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We study the Cauchy problem for the integrable coupled Hirota system with a 4 × 4 Lax pair on the line with decaying initial data . By deriving a Riemann–Hilbert representation for the solution, we compute the precise leading-order terms for long-time asymptotics based on the nonlinear steepest descent arguments. Introduction We consider the following coupled Hirota system [10], which also can be called high-order new type coupled NLS (nonlinear Schrödinger) system: i q 1 , t + q 1 , x x − 2 ( | q 1 | 2 + 2 | q 2 | 2 ) q 1 − 2 q 1 ⁎ q 2 2 + i ϵ q 1 , x x x − 6 i ϵ ( | q 1 | 2 + | q 2 | 2 ) q 1 , x − 6 i ϵ ( q 1 q 2 ⁎ + q 2 q 1 ⁎ ) q 2 , x = 0 , i q 2 , t + q 2 , x x − 2 ( | q 2 | 2 + 2 | q 1 | 2 ) q 2 − 2 q 2 ⁎ q 1 2 + i ϵ q 2 , x x x − 6 i ϵ ( | q 1 | 2 + | q 2 | 2 ) q 2 , x − 6 i ϵ ( q 2 q 1 ⁎ + q 1 q 2 ⁎ ) q 1 , x = 0 , where q 1 and q 2 denote the slowly varying complex amplitudes in two interacting optical modes, ϵ is a real parameter, and the asterisk stands for the complex conjugate. If ϵ = 0 , the coupled model (1) can be used to describe the propagation of orthogonally polarized optical waves in an isotropic medium [25], and dynamics of a one-dimensional two-component Bose-Einstein condensate system with particle transition [18]. While ϵ ≠ 0 , the coupled system (1) includes higher-order effects such as the third-order dispersion and self steepening effects. It follows that System (1) possesses potential physical applications in describing the propagation of ultrashort optical pulses in a nonlinear media. System (1) is integrable and admits a 4 × 4 matrix Lax pair, while the classical coupled Hirota system [22] involves 3 × 3 matrices. Through constructing N th-order Darboux transformation based on Lax pair, the beak-shaped rogue wave solutions for (1) have been derived in [10]. As we all known, the inverse scattering transform provides a powerful method to investigate properties of solutions to some integrable nonlinear PDEs (partial differential equations) with initial value problems. Moreover, the long-time asymptotic analysis for the integrable PDEs can be, via inverse scattering, formulated as a problem of finding asymptotics of certain oscillatory Riemann–Hilbert problems. The most influential is the nonlinear steepest descent method [7] introduced by Deift and Zhou in 1993 to study the long-time asymptotic behavior of solutions for the Cauchy problem of modified KdV (Korteweg-de Vries) equation. Subsequently, this powerful approach is successfully applied to investigate the large-time asymptotics of solutions for the initial or initial-boundary value problems of integrable nonlinear PDEs associated with 2 × 2 -matrix spectral problems with vanishing boundary conditions, for example, NLS equation [8], KdV equation [14], sine-Gordon equation [9], the Camassa–Holm equation [4], Hirota equation [15], [16], derivative NLS equation [1], fifth-order modified KdV equation [21], [23], Fokas–Lenells equation [28], short pulse equation [6], [27], and so on. In general, it is difficult and complicated to study the long-time asymptotic behavior of solutions to integrable nonlinear PDEs for the case of 3 × 3 -matrix spectral problems through the nonlinear steepest descent method. The challenge comes from the explicit unsolvability of a function δ ( k ) which satisfies a 2 × 2 -matrix Riemann–Hilbert problem and is introduced to remove the middle matrix term while splitting the jump matrix into an appropriate upper/lower triangular form. However, in literature [11], the authors developed a nature strategy to use the available function det [ δ ( k ) ] which can be solved explicitly by the Plemelj formula to approximate δ ( k ) by error control when deriving the long-time asymptotic formula for the Cauchy problem of the coupled NLS equation. Then the Deift–Zhou method was successfully generalized to derive the long-time asymptotics of the initial value problems of SaSa–Satsuma equation [19], [20], the coupled modified KdV equation [13], three-component coupled nonlinear Schrödinger system [24] and the initial-boundary value problem of coupled Hirota equation [22] associated with the high-order matrix spectral problem. Moreover, long-time asymptotics for the Degasperis–Procesi equation on the half-line and a generalizations of the NLS equation with 3 × 3 Lax pair were also studied in [5] and [2], respectively. In this paper, motivated by the work [12], we focus on the long-time asymptotic behavior of the Cauchy problem for the coupled Hirota system (1) with a 4 × 4 Lax pair by generalizing the nonlinear steepest descent method with the initial value conditions q 1 ( x , 0 ) = q 1 , 0 ( x ) , q 2 ( x , 0 ) = q 2 , 0 ( x ) belonging to the Schwartz space S ( R ) . The most outstanding structure of this system is that it admits a 4 × 4 -matrix spectral problem, which present some difficulties and novelties. (a) Compared with the 2 × 2 or 3 × 3 -matrix spectral problem, the spectral analysis of 4 × 4 -matrix spectral problem is more complicated since it characterized by a higher order. Observe that every two of eigenvalues of the 4 × 4 Lax pair for System (1) are equal, and thus we adopt the 2 × 2 block notations for the 4 × 4 -matrices as in [12], and each element of the block is a 2 × 2 -matrix. Then the analyticity and symmetry of the eigenfunctions to the Lax equations is detailed analyzed. We directly construct the associated 4 × 4 -matrix Riemann–Hilbert problem by the Jost solutions instead of using the Fredholm integral equation [5], [17]. (b) In the first step of the steepest descent analysis, i.e., triangular factorization of the jump matrix, we must introduce two different 2 × 2 -matrix valued functions δ j ( k ) ( j = 1 , 2 ) which satisfy two corresponding 2 × 2 -matrix Riemann–Hilbert problems to absorb the middle matrix terms. The solutions of these Riemann–Hilbert problems cannot be given explicitly in general, which sets an obstacle to perform scaling transformation to reduce the Riemann–Hilbert problem to a model one. Note that the determinants of the two solution matrices δ j ( k ) satisfy a same scalar Riemann–Hilbert problem, which solution denoted δ ( k ) can be solved explicitly by Plemelj formula. However, we can approximate the functions δ j ( k ) by δ ( k ) with the controlled error terms because our topic of present work is concerning the asymptotic behavior of solution. (c) To solve the model Riemann–Hilbert problem, however, we obtain four relevant ordinary differential equations in the form of 2 × 2 -matrix valued functions, which can not be directly transformed to parabolic-cylinder equation. Fortunately, by a careful observation and analysis, the relevant 2 × 2 -matrix functions can be reduced to diagonal forms. And thus they can be expressed by the parabolic-cylinder functions by solving the corresponding parabolic-cylinder equation. Finally, by the asymptotic expansion and special properties of the parabolic-cylinder function, the model Riemann–Hilbert problem is solved. (d) The spectral curve for System (1) possesses two non-symmetric stationary points, which different from the case considered in [12] where the phase function has a single critical point. Through a key symmetry observation for the two model Riemann–Hilbert problems near stationary points, the long-time asymptotic formula for the solution of Cauchy problem to coupled Hirota system (1) is obtained. The primary result of this paper is as follows. Theorem 1 Let ϵ > 0 and the initial data q 1 , 0 ( x ) , q 2 , 0 ( x ) ∈ S ( R ) . Suppose that the determinants of the 2 × 2 -matrix valued spectral functions S 1 ( k ) , S 2 ( k ) and S 4 ( k ) defined in (27) have no zeros in C + , R and C − , respectively. Then, for a given constant κ > 0 , the solution of the Cauchy problem for the coupled Hirota system (1) as t → ∞ satisfies the following asymptotic formulae: (i) For ξ = x t < 1 3 ϵ − κ , ( q 1 ( x , t ) q 2 ( x , t ) ) = 1 t ( q 1 , a s ( x , t ) q 2 , a s ( x , t ) ) + O ( ln t t ) , the leading-order coefficient ( q 1 , a s ( x , t ) q 2 , a s ( x , t ) ) is given by ( q 1 , a s ( x , t ) q 2 , a s ( x , t ) ) = e π ν ( k 1 ) 2 ν ( k 1 ) Γ ( i ν ( k 1 ) ) 2 π ( 1 − 6 ϵ k 1 ) ( γ 11 ⁎ ( k 1 ) γ 12 ⁎ ( k 1 ) ) e i ϕ a + e π ν ( k 2 ) 2 ν ( k 2 ) Γ ( − i ν ( k 2 ) ) 2 π ( 6 ϵ k 2 − 1 ) ( γ 11 ⁎ ( k 2 ) γ 12 ⁎ ( k 2 ) ) e i ϕ b , where Γ ( ⋅ ) denotes the Gamma function, ϕ a = − π 4 − ν ( k 1 ) ln ( 8 t ( k 2 − k 1 ) 2 ( 1 − 6 ϵ k 1 ) ) − 4 i t k 1 2 ( 8 ϵ k 1 − 1 ) + 1 π ∫ k 1 k 2 ln ( 1 + det [ γ ( s ) γ † ( s ) ] − | γ ( s ) | 2 1 + det [ γ ( k 1 ) γ † ( k 1 ) ] − | γ ( k 1 ) | 2 ) d s s − k 1 , ϕ b = − 3 π 4 + ν ( k 2 ) ln ( 8 t ( k 2 − k 1 ) 2 ( 6 ϵ k 2 − 1 ) ) − 4 i t k 2 2 ( 4 ϵ k 2 − 1 ) + 1 π ∫ k 1 k 2 ln ( 1 + det [ γ ( s ) γ † ( s ) ] − | γ ( s ) | 2 1 + det [ γ ( k 2 ) γ † ( k 2 ) ] − | γ ( k 2 ) | 2 ) d s s − k 2 , and k 1 , k 2 , ν ( k 1 ) , ν ( k 2 ) are defined by (66) , (67) , (131) and (133) , respectively. γ i j ( k ) is the ( i , j ) entry of the 2 × 2 -matrix valued function γ ( k ) defined in (33) . (ii) For ξ = x t > 1 3 ϵ + κ , q 1 ( x , t ) and q 2 ( x , t ) decay rapidly as t → ∞ . Remark 1 For ξ = x t > 1 3 ϵ + κ , there is no real critical points for the phase function θ ( k ) , and thus, it is easy to prove that the solution q 1 ( x , t ) and q 2 ( x , t ) of System (1) is rapidly decreasing as t → ∞ . The outline of this paper is as follows. In Section 2, we study the direct scattering, and present a detailed analysis for the analyticity, symmetry, asymptotics of Jost solutions and scattering matrix. We set up the inverse problem, formulate the associated 4 × 4 -matrix Riemann–Hilbert problem and give a representation formula for the solution of Cauchy problem of the coupled Hirota system (1) in Section 3. In Section 4, begin with the corresponding matrix Riemann–Hilbert problem, we use the ideas of the nonlinear steepest descent method to transform the Riemann–Hilbert problem to a form suitable for determining the long-time asymptotics. By solving the local models for the Riemann–Hilbert problem near the critical points, we find the asymptotic formula of solution pair q 1 ( x , t ) , q 2 ( x , t ) . Section snippets Lax pair System (1) is the compatibility condition of the following Lax pair (see [10]) Ψ x ( x , t ; k ) = U ( x , t ; k ) Ψ ( x , t ; k ) , Ψ t ( x , t ; k ) = V ( x , t ; k ) Ψ ( x , t ; k ) , where Ψ ( x , t ; k ) is a 4 × 4 -matrix valued function, k ∈ C is the spectral parameter. The 4 × 4 -matrix valued functions U and V are given by U ( x , t ; k ) = − i k Σ + i Q ( x , t ) , V ( x , t ; k ) = ( − 4 i ϵ k 3 + 2 i k 2 ) Σ + P ( x , t ; k ) , where Σ = ( I 2 × 2 0 2 × 2 0 2 × 2 − I 2 × 2 ) , Q = ( 0 2 × 2 − W ⁎ W 0 2 × 2 ) , I 2 × 2 = ( 1 0 0 1 ) , W = ( q 1 q 2 q 2 q 1 ) , P = 4 i ϵ Q k 2 + 2 ( i ϵ Σ Q 2 − i Q + ϵ Q x Σ ) k − 2 i ϵ Q 3 − i Σ Q 2 − i ϵ Q x x − Q x Σ − ϵ ( Q x Q − Q Q x ) . Note that Q and P are rapidly decaying as | x | → ∞ if The inverse problem According to the analyticity of Ψ ± ( x , t ; k ) , we study the inverse problem by means of a Riemann–Hilbert problem. Define the sectionally meromorphic 4 × 4 -matrix valued function as M ( x , t ; k ) = { ( μ L − ( x , t ; k ) S 1 − 1 ( k ) , μ R + ( x , t ; k ) ) , k ∈ C + , ( μ L + ( x , t ; k ) , μ R − ( x , t ; k ) S 4 − 1 ( k ) ) , k ∈ C − , and M ± ( x , t ; k ) = lim ε → 0 M ( x , t ; k ± i ε ) . Theorem 2 The sectionally analytic function M(x,t;k) satisfies the following 4 × 4 -matrix Riemann–Hilbert problem: • Analyticity: M ( x , t ; k ) is analytic off the real line R ; • Jump condition: For k ∈ R , M + ( x , t ; k ) = M − ( x , t ; k ) J ( x , t ; Long-time asymptotic analysis In this section, based on the associated Riemann–Hilbert problem established in Theorem 2, we study the long-time asymptotics of solution to initial value problem of the coupled Hirota system (1) with the help of the nonlinear steepest decent method. Without loss of generality, we assume ϵ > 0 and restrict our attention for the asymptotic analysis to the region ξ = x t < 1 3 ϵ − κ for a given constant κ > 0 . In this case, by calculating that d θ d k = 0 , we get two real stationary points k 1 = 1 6 ϵ ( 1 − 1 − 3 ϵ ξ ) , k 2 = 1 6 ϵ ( 1 + 1 − Acknowledgements Nan Liu was supported by the Natural Science Foundation of Jiangsu Province under Grant No. BK20220434 , the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant No. 22KJB110002 , and the Startup Foundation for Introducing Talent of NUIST under Grant No. 2022r028 . References (29) C. Charlier et al. Long-time asymptotics for an integrable evolution equation with a 3 × 3 Lax pair Physica D (2021) Z. Du et al. Beak-shaped rogue waves for a higher-order coupled nonlinear Schrödinger system with 4 × 4 Lax pair Appl. Math. Lett. (2021) X. Geng et al. Long-time asymptotics of the coupled modified Korteweg–de Vries equation J. Geom. Phys. (2019) B. Guo et al. Long-time asymptotics for the Hirota equation on the half-line Nonlinear Anal. TMA (2018) L. Huang et al. Long-time asymptotic for the Hirota equation via nonlinear steepest descent method Nonlinear Anal., Real World Appl. (2015) J. Lenells Initial-boundary value problems for integrable evolution equations with 3 × 3 Lax pairs Physica D (2012) H. Liu et al. The Deift–Zhou steepest descent method to long-time asymptotics for the Sasa–Satsuma equation J. Differ. Equ. (2018) N. Liu et al. Painlevé-type asymptotics of an extended modified KdV equation in transition regions J. Differ. Equ. (2021) W. Ma Long-time asymptotics of a three-component coupled nonlinear Schrödinger system J. Geom. Phys. (2020) J. Xu Long-time asymptotics for the short pulse equation J. Differ. Equ. (2018) J. Xu et al. Long-time asymptotics for the Fokas–Lenells equation with decaying initial value problem: without solitons J. Differ. Equ. (2015) L.K. Arruda et al. Long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line Nonlinearity (2017) R. Beals et al. Scattering and inverse scattering for first order systems Commun. Pure Appl. Math. (1984) A. Boutet de Monvel et al. Painlevé-type asymptotics for the Camassa–Holm equation SIAM J. Math. Anal. (2010) View more references Cited by (0) Recommended articles (6) Research article On toral posets and contact Lie algebras Journal of Geometry and Physics, Volume 190, 2023, Article 104861 Show abstract A ( 2 k + 1 ) -dimensional Lie algebra is called contact if it admits a one-form φ such that φ ∧ ( d φ ) k ≠ 0 . Here, we extend recent work to describe a combinatorial procedure for generating contact, type-A Lie poset algebras whose associated posets have chains of arbitrary cardinality, and we conjecture that our construction leads to a complete characterization. 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