + i γ 2 2 σ K Q 4 − γ 2 K ( Q Q t − Q t Q ) + λ 2 σ λ − σ γ Q − i λ σ 3 Q 2 + γ Q 3 + i σ σ 3 Q t , (3) with K = 0 0 0 0 1 0 0 0 1 , σ 3 = 1 0 0 0 − 1 0 0 0 − 1 , Q = 0 u 1 u 2 − u 1 * 0 0 − u 2 * 0 0 . The asterisk denotes the complex conjugation operation for any spectral parameter λ . Equation 1 can be easily derived from the compatibility condition of U z − V t + U V − V U = 0 .

Based on the Lax pair (Equations 2, 3), one can construct a nonrecursive DT [44, 45, 52] by which the n th-order rogue wave solutions can be obtained u 1 [ n ] = u 10 1 − i | u 10 | Y 1 M − 1 Y 2 † d e t ( M ) d e t ( M † ) , u 2 [ n ] = u 20 1 − i | u 20 | Y 1 M − 1 Y 3 † d e t ( M ) d e t ( M † ) . (4) Here the sign † indicates the complex conjugate transpose and d e t denotes the determinant of matrices. The non-recursive DT fundamentally differs from recursive approaches. It constructs the n th-order solution analytically in one step through the matrix M and vectors Y j ( j = 1,2,3 , the same below) avoiding the error-prone iterative dressing of recursive DT. The solution’s degrees of freedom are globally parameterized by 3 n complex γ s ( s = 1 , … , 3 n ), enabling direct control over rogue wave states. u ı 0 represent the seeding plane-wave solutions u ı 0 = a ı ⁡ exp ( i ω ı t − i k ı z ) , (5) the amplitude ( a ı ) , the wavenumber ( k ı ) , and the frequency ( ω ı ) follow a dispersion relationship k ı = ( a 1 2 + a 2 2 ) υ ı + 1 2 σ ω ı 2 , υ ı = γ ω ı − 1 . (6) Y j are 1 × n row vectors, defined as Y 1 Y 2 Y 3 = [ Φ ( 0 ) , Φ ( 1 ) , Φ ( 2 ) , … , Φ ( n − 1 ) ] , (7) here Φ ( m ) are the series coefficients of the Taylor expansion of the factorized column vector Φ ( λ ) = G − 1 R ( λ ) , where the given spectral parameter λ = λ 0 . M is an n × n matrix whose matrix elements M ı j can be obtained by the Taylor expansion of Φ † X Φ / ( λ − λ * ) , and G = 1 0 0 0 u 10 * / a 1 0 0 0 u 20 * / a 2 , X = λ * 0 0 0 λ 0 0 0 λ . (8) The key of the above derivation process is to solve the expression for R and to perform a Taylor expansion of it. Then, by substituting the plane-wave solution (5) into the Lax pair (2), the characteristic function R ( λ ) can be obtained, R ( λ ) = G Φ ( λ ) , Φ ( λ ) = Γ 1 N 1 + Γ 2 N 2 + Γ 3 N 3 , (9) where Γ j are arbitrary complex constants, and the expressions for N j are as follows N j = 1 c 1 j c 2 j e i φ j , φ j = μ j t + ν j z , (10) with c ı j = − i a ı γ ψ j λ ( ψ j − 2 σ υ ı ) , ψ j = 2 μ j σ γ − λ − σ , ν j = 2 A γ 2 λ − ( λ − σ ) 2 4 σ γ 2 + i λ 2 ( ω 1 a 1 c 1 j + ω 2 a 2 c 2 j ) + i λ 2 γ σ ( A γ 2 − λ + σ ) ( a 1 c 1 j + a 2 c 2 j ) , (11) the parameter μ j in Equation 10 are the three roots of the cubic equation of μ , μ − β 2 + A γ 2 σ − κ 4 3 − 3 ς μ − β 2 + A γ 2 σ − κ 4 + 2 ϱ = 0 , (12) where the coefficients are defined by the following equations ς = β 2 + δ 2 12 − ι 3 σ , ϱ = β 3 − β ( 2 ι + σ δ 2 ) 4 σ + δ ϖ 4 σ , (13) with β = A γ 2 + λ − σ 3 σ γ + κ 6 , ι = a 1 2 υ 1 + a 2 2 υ 2 , ϖ = a 1 2 υ 1 − a 2 2 υ 2 , A = a 1 2 + a 2 2 , κ = ω 1 + ω 2 , δ = ω 1 − ω 2 . (14)

From Equations 12–14, the spectral parameter λ satisfies the following equation ς 3 − ϱ 2 = 0 , (15) and it is easy to see that under the condition (Equation 15), Equation 12 will allow for the double-root μ 1 = μ 2 = μ 0 , or when ς = ϱ = 0 , allow triple-root μ 1 = μ 2 = μ 3 = μ 0 . Under both conditions, a particular value of λ 0 can be obtained for λ .

Now, let’s focus on the triple-root case. According to cubic Equation 12 there will be a triple-root μ j = μ 0 = − γ A 2 σ + κ 4 + i 3 δ 4 , (16) at the point of A = − σ δ 2 ( υ 1 + υ 2 ) / υ 1 υ 2 , where σ υ 1,2 < 0 .

To simplify the analytic derivation, we split λ 0 into a real part λ r and an imaginary part λ m , i.e., λ 0 = λ r + i λ m , in the following form λ r = − A γ 2 − 1 2 σ γ κ + σ , λ m = 3 3 2 γ δ σ . (17)

When choosing Γ j appropriately, it is easy to check that the factorized eigenfunction Φ ( λ ) given by Equation 9 will disappear at λ = λ 0 . Therefore, the Taylor expansion of Φ ( λ ) can be performed at the specific point λ = λ 0 , thereby obtaining the row vectors Y j and matrix M . Thus, in the case of triple-root, let λ = λ 0 + χ ϵ 3 [ χ = λ 0 − λ 0 * = 2 i I m ( λ 0 ) ] Γ 1 = 1 3 ∑ j = 1 n γ 3 j − 2 − γ 3 j − 1 ϕ 2 ϵ + γ 3 j ϕ 1 ϵ 2 ϵ 3 ( j − 1 ) , Γ 2 = 1 3 ∑ j = 1 n γ 3 j − 2 − γ 3 j − 1 ϕ 1 ϵ + γ 3 j ϕ 2 ϵ 2 ϵ 3 ( j − 1 ) , Γ 3 = 1 3 ∑ j = 1 n γ 3 j − 2 + γ 3 j − 1 ϵ − γ 3 j ϵ 2 ϵ 3 ( j − 1 ) , where ϵ is a complex perturbation parameter, γ s are 3 n arbitrary complex constants called structural parameters, ϕ 1 = 1 / 2 + i 3 / 2 and ϕ 2 = 1 / 2 − i 3 / 2 , which can be expanded to a power of ϵ 3 by Φ ( λ ) , Φ ( λ ) = Φ ( 0 ) + Φ ( 1 ) ϵ 3 + Φ ( 2 ) ϵ 6 + ⋅ ⋅ ⋅ + O ( ϵ 3 n ) . (18) Here, O stands for infinitesimals of the same order. Of course, with respect to the case of double-root, it is sufficient to simply replace ϵ 3 n → ϵ 2 n .

Therefore, the matrix elements of M in Equation 4 are given by the following equation Φ † X Φ λ − λ * = ∑ ı j n M ı j ϵ * 3 ( ı − 1 ) ϵ 3 ( j − 1 ) + O ( | ϵ | 6 n ) . (19) From the above derivation procedure, the three one-row matrices Y j are obtained from Equation 7, and the matrix X is obtained from Equation 8. In this way, we obtain the n th-order rogue wave solutions for Equation 4.

The triple-root condition ( ς = ϱ = 0 in Equation 15) induces critical structural changes in the DT: the eigenfunction expansion (Equation 18) requires a cubic perturbation ϵ 3 (rather than quadratic for double roots), scaling the solution space as ϵ 3 n for n th-order solutions. This necessitates the incorporation of 3 n complex parameters γ s in Γ j , effectively tripling the system’s degrees of freedom compared to the double-root case. These expanded parametric capabilities directly enable the matrix M (Equation 19) to accommodate higher-order polynomial roots, thereby unlocking rogue wave states that can simultaneously host up to n ( n +1) fundamental Peregrine solitons. The unique physical manifestation of this phenomenon originates from the nonlinear self-steepening term ( γ ≠ 0 ) in Equation 1, which amplifies the system’s capacity for extreme wave interactions while preserving the mathematical integrity of the triple-root configuration.

When the appropriate Γ j are input, the eigenfunction Φ ( λ ) in Equation 9 can be precisely extended to Equation 18. The three expansion terms Φ ( 0 ) , Φ ( 1 ) , and Φ ( 2 ) can be given by the following equations Φ ( 0 ) = T R 0 R 1 R 2 e i φ 0 , Φ ( 1 ) = T S 0 S 1 S 2 e i φ 0 , Φ ( 2 ) = T W 0 W 1 W 2 e i φ 0 , (20) where T = diag ( 1 , c 10 , c 20 ) (diag means the diagonal matrix), ψ 0 = σ ( ν − i 3 δ γ ) ( ν = υ 1 + υ 2 ) . φ 0 , c ı 0 , and ν 0 are given by Equations 10, 11 (let j = 0 ), respectively. The polynomial representations of R 0,1,2 , S 0,1,2 , and W 0,1,2 are as follows R 0 = γ 1 − i γ 2 2 τ + i γ 3 4 ( τ 2 + 2 τ + 3 i ξ ) , R ı = R 0 − 3 δ 2 α ı γ 2 − γ 3 τ − ϕ ı , S 0 = γ 1 b 0 + γ 2 e 0 + γ 3 h 0 + γ 4 − i γ 5 2 τ + i γ 6 4 τ 2 + 2 τ + 3 i ξ , S ı = S 0 − 3 δ 2 α ı γ 1 b ı + γ 2 e ı + γ 3 h ı + γ 5 − γ 6 ( τ − ϕ ı ) , W 0 = γ 1 l 0 + γ 2 p 0 + γ 3 q 0 + γ 4 b 0 + γ 5 e 0 + γ 6 h 0 + γ 7 − i γ 8 2 τ + i γ 9 4 ( τ 2 + 2 τ + 3 i ξ ) , W ı = W 0 − 3 δ 2 α ı γ 1 l ı + γ 2 p ı + γ 3 q ı + γ 4 b ı + γ 5 e ı + γ 6 h ı + γ 8 − γ 9 ( τ − ϕ ı ) . (21) where α ı = ψ 0 ( 2 σ υ ı − ψ 0 ) 4 υ ı γ σ 2 , β 2 = i ( 3 δ − i κ ) 8 σ γ 2 , ϑ = t − A γ z 2 + ( λ 0 − σ ) z 2 γ − 2 σ 2 γ 2 z β 2 , τ = 3 δ ϑ , ξ = δ 2 σ z . (22) γ s are complex structure parameters, the function ϑ and the parameters α_τ , β₂ , τ, and ξ are given by Equation 22. According to Equation 21, R 0,1,2 , S 0,1,2 , and W 0,1,2 have 3, 6, and 9 structure parameters γ s , corresponding to the first-, second-, and third-order solutions, respectively. The other polynomials b 0,1,2 , e 0,1,2 , h 0,1,2 , l 0,1,2 , p 0,1,2 , and q 0,1,2 are given in Supplementary Appendix A.

Obviously, the expressions for Y j and M can be obtained by substituting Equation 20 into Equations 7, 19, respectively. Then, we can substitute the obtained expressions for Y j and M into Equation 4 to obtain up to third-order rogue wave solutions.