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The inverse scattering transform is used to study the Sasa–Satsuma equation with the associated transmission coefficient consisting of N (an arbitrary finite number) pairs of higher-order poles. The direct problem is introduced by analyzing the discrete spectrum associated with N pairs of multiple zeros, and the inverse problem is characterized in terms of a 3 × 3 matrix Riemann–Hilbert problem equipped with several residue conditions at N pairs of multiple poles. In the reflectionless case, it is shown that N-multipole solutions of the Sasa–Satsuma equation can be constructed by a linear algebraic system, whose solution is unique. As an application, several explicit solutions of the Sasa-Satsuma equation are obtained by solving the linear algebraic system, including higher-order single-humped and double-humped solitons, higher-order breathers, and others. [ABSTRACT FROM AUTHOR]

This study delves into utilizing the Kudryashov auxiliary equation method on the time-fractional (3+1)-dimensional Sasa–Satsuma equation, pivotal for transmitting ultra-fast pulses in optical fibers. The main aim is to generate a variety of optical solitons for the time-fractional nonlinear (3+1)-dimensional Sasa–Satsuma model, including dark, bright, singular, and straddled solitons. These solitons are derived from exponential and hyperbolic function-type solutions, presenting novel findings not previously reported. The study demonstrates the effectiveness of the implemented method in deriving analytical solutions for fractional differential equations. The (3+1)-dimensional Sasa–Satsuma model's potential in optical fiber transmission is highlighted, offering significant insights into generating optical solitons and their applications. [ABSTRACT FROM AUTHOR]

In this paper, the extended Jacobi elliptic function expansion method is applied to conformable time-fractional Sasa-Satsuma equation. Variety of new traveling wave solutions are constructed. Thanks to the Mathematica software package, to interprete the behivor of some particular exact solutions, surfaces and contour plots are ploted. [ABSTRACT FROM AUTHOR]

In this article, the phase portraits, chaotic patterns, and traveling wave solutions of the stochastic Sasa–Satsuma equation are investigated. Firstly, the stochastic Sasa–Satsuma equation is transformed into an ordinary differential equation through traveling wave transformation. Secondly, two-dimensional planar dynamical system is presented by using the theory of planar dynamical systems. Then, the three-dimensional and two-dimensional phase portraits of the dynamical system are drawn by using Maple software. Finally, the complete discriminant system method is used to solve the stochastic Sasa-Satsuma equation, resulting in many solutions that other methods cannot obtain, including rational, trigonometric, hyperbolic, and Jacobi elliptic function solutions. Moreover, three-dimensional-surface plots and two-dimensional-shape plots for the module length of some solutions under different parameters are drawn by using Maple software. The innovation of this article lies in introducing stochastic parameters into the Sasa–Satsuma equation, obtaining more diverse and comprehensive conclusions. [ABSTRACT FROM AUTHOR]

This article investigates novel exact solutions of the Sasa-Satsuma equation for describing the nonlinear characteristics of pulse transmission in optical fiber communication. Various new exact solutions of the Sasa-Satsuma equation were derived employing both the $ \left({\frac{{{G^{\prime}}}}{{{G^{\prime}} + G + A}}} \right) $ ( G ′ G ′ + G + A ) expansion method and the Bernoulli $ \left({\frac{{G^{\prime}}}{G}} \right) $ ( G ′ G) expansion method. To provide a more intuitive depiction of the optical solitons' behaviour during transmission, we generated two-dimensional graphs, three-dimensional graphs, and contour maps corresponding to the solutions. These images enable us to showcase the dynamic behaviour of this theoretical model and delve into the specific influence of the self-steepening effect on the transmission characteristics of optical solitons described by the new exact solutions to the Sasa-Satsuma equation. [ABSTRACT FROM AUTHOR]

The nonlinear space–time fractional Sasa–Satsuma and Schrödinger–Hirota equations with beta derivative describe optical soliton, photonics, plasmas, neutral scalar masons, and long-surface gravitational waves in the real world. Through the fractional wave transform, the models are converted into a single wave variable equation. In this article, we examine a range of compatible, useful, and typical wave solutions expressed in the forms of hyperbolic, trigonometric, and rational functions uniformly through the ( Q ′ / Q , 1 / Q )-expansion approach. When specific parameter values are set, the generalized wave solutions exhibit a wide range of shapes, including asymptotic, anti-asymptotic, dark-optical, breather, lump-periodic, kink, kink-bell-shaped, homoclinic-breather, bright, dark, and periodic solitons that resemble periodic breathing patterns. We also investigate the effect of the fractional parameter δ into the wave profile, revealing a clear correlation between changes in the fractional order derivative δ and variation in the soliton's shape. The results underscore the use of this approach for the exploration of diverse nonlinear fractional systems within the context of beta derivatives. Varying the fractional-order δ and maintaining specific fixed parameter values, we depict 3D-surface, 2D-surface, density, and contour plots to visualize some of the derived solutions. [ABSTRACT FROM AUTHOR]

The exploration of higher-order and multicomponent extensions of the nonlinear Schrödinger equation holds significant importance across diverse applications, notably within the field of optics. Among these equations, the integrable Sasa–Satsuma equation stands out for its captivating soliton solutions. The Sasa–Satsuma equation serves as a mathematical representation for describing the propagation of femtosecond light pulses through optical fibers. The breather, multiwave, combined dark–bright, singular, dark, bright and periodic singular optical soliton solutions are derived in this paper by using the 1φ(ϑ),φ′(ϑ)φ(ϑ) and multivariate generalized exponential rational integral function approach. Furthermore, the paper emphasizes the essential fiber parameters essential for the emergence of these structures and offers visual representations of chosen solutions to elucidate their physical characteristics. The novelty of this work lies in the inaugural application of these methods to the Sasa–Satsuma equation and gives a significant advancement in the understanding of optical phenomena. This study not only opens avenues for further research but also introduces a fresh perspective on the Sasa–Satsuma model in the field of nonlinear physics with its applications in optical systems.

A key objective of the paper is to study the dynamical system for the two types of Sasa–Satsuma equations, namely; Sasa–Satsuma equation and higher-order Sasa–Satsuma equation. A Sasa–Satsuma equation is used to describe the propagation of femtosecond pulses through optical fiber systems. The bifurcation and chaotic characteristic of the Sasa–Satsuma equation and higher-order Sasa–Satsuma equation that arises in fluid dynamics and nonlinear optics are studied. For both models, by using the theory of planar dynamical system the bifurcation and chaotic characteristic of the Sasa–Satsuma equation and higher-order Sasa–Satsuma equation that arises in fluid dynamics and nonlinear optics are studied. For a better understanding of these dynamical behaviors, phase portraits in 2D and 3D figures are dawn. For both equations, the equilibrium points and their effects on the bifurcation behavior are analyzed. Moreover, from the presented results, both models have different dynamical behavior. [ABSTRACT FROM AUTHOR]

Optical soliton solutions in birefringent fibers and other analytic solutions for Sasa-Satsuma equation with Kerr law nonlinearity are investigated using extended F-expansion method. Bright, dark, singular, combo bright-singular, combo dark-singular and combo singular soliton solutions are extracted. Also, periodic, rational and Jacobi elliptic functions solutions are derived. Finally, we offer some 3D, 2D and contour plots to illustrate the nature of the provided solutions.

In this paper, the Sasa-Satsuma equation in fluid dynamics and nonlinear optics is investigated. Starting from the first-order Darboux transformation, we construct an N-fold generalized Darboux transformation (GDT), where N is a positive integer. Through the obtained N-fold GDT, we derive three kinds of the semirational solutions, which describe the second-order degenerate solitons, third-order degenerate solitons and interaction between the second-order degenerate solitons and one soliton, respectively. We graphically illustrate the above three kinds of semirational solutions and investigate them through the asymptotic analysis, from which we find that the characteristic lines of the semirational solutions are composed of the straight lines and curves. Expressions of the characteristic lines, positions, amplitudes, slopes, positions and phase shifts of the asymptotic solitons are presented through the asymptotic analysis. The above discussions might be extended to the higher-order solitons, and to the relevant analysis on the degenerate breathers. [ABSTRACT FROM AUTHOR]

Here, we look at the Sasa-Satsuma equation with M-truncated derivative (SSE-MTD). The analytical solutions in the form of trigonometric, hyperbolic, elliptic, and rational functions are constructed using the Jacobi elliptic function and generalizing Riccati equation mapping methods. Because the Sasa–Satsuma equation is applied to explain the propagation of femtosecond pulses in optical fibers, the acquired solutions can be employed to explain a wide range of important physical phenomena. Moreover, we apply the MATLAB tool to generate a series of graphs to address the effect of the M-truncated derivative on the exact solution of the SSE-MTD.

This current research is about the exact solitons of time-fractional Sasa-Satsuma higher order non-linear Schrödinger equation with novel definition of derivative that are achieved with the use of three analytical methods i.e extended Jacobi's elliptic expansion function method (EJEEFM), exp a function method and Kudryashov method. The obtained optical solitons are verified with the use of Mathematica software. Finally, different graphs are also drawn of the gained solutions. The main purpose of our work is to explain the effect of beta-time fractional derivative on the solutions of Sasa-Satsuma equation that are gained with the use of EJEEF, exp a function, Kudryashov methods. These exact solitons show that these three techniques are efficient, simple and reliable than other techniques. [ABSTRACT FROM AUTHOR]

The Sasa-Satsuma equation with 3 × 3 matrix spectral problem is one of the integrable extensions of the nonlinear Schrödinger equation. In this paper, we consider the Cauchy problem of the Sasa-Satsuma equation with generic decaying initial data. Based on the Rieamnn-Hilbert problem characterization for the Cauchy problem and the ∂ ‾ -nonlinear steepest descent method, we find qualitatively different long time asymptotic forms for the Sasa-Satsuma equation in three solitonic space-time regions: (1) For the region x < 0 , | x / t | = O (1) , the long time asymptotic is given by q (x , t) = u s o l (x , t | σ d (I)) + t − 1 / 2 h + O (t − 3 / 4) , in which the leading term is N (I) solitons, the second term the second t − 1 / 2 order term is soliton-radiation interactions and the third term is a residual error from a ∂ ‾ -equation. (2) For the region x > 0 , | x / t | = O (1) , the long time asymptotic is given by u (x , t) = u s o l (x , t | σ d (I)) + O (t − 1) , in which the leading term is N (I) solitons, the second term is a residual error from a ∂ ‾ -equation. (3) For the region | x / t 1 / 3 | = O (1) , the Painlevé asymptotic is found by u (x , t) = 1 t 1 / 3 u P (x t 1 / 3 ) + O (t 2 / (3 p) − 1 / 2) , 4 < p < ∞ , in which the leading term is a solution to a modified Painlevé II equation, the second term is a residual error from a ∂ ‾ -equation. [ABSTRACT FROM AUTHOR]

Sasa–Satsuma equation is proposed to model the propagation and interaction of the sub-picosecond or femtosecond pulses in a monomode optical fiber. Different from several integrable equations in the Ablowitz–Kaup–Newell–Segur system, the higher-order zeros of Riemann–Hilbert problem for the Sasa–Satsuma appear in quadruples. A new approach to study the multi-geometric discrete spectral problem with several pairs of zeros for the Sasa–Satsuma equation is proposed. Thus, the complete soliton solutions corresponding to the higher-order zeros with arbitrary geometric and algebraic multiplicities are derived. Moreover, the inelastic interactions between or among the solitons corresponding to the higher-order non-elementary zeros exhibit the shape-changing phenomena. [ABSTRACT FROM AUTHOR]

Higher order and multidimensional generalizations of the nonlinear Schrödinger equation are useful in a variety of applications. A generalized (3+1)-dimensional nonlinear Sasa–Satsuma equation is studied via the improved generalized Riccati equation mapping method and its Bäcklund transformation and the extended hyperbolic function method. This nonlinear model can be use to describe the physical processes of wave propagation in optical fibers. We explicitly retrieved varieties of optical solitons like bright, dark, singular, combined bright–dark and combined singular-bright to this multidimensional generalized nonlinear model. Other solutions, such as periodic, rational, and exponential solutions of different structures, are also produced as a result of the evolution of this improved technique. Furthermore, the graphs in two and three dimensions are provided to visualize the structure of the solutions acquired by choosing certain parameter values.

The exploration of higher-order and multicomponent extensions of the nonlinear Schrödinger equation holds significant importance across diverse applications, notably within the field of optics. Among these equations, the integrable Sasa–Satsuma equation stands out for its captivating soliton solutions. The Sasa–Satsuma equation serves as a mathematical representation for describing the propagation of femtosecond light pulses through optical fibers. The breather, multiwave, combined dark–bright, singular, dark, bright and periodic singular optical soliton solutions are derived in this paper by using the 1 φ (ϑ) , φ ′ (ϑ) φ (ϑ) and multivariate generalized exponential rational integral function approach. Furthermore, the paper emphasizes the essential fiber parameters essential for the emergence of these structures and offers visual representations of chosen solutions to elucidate their physical characteristics. The novelty of this work lies in the inaugural application of these methods to the Sasa–Satsuma equation and gives a significant advancement in the understanding of optical phenomena. This study not only opens avenues for further research but also introduces a fresh perspective on the Sasa–Satsuma model in the field of nonlinear physics with its applications in optical systems. • Nonlinear dynamics of optical phenomena. • Soliton Structures through the Sasa–Satsuma Equation. • The increasing relevance of the Sasa–Satsuma equation. [ABSTRACT FROM AUTHOR]

The Sasa-satsuma (SS) dynamical equation interpret propagation of ultra-short and femto-second pulses in optical fibers. This dynamical model has important physical significance. In this article, two mathematical techniques namely, improved F-expansion and improved auxiliary methods are utilized to construct the several types of solitons such as dark soliton, bright soliton, periodic soliton, Elliptic function and solitary waves solutions of Sasa-satsuma dynamical equation. These results have imperative applications in sciences and other fields, and constructive to recognize the physical structure of this complex dynamical model. The computing work and obtained results show the influence and effectiveness of current methods. [ABSTRACT FROM AUTHOR]

This article is about the study of several types of solitions such as periodic soliton and solitary waves have been derived for sasa-satsuma equation by employing Riccati equation mapping technique. The significant model discusses the prolongation of femto-second pulses in optical fibers and also analyzes the prolongation, and connection of the ultra-short pulses in the sub-pico second or femto-second regime. We have also discussed Modulation instability (MI) analysis with the help of linear analysis of standard stability that illustrate all obtained outcomes are stable and exact. The computational work and accomplished outcomes portray the power and efficiency of this method. Furthermore, various other higher order dynamical models can be solved with the help of this simple, effective and powerful technique. [ABSTRACT FROM AUTHOR]

We consider the initial-value problem for the Sasa–Satsuma equation on the line with decaying initial data. Using a Riemann–Hilbert formulation and steepest descent arguments, we compute the long-time asymptotics of the solution in the sector | x | ≤ M t 1 / 3 , M constant. It turns out that the asymptotics can be expressed in terms of the solution of a modified Painlevé II equation. Whereas the standard Painlevé II equation is related to a 2 × 2 matrix Riemann–Hilbert problem, this modified Painlevé II equation is related to a 3 × 3 matrix Riemann–Hilbert problem. [ABSTRACT FROM AUTHOR]

The solutions u (x , t) of the Sasa–Satsuma equation was constructed in terms of the solution of a 3 × 3 Riemann–Hilbert problem by Fokas method. To formulate the associated Riemann–Hilbert problem, we need know all of the boundary value data, i.e. u (0 , t) , u x (0 , t) , u x x (0 , t). However, for a well-posed problem, not all of these boundary data were prescribed, the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In this paper, we analyze particular linearizable boundary conditions, which means all the spectral functions can be constructed by an algebraic system in terms of the initial value data and the predicted boundary value data. [ABSTRACT FROM AUTHOR]

Optical soliton solutions in birefringent fibers and other analytic solutions for Sasa-Satsuma equation with Kerr law nonlinearity are investigated using extended F-expansion method. Bright, dark, singular, combo bright-singular, combo dark-singular and combo singular soliton solutions are extracted. Also, periodic, rational and Jacobi elliptic functions solutions are derived. Finally, we offer some 3D, 2D and contour plots to illustrate the nature of the provided solutions. [ABSTRACT FROM AUTHOR]

Abstract The initial value problem for the Sasa–Satsuma equation is transformed to a 3 × 3 matrix Riemann–Hilbert problem with the help of the corresponding Lax pair. Two distinct factorizations of the jump matrix and a decomposition of the vector-valued function ρ (k) are given, from which the long-time asymptotics for the Sasa–Satsuma equation with decaying initial data is obtained by using the nonlinear steepest descent method. [ABSTRACT FROM AUTHOR]

In this work, we utilize the improved modified extended tanh method to get optical solitons and other exact solutions for (3+1) dimensional Sasa–Satsuma model with third order dispersion that describe the dynamics of ultrashort pulses in the optical fibers. Many exact solutions are raised with the aid of the proposed method. These solutions include bright solitons, dark solitons, singular solitons, periodic wave solutions, hyperbolic wave solutions, Weierstrass elliptic solutions and Jacobi elliptic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we reestablish the elementary Darboux transformation for Sasa-Satsuma equation with the aid of loop group method. Furthermore, the generalized Darboux transformation is given with the limit technique. As direct applications, we give the single solitonic solutions for the focusing and defocusing case. The general high order solution formulas with the determinant form are obtained through generalized DT and the formal series method. [ABSTRACT FROM AUTHOR]

Sasa–Satsuma equation (SSE) is one of the nontrivial integrable extensions of nonlinear Schrödinger equation including third order dispersion, self-frequency shift and self-steepening. The hierarchy of N t h -order semi-rational solutions with 3 N free parameters of SSE can be calculated theoretically through a modified dressing transformation associated with a novel expansion technique. The Bloch eigenfunction is introduced and Schur multinomial is invoked. When the free parameters satisfy some constraints, these solutions can be reduced to pure rational solutions so as to study the dynamics of rogue waves. In addition, triangular and elliptical multi-rogue wave patterns in either single-peak or double-peak case are examined smoothly. On the other hand, the semi-rational solutions allow us to investigate various interesting superimposed scenes between rogue waves and breathers. These results may contribute to demonstrating the rogue wave phenomenon emerging in water waves, optical fibers and generally in dispersive nonlinear media. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs) can be used to predict various solutions of nonlinear partial differential equations in diverse physical fields. We use the PINNs to predict the (single-) double-hump soliton, multi-hump breather, anti-dark soliton, Mexican-hat soliton, twisted rogue-wave pairs and rational W-shaped soliton of the Sasa–Satsuma equation in optical fibers and deep water wave. The L 2 relative error of the prediction results can achieve high accuracy even with only less training data and training time while the results show that the prediction error will increase with the increase of prediction time. In addition, we employ the PINNs to investigate the parameter discovery of the Sasa–Satsuma equation via the Mexican-hat soliton solution. For the anti-dark soliton, we compare the predicted results and the amount of data for the PINNs method and the traditional split-step Fourier technique, and show that the PINNs method requires less data with the same accuracy of prediction results. • Data-driven solutions of the Sasa–Satsuma equation. • Data-driven parameters discovery of the Sasa–Satsuma equation. • Compared with the split-step Fourier technique, the PINNs method requires less data with the same accuracy of prediction results. [ABSTRACT FROM AUTHOR]

This paper mainly studies the optical soliton solutions of the coupled cubic-quartic Sasa–Satsuma equation with Kerr law nonlinearity describing the propagation of femtosecond pulses in optical fibers. Firstly, using traveling wave transformation, the coupled cubic-quartic Sasa–Satsuma equation with Kerr law nonlinearity in birefringent fibers is simplified into the coupled nonlinear ordinary differential equations. Secondly, according to the trial method of polynomial of rank homogeneous equation, the optical soliton solutions of coupled cubic-quartic Sasa–Satsuma equation with Kerr law nonlinearity can be obtained. Finally, in order to explain the propagation of optical solitons, three-dimensional and two-dimensional diagrams of the obtained solutions are drawn. [ABSTRACT FROM AUTHOR]

In this article, Painlevé integrability of the multi-component Sasa-Satsuma equation is confirmed by using the standard WTC approach and the Kruskal simplification. Then, by means of the multi-linear variable separation approach, a new exact solution with lower-dimensional arbitrary functions is constructed. For the physical quantity new coherent structure which possesses peakons at x-axis and compactons at y-axis is illustrated both analytically and graphically. [ABSTRACT FROM AUTHOR]

We analyze the rogue wave spectra of the Sasa–Satsuma equation and their appearance in the spectra of chaotic wave fields produced through modulation instability. Chaotic wave fields occasionally produce high peaks that result in a wide triangular spectrum, which could be used for rogue wave detection. [ABSTRACT FROM AUTHOR]

Higher order and multidimensional generalizations of the nonlinear Schrödinger equation are useful in a variety of applications. A generalized (3 + 1) -dimensional nonlinear Sasa–Satsuma equation is studied via the improved generalized Riccati equation mapping method and its Bäcklund transformation and the extended hyperbolic function method. This nonlinear model can be use to describe the physical processes of wave propagation in optical fibers. We explicitly retrieved varieties of optical solitons like bright, dark, singular, combined bright–dark and combined singular-bright to this multidimensional generalized nonlinear model. Other solutions, such as periodic, rational, and exponential solutions of different structures, are also produced as a result of the evolution of this improved technique. Furthermore, the graphs in two and three dimensions are provided to visualize the structure of the solutions acquired by choosing certain parameter values. • The higher-order Sasa–Satsuma equation is considered. • The improved generalized Riccati equation mapping method and Backlund transformation are used. • The obtained results are given by 2D and 3D figures. • The bright, dark, singular and combined bright–dark optical solitons are obtained. [ABSTRACT FROM AUTHOR]

Traveling wave reduction of the generalized Sasa–Satsuma equation with arbitrary refractive index is considered. The system of equations corresponding to the real and imaginary parts is presented. The two compatibility conditions to this system equations are given. Methods for finding exact solutions of nonlinear differential equations with arbitrary powers of nonlinearity are presented and discussed. Periodic and solitary wave solutions of the generalized Sasa–Satsuma equation with arbitrary refractive index are found. [ABSTRACT FROM AUTHOR]

This paper carries out the numerical analysis of perturbed optical solitons that are governed by Sasa–Satsuma equation. Laplace–Adomian decomposition is the integration scheme that is implemented to recover bright and dark optical solitons to the model. The surface plots of bright and dark optical solitons are exhibited. The contour plots and error plots are also displayed. The error measure that emerged from the integration scheme is O (1 0 − 7) and is an impressive measure! [ABSTRACT FROM AUTHOR]

This paper studies the Sasa-Satsuma equation, which is used to describe the nonlinear phenomenon of pulse propagation in optical fiber communication. By using the direct integration method, all envelope traveling wave patterns are given, including singular solutions, solitary wave solutions and elliptic function double periodic solutions. These patterns show completely the behaviors of dynamics of the model. [ABSTRACT FROM AUTHOR]

The main goal of the present paper is to steer an investigation on the dynamics of soliton solutions of a nonlinear model in the monomode optical fibers. In this respect, first, soliton solutions of the model termed as the Sasa–Satsuma (SS) equation describing the propagation of short light pulses are derived in the presence of the beta-derivatives; then, the dynamics of soliton solutions in the monomode optical fibers is analyzed for different values of the parameter β. The soliton solutions presented in this study are categorized in the most common classes of solitons namely bright and dark soliton solutions. [ABSTRACT FROM AUTHOR]

This paper obtains soliton solutions of the Sasa–Satsuma model which governs the propagation of solitons through an optical fiber. Bright, dark, singular and combo-singular soliton solutions are recovered by the application of four integration algorithms. [ABSTRACT FROM AUTHOR]

• Sasa–Satsuma equation that carries nonlinear chromatic dispersion is studied. • Stationary optical solitons are revealed. • The solutions are in terms of Gauss' hypergeometric functions. • Convergence criteria are also listed. This paper retrieves stationary optical solitons to Sasa–Satsuma equation that carries nonlinear chromatic dispersion. The solutions are in terms of Gauss' hypergeometric functions and consequently the convergence criteria are also listed. [ABSTRACT FROM AUTHOR]

We present the second order rogue wave solutions of the Sasa–Satsuma evolution equation that have significantly more complicated structure than in the case of the nonlinear Schrödinger equation. This equation takes into account the third order dispersion, self-frequency shift and self-steepening effects, thus making it closer to realistic physical applications. These effects are important both for water waves and pulses in optical fibres. Although the Sasa–Satsuma equation is integrable, the degree of complexity of these new solutions has prevented them being found previously. Here, we present in explicit form the second-order rogue wave solutions of the Sasa–Satsuma equation for three free parameter values of the family of rogue wave solutions. Special choices of the free parameters allow us to present these solutions in the simplest possible way and to illustrate them graphically. • SSE has the degree of complexity of new solutions. • We present in explicit form the second-order rogue wave solutions of SSE. • The solutions have three free parameter values of the family of rogue wave. • Special choices of the free parameters illustrate solutions graphically. [ABSTRACT FROM AUTHOR]