Dynamics and stability of soliton solutions for the seventh-order Sawada-Kotera-Ito equation with applications

Abstract

The seventh-order Sawada-Kotera-Ito equation is a fundamental nonlinear partial differential equation that arises in modeling complex wave phenomena in fluid dynamics and other physical systems. In this study, two analytical techniques, namely, the $\phi ^6$-model expansion method and the extended simplest equation method are employed to derive exact analytical solutions to the seventh-order Sawada-Kotera-Ito equation. As a result, we construct explicit solutions that describe solitary waves, kink and anti-kink waves, breather-type waves, and other wave structures. The efficiency and versatility of these methods are demonstrated through the systematic derivation of solutions, which are further validated through graphical representations. The obtained solutions have diverse applications in various areas of applied sciences, and the graphical structures assist researchers in understanding the physical phenomena underlying this dynamical model. Modulational instability (MI) analysis is performed to investigate the stability of the model and its solutions. The computational results confirm that these methods are simple, straightforward, and efficient. These findings provide new insights into the dynamics of higher-order nonlinear wave equations and broaden their potential applications in mathematical physics and engineering. Moreover, these methods can be applied to other nonlinear wave equations arising in mathematical physics and related fields of applied sciences.

Introduction

Nonlinear partial differential equations (NLPDEs) occupy a central position as one of the most widely used models for describing complex phenomena in various scientific fields, including physics, biology, and chemistry^1,2,3,4,5. Among the many aspects studied within the framework of NLPDEs, traveling waves hold a particularly significant role. These waves are crucial because, under different conditions, each type represents specific nonlinear physical phenomena. Over the years, researchers have made substantial progress in deriving and solving such equations, employing numerical, analytical, and semi-analytical techniques to obtain traveling wave solutions^6,7,8,9.

Several well-established mathematical approaches have been employed to tackle NLPDEs and to extract exact or approximate solutions. Among these, the inverse scattering transform method is highly effective for solving integrable nonlinear equations¹⁰. Transformation-based techniques such as Bäcklund and Darboux transformations also play a crucial role by enabling the construction of new solutions from known ones¹¹. Additionally, perturbation and transformation techniques, including the homotopy perturbation and reduced differential methods, serve as powerful tools for approximating solutions in nonlinear systems^12,13,14. Another widely used approach is the first integral method, which involves integrating differential equations to obtain exact solutions^15,16,17,18. The $\left( \frac{G'}{G}\right)$-expansion method has also proven effective in constructing traveling wave solutions by simplifying the treatment of nonlinear equations^19,20,21. Furthermore, the sub-equation method is a commonly used technique that has successfully produced numerous soliton and periodic wave solutions^22,23,24.

Moreover, Hirota’s direct method provides a systematic procedure for constructing multi-soliton solutions^25,26,27. The homogeneous balance method is also considered highly effective, as it helps balance the nonlinear and derivative terms in an equation, thereby guiding the construction of analytical solutions^28,29. Other significant approaches include the variational iteration method, which refines approximations through successive iterations³⁰, and the tanh-sech method, known for its particular effectiveness in deriving soliton-type solutions³¹. Various methods have been applied to study nonlinear wave equations^32,33,34, including the Jacobi elliptic function method, the modified simple equation method³⁵, the exp($\phi (\xi )$)-expansion method³⁶, and the alternative functional variable method³⁷. Each of these techniques and many others constitute an essential part of the analytical toolkit for understanding nonlinear wave dynamics and their wide-ranging applications in different scientific fields^38,39,40.

The $\phi ^6$-model expansion method and the extended simplest equation method applied in this study are effective analytical techniques for transforming NLPDEs into solvable ODEs⁴¹. These methods utilize different forms of nonlinear ODEs that enable the derivation of exact analytical solutions for complex NLPDEs⁴². The strength of these approaches lies in their ability to handle a wide range of nonlinearities and higher-order derivatives, making them applicable to a broad spectrum of problems involving wave propagation, fluid dynamics, and other systems characterized by nonlinear interactions⁴³. In particular, the Riccati-Bernoulli equation, which is incorporated within the $\phi ^6$-model expansion method, proves especially useful due to its capacity to address both polynomial and exponential nonlinearities. This flexibility allows for the modeling of various nonlinear phenomena across different scientific domains. To apply these methods, a suitable ansatz or wave transformation is employed to reduce the NLPDE by minimizing the number of independent variables. Typically, this transformation reduces the PDE to an ODE with respect to a similarity variable, which links the spatial and temporal components of the system. This reduction significantly simplifies the structure of the PDE, making it compatible with the chosen ODE form⁴⁴. Once transformed, specific solution forms usually expressed in terms of hyperbolic or trigonometric functions are assumed for the unknown function⁴⁵. These assumed forms help further simplify the resulting ODE, enabling the derivation of exact solutions through systematic substitution and algebraic manipulation.

Furthermore, we present a Bäcklund transformation for the Riccati-Bernoulli equations, which plays a pivotal role in generating new solutions from known ones. This transformation enables the construction of an infinite hierarchy of novel solutions, thereby significantly expanding the solution space of the original NLPDE³. The ability to systematically derive multiple solutions from a single initial solution highlights the strength of this method in exploring the rich and diverse solution structures of NLPDEs. Such transformations are particularly valuable in the field of mathematical physics, where the existence of multiple exact solutions often leads to deeper insights into the stability, bifurcation behavior, and interaction dynamics of nonlinear wave phenomena^46,47.

The Sawada-Kotera (SK) equation was introduced as a multi-Korteweg-de Vries (mKdV) equation by Wazwaz⁴⁸ and has since been studied through various analytical approaches. The condensed Hirota method was applied to obtain multi-soliton solutions⁴⁹, after which several researchers explored the mKdV model using different techniques, such as the sine-cosine function method⁵⁰ and Kudryashov’s approach⁵¹, leading to the discovery of various types of singular solutions, including kink, bright (brilliant), and periodic waveforms. These methods were further extended to derive new dual wave solutions using modified Kudryashov and auxiliary equation methods⁵². As such, the SK equation is recognized as a special case of the mKdV equation. In the context of nonlinear partial differential equations with variable coefficients, the mKdV equation forms a crucial class of nonlinear models, particularly in optical fiber communications. In these models, the variable coefficients represent key physical parameters, such as group velocity dispersion, Kerr nonlinearity, third-order dispersion, cubic time-delay effects, tidal and wave energy variations, and the influence of external potentials^53,54.

Recent studies on nonlinear wave propagation have highlighted the significant role of higher-order NLPDEs in modeling complex physical phenomena, particularly within the frameworks of soliton theory and integrable systems. Among these, the seventh-order Sawada Kotera Ito (SKI) equation stands out due to its high level of generality⁵⁵. Building upon the fifth-order model, this equation incorporates additional dispersive terms that significantly influence the stability and evolution of solitary wave solutions⁵⁶. The inclusion of these higher-order dispersive terms enhances the model’s ability to accurately describe complex, multi-scale wave interactions capabilities that are often lacking in lower-order equations. The seventh-order SKI equation has found important applications in fields such as fluid dynamics, plasma physics, and nonlinear optics, where it provides valuable insight into wave modulation, stability properties, and nonlinear wave interaction dynamics⁵⁷.

The seventh-order Sawada-Kotera-Ito (SKI) equation is considered in this study because it serves as a powerful model for describing complex nonlinear wave phenomena in fluid dynamics, plasma physics, and other areas of applied science. Higher-order nonlinear partial differential equations like the SKI equation provide a more accurate representation of wave propagation, dispersion, and nonlinear interactions in various physical media⁵⁸. Despite its mathematical complexity, this model captures essential dynamical features that lower-order equations cannot, such as multi-peak solitons, breathers, and kink-type structures. Therefore, investigating its exact analytical solutions not only enhances our understanding of nonlinear wave behavior but also contributes to the development of robust analytical methods that can be extended to other advanced models in mathematical physics and engineering applications. The seventh-order SKI equation has attracted considerable attention in mathematical physics due to its rich structure and integrability properties. This has motivated researchers to study it using various analytical and computational techniques to explore its soliton solutions, conservation laws, and symmetries⁵⁹.

In this study, exact solutions of the seventh-order SKI equation are constructed using two analytical techniques: the $\phi ^6$-model expansion method and the extended simplest equation method. These methods yield explicit solutions that describe various nonlinear wave structures, including solitary waves, kink and anti-kink waves, breather-type waves, and other complex waveforms associated with the model. The effectiveness and flexibility of these techniques are demonstrated through the systematic derivation of solutions, which are further validated using graphical representations. To assess the stability of the model and its solutions, a modulational instability analysis is conducted. The computational findings confirm that both methods are simple, efficient, and robust, making them valuable tools for analyzing higher-order nonlinear wave equations.

The rest of the article is organized as follows: Section 2 presents the methods used in this study. Section 3 is devoted to the construction of exact solutions. Section 4 discusses the stability of the model. The discussion of the obtained results is provided in Section 5, and the conclusion is presented in the final section.

Methodology

Here, we introduce the extended simplest equation method (ESEM) in finding exact solutions to nonlinear evolution equations. This study considers one of the higher order nonlinear evolution equations as follows:

$$\begin{aligned} P\left( u,u_t,u_x,u_{\text {tt}},u_{\text {xx}},\text {...}\right) =0, \end{aligned}$$

(1)

Here, P represents a polynomial expression involving u(x,t) and its partial derivatives, incorporating both nonlinear terms and even-order highest derivatives.

Stage 1st: The travelling wave transformation

$$\begin{aligned} u(x,t)=F(\phi ), \phi =k x - \omega t, \end{aligned}$$

(2)

where k and $\omega$ are positive constant. From our last equation (1) under transformation is given the following nonlinear ODE:

$$\begin{aligned} G\left( F, -\omega F^{'}, k F^{'}, \omega ^2 F^{''}, k^2 F^{''}, \text {...} \right) =0. \end{aligned}$$

(3)

The phi-model expansion approach

The key stages of this technique are as

Stage 2nd: It is considered that equation (3) has solution in the following form:

$$\begin{aligned} F\left( \phi \right) = \sum _{i=-M}^{M} B_i \left( h(\phi ) \right) ^i \end{aligned}$$

(4)

where $B_j$ are constants, and $h\left( \phi \right)$ is derived from the Backlund transformation, $h\left( \phi \right) = \frac{ -\zeta p_2+ p_1 Z\left( \phi \right) }{p_1+p_2 Z \phi }$. In this context, $\zeta$, $p_1$, and $p_2$ are constants with $p_2 \ne 0$, and $Z\left( \phi \right)$ serves as the solution to the ensuing ODE.

$$\begin{aligned} \frac{\text {dZ}}{\text {d}}\phi = \zeta + Z \left( \phi \right) ^2. \end{aligned}$$

(5)

Stage 3rd: Applying the homogeneous balance principle on equation (3) yields the positive integer M, as shown in equation (4). First, it must be recognized that the balance number of a processor can be determined as follows

$$\begin{aligned} D\left[ \frac{ d^k f}{\text {d}\psi ^k}\right] = M + k, D\left[ f^J \frac{ \left( f d^k\right) }{\text {d}\psi ^k}\right] ^s=M\text {J} + s (k+M). \end{aligned}$$

(6)

Stage 4th: In equation (3), we substitute equation (4). Thereafter, we collect all the terms involving g($\phi$). Setting polynomial coefficients equal to zero yields a system of algebraic equations.

Stage 5th: These equations have been resolved by the Mathematica computational facilities, from which wave solutions for the equation (1) were discovered. Mathematica version 13.2 (https://www.wolfram.com/mathematica) was uesd to plot the figures of the soliton solutions.

Extended simplest equation approach

The key stages of this technique are as

Stage 2nd: The solution to Eq. (4) takes the form:

$$\begin{aligned} F(\phi ) = \sum _{i=-M}^M B_i \psi ^i (\phi ), \end{aligned}$$

(7)

here $B_i$ are arbitrary constants and the integer M arises as a positive value when balancing principle is applied to Eq. 3, and $\psi (\phi )$ satisfies the following equation.

$$\begin{aligned} \psi ' (\phi ) = a_0 + a_1 \psi + a_2 \psi ^2, \end{aligned}$$

(8)

where the arbitrary constants are $a_0,a_1$ and $a_2$.

Stage 3rd: Merging Equations (7) and (8) into equation (3) and setting the coefficients of powers of $\psi ^i$ to zero, yields an algebraic system of equations with parameters $B_{i},a_0, a_1, a_2, k$ and $\omega$. Mathematica solves the algebraic system and the values of the parameters to be determined.

Stage 4th: By replacing the values of the parameters obtained in Stage 3rd and $\psi (\phi )$ into Eq. (7), one can derive the solution to Eq. (1).

Exact solution formulation of seventh-order Sawada-Kotera-Ito equation

By using phi-model expansion approach

Acquiring wave solutions for the problem tackled here is feasible by the Riccati-Bernoulli sub-ode technique and through the B acklund transformation process. The seventh-order Sawada-Kotera-Ito equation^53,54 possesses the general form as

$$\begin{aligned} u_t + 252 u^3 u_x + 63 u_x^3 + 378 u u_x u_{\text {2x}} + 126 u^2 u_{3x} + 63 u_{2x} u_{3x} + 42 u_x u_{4x} + 21 u u_{5 x} + u_{7 x}=0. \end{aligned}$$

(9)

The Sawada-Kotera-Ito equation of seventh order is a high-order nonlinear partial differential equation that models the complexity of wave dynamics in physical systems including fluid mechanics, plasma physics, and nonlinear optics. It describes propagation and interaction of nonlinear waves so as to give us insight into the behavior of solitary and periodic wave phenomena. The unknown function u(x, t) represents an undetermined wave front. By using the transformation (2) on equation (9), the following ODE is obtained as

$$\begin{aligned} 252 k F^3 F' + k^3 \left( 63 (F')^3 + 378 F F' F'' + 126 F^2 F''' \right) + k^5 \left( 63 F'' F''' + 42 F' F^{(4)} + 21 F F^{(5)} \right) \nonumber \\ + k^7 F^{(7)}- \omega F' =0. \end{aligned}$$

(10)

By integrating with respect to $\phi$ once, we get.

$$\begin{aligned} 63kF4 + 63k^3\left( F\left( F'\right) ^2+2F^2F''\right) +21k^5\left( \left( F''\right) ^2+F'F'''+FF^4\right) +k^7F^6-\omega F=0 \end{aligned}$$

(11)

We use the proposed approach, which takes advantage of intrinsic properties in the system balancing equations, to minimize and solve for wave structures. Integrating specific phrases enables the extraction of individual features of wave events. Substituting equations (4) and (5) into equation (11) and collecting the coefficients of Z $\left( \phi \right)$ provides the system of equations. Solving the obtained system of equation via Mathematica software and solutions sets are obtained which shown below.

$$\begin{aligned} B_{-2} = -\frac{B_0^2}{2 k^2}, B_{-1} = 0, B_1 = B_2 = 0, \zeta = -\frac{B_0}{2 k^2}, \omega = 8 B_0^3 k. \end{aligned}$$

(12)

$$\begin{aligned} B_{-2} = B_{-1} = 0, B_0 = -2 \zeta k^2, B_1 = 0, B_2= -2 k^2, \omega = -64 k^7 \zeta ^3 . \end{aligned}$$

(13)

$$\begin{aligned} B_{-2} = -2 \zeta ^2 k^2, B_{-1} = 0, B_0 = -4 \zeta k^2, B_1 = 0, B_2 = -2 k^2, \omega = -4096 \zeta ^3 k^7. \end{aligned}$$

(14)

From solution sets (12), (13) and (14), The families of solutions for equation (9) are listed as follows:

Family 1

From solution set (12), we got

$$\begin{aligned} u_{11}(x,t) = B_0 + \frac{B_0^2 \tanh ^2\left( \sqrt{-\zeta } \phi \right) }{2 \zeta k^2},~~\zeta <0. \end{aligned}$$

(15)

$$\begin{aligned} u_{12} (x,t) = B_0 + \frac{B_0^2 \coth ^2\left( \sqrt{-\zeta } \phi \right) }{2 \zeta k^2},~~ \zeta <0. \end{aligned}$$

(16)

Fig. 1

The solutions (15) and (16) are represented through the assignment of suitable parameters as follows: (A) shows peakon periodic soliton with contourplot (CP) and its 2-dimensional (2-D) in (B), (C) depicts multi-peak solitons with CP and its 2-D in Fig(1.D), respectively.

From solution set (13), we got

$$\begin{aligned} u_{13} (x,t) = 2 \zeta k^2 \text {csch}^2\left( \sqrt{-\zeta } \phi \right) ,~~ \zeta < 0. \end{aligned}$$

(17)

$$\begin{aligned} u_{14} (x,t) = -2 \zeta k^2 \text {sech}^2\left( \sqrt{-\zeta } \phi \right) ,~~ \zeta <0. \end{aligned}$$

(18)

Fig. 2

With appropriate values assigned to the parameters, the solutions (17) and (18) are visualized: (A) shows singular dark solitons with CP, and their respective 2D is in (B), whereas (C) depicts bright solitons with CP, and its 2-D image is represented in (D).

From solution set (14), we got

$$\begin{aligned} u_{15} (x,t) = 2 \zeta k^2 \left( \tanh ^2\left( \sqrt{-\zeta } \phi \right) +\coth ^2\left( \sqrt{-\zeta } \phi \right) -2\right) ,~~ \zeta <0. \end{aligned}$$

(19)

$$\begin{aligned} u_{16} (x,t)= 2 \zeta k^2 \left( \tanh ^2\left( \sqrt{-\zeta } \phi \right) +\coth ^2\left( \sqrt{-\zeta } \phi \right) -1\right) ,~~ \zeta < 0. \end{aligned}$$

(20)

Family 2

From solution set (12), we got

$$\begin{aligned} u_{21}(x,t) = B_0-\frac{B_0^2 \tan ^2\left( \sqrt{\zeta } \phi \right) }{2 \zeta k^2},~~ \zeta >0. \end{aligned}$$

(21)

$$\begin{aligned} u_{22} (x,t) = B_0 - \frac{B_0^2 \cot ^2\left( \sqrt{\zeta } \phi \right) }{2 \zeta k^2},~~ \zeta >0. \end{aligned}$$

(22)

Fig. 3

The solutions (21) and (22) are represented through the assignment of suitable parameters as follows: Multi-peak periodic solitons in (A) periodic wave with CP on 2-D are shown in (B), deficiencies to aforesaid by 4-vector periodic solitons in (C), and the CP and its 2D are depicted in (D).

From solution set (13), we got

$$\begin{aligned} u_{23}(x,t) = -2 \zeta k^2 \csc ^2\left( \sqrt{\zeta } \phi \right) ,~~ \zeta > 0. \end{aligned}$$

(23)

$$\begin{aligned} u_{24}(x,t) = -2 \zeta k^2 \sec ^2\left( \sqrt{\zeta } \phi \right) ,~~ \zeta > 0. \end{aligned}$$

(24)

From solution set (14), we got

$$\begin{aligned} u_{25} (x,t) = -2 \zeta k^2 \left( \tan ^2\left( \sqrt{\zeta } \phi \right) + \cot ^2\left( \sqrt{\zeta } \phi \right) + 2 \right) ,~~ \zeta > 0. \end{aligned}$$

(25)

$$\begin{aligned} u_{26} (x,t) = -2 \zeta k^2 \left( \cot ^2\left( \sqrt{\zeta } \phi \right) +\sec ^2\left( \sqrt{\zeta } \phi \right) \right) ,~~~~~~~ \zeta > 0. \end{aligned}$$

(26)

Fig. 4

Equations (24) and (25) represent the solutions shown by assigning appropriate values for parameters as follows: (A) reveals the singular periodic solitary waves with CP and that one in (B) is 2-D. (C) shows a multi-peakon solitary wave having optional amplitudes with CP and that one in (D) is 2-D.

By using extended simple equation approach

By employing the balance principle on Eq. (11), obtain $M=2$. Thus, the solution of equation (11) is as

$$\begin{aligned} F(\phi ) = \frac{B_{-2}}{\psi ^2} + \frac{B_{-1}}{\psi } + B_0 + B_1 \psi + B_2 \psi ^2. \end{aligned}$$

(27)

By substitute Eq. (27) and Eq. (8) into Eq. (11) and equate coefficients of $\psi ^i$ to zero, this gives rise to a system of algebraic equations on $B_i, a_0, a_1, a_2, \omega$ and k. The families of solutions follow from the solution of the corresponding equations:

Family-I If we take $a_1=0$

$$\begin{aligned} B_{-2} = -2 a_0^2 k^2, B_{-1} = 0, B_0 = -2 a_0 a_2 k^2, B_1 = B_2 = 0, \omega = -64 a_0^3 a_2^3 k^7. \end{aligned}$$

(28)

$$\begin{aligned} B_{-2} = -2 a_0^2 k^2, B_{-1} = 0, B_0 = -4 a_0 a_2 k^2, B_1 = 0, B_2 = -2 a_2^2 k^2, \omega = -4096 a_0^3 a_2^3 k^7. \end{aligned}$$

(29)

The solitary wave results of Eq. (9) can be attained from solutions sets (28) and (29) in the form as

$$\begin{aligned} u_{211} \left( \phi \right) = -2 a_0 a_2 k^2 \csc ^2\left( \sqrt{a_0 a_2} (\phi + \phi _0 )\right) , a_0 a_2>0. \end{aligned}$$

(30)

$$\begin{aligned} u_{212} \left( \phi \right) = 2 a_0 a_2 k^2 \text {csch}^2\left( \sqrt{-a_0 a_2} \phi +\frac{\rho \log \left( \phi _0 \right) }{2} \right) , \phi _0>0, \rho = \pm 1, a_0 a_2<0. \end{aligned}$$

(31)

$$\begin{aligned} u_{213} (\phi ) = -8 a_0 a_2 k^2 \csc ^2\left( 2 \sqrt{a_0 a_2} (\phi + \phi _0 )\right) , a_0 a_2>0. \end{aligned}$$

(32)

$$\begin{aligned} u_{214} (\phi ) = 8 a_0 a_2 k^2 \text {csch}^2\left( 2 \sqrt{-a_0 a_2} \pi + \rho \log (\phi _0 )\right) , \phi _0>0, \rho =\pm 1, a_0 a_2<0. \end{aligned}$$

(33)

Fig. 5

The solutions (30) and (32) are depicted by assigning suitable parameter values as follows: (A) shows a singular periodic solitary wave with CP and its 2-D representation in (B), (C) depicts a periodic solitary wave with diverse amplitudes and CP, and its 2-D representation is provided in (D), respectively.

Family-II if we take $a_0=0$

$$\begin{aligned} B_{-2} = B_{-1} = B_0 = 0, B_1 = -2 a_1 a_2 k^2, B_2 = -2 a_2^2 k^2, \omega = a_1^6 k^7. \end{aligned}$$

(34)

The solitary wave results of Eq. (9) can be attained from solutions set (34) in the form as

$$\begin{aligned} u_{221} (\phi ) = -\frac{2 a_1^2 a_2 k^2 e^{a_1 (\phi +\phi _0 )}}{\left( a_2 e^{a_1 (\phi +\phi _0 )}-1\right) ^2}, a_1>0. \end{aligned}$$

(35)

$$\begin{aligned} u_{222} (\phi ) = \frac{2 a_1^2 a_2 k^2 e^{a_1 (\phi +\phi _0 )}}{\left( a_2 e^{a_1 (\phi +\phi _0 )}+1\right) ^2}, a_1<0. \end{aligned}$$

(36)

Fig. 6

The solutions (35) and (40) are defined by assigning suitable parameter values as follows: (A) shows a Dark soliton with CP and its 2-D representation in (B), and (C) shows a anti-kink soliton with CP and its 2-D representation in (D) respectively.

Family-III

$$\begin{aligned} B_{-2} = B_{-1} = 0, B_0 = -2 a_0 a_2 k^2, B_1 = -2 a_1 a_2 k^2, B_2 = -2 a_2^2 k^2, \omega = \left( a_1^2-4 a_0 a_2\right) ^3 k^7. \end{aligned}$$

(37)

$$\begin{aligned} B_{-2} = -2 a_0^2 k^2, B_{-1} = -2 a_0 a_1 k^2, B_0 = -2 a_0 a_2 k^2, B_1 = B_2 = 0, \omega = \left( a_1^2-4 a_0 a_2\right) ^3 k^7. \end{aligned}$$

(38)

The solitary wave solutions of Eq. (9) are derived from solution sets (37) and (38) as:

$$\begin{aligned} u_{331} (\phi )&= \frac{\left( a_1^2-4 a_0 a_2\right) k^2}{2} \sec ^2\left( \frac{\sqrt{4 a_0 a_2-a_1^2}}{2} (\phi +\phi _0 )\right) ,~~4 a_0 a_2>a_1^2,~~ a_1>0. \end{aligned}$$

(39)

$$\begin{aligned} u_{332} (\phi )&= -\frac{k^2 \sec ^2\left( \frac{1}{2} \sqrt{4 a_0 a_2-a_1^2} (\phi +\phi _0 )\right) }{2} \left( 2 \sqrt{4 a_0 a_2-a_1^2} a_1 \sin \left( \sqrt{4 a_0 a_2-a_1^2} (\phi +\phi _0 )\right) \right. \nonumber \\&\quad \left. +a_1^2 \left( 2 \cos \left( \sqrt{4 a_0 a_2-a_1^2} (\phi +\phi _0 )\right) + 1\right) + 4 a_0 a_2\right) , ~~~~~ 4 a_0 a_2>a_1^2 ,~ a_1<0. \end{aligned}$$

(40)

$$\begin{aligned} u_{333} (\phi )&= \frac{2 a_0 a_2 \left( a_1^2-4 a_0 a_2\right) k^2}{\left( a_1 \cos \left( \frac{\sqrt{4 a_0 a_2-a_1^2}}{2} (\phi +\phi _0 )\right) -\sqrt{4 a_0 a_2-a_1^2} \sin \left( \frac{\sqrt{4 a_0 a_2-a_1^2}}{2} (\phi +\phi _0 )\right) \right) ^2},4 a_0 a_2>a_1^2, a_1>0. \end{aligned}$$

(41)

$$\begin{aligned} u_{334} (\phi )&= - 2 a_0 a_2 k^2 \left( 1 + \frac{2 a_1}{a_1 + \sqrt{4 a_0 a_2-a_1^2} \tan \left( \frac{\sqrt{4 a_0 a_2-a_1^2}}{2} (\phi +\phi _0 )\right) } \right) . \nonumber \\&\quad\left. + \frac{4 a_0 a_2}{\left( a_1 + \sqrt{4 a_0 a_2-a_1^2} \tan \left( \frac{\sqrt{4 a_0 a_2-a_1^2}}{2} (\phi +\phi _0 )\right) \right) ^2}\right) , ~~ 4 a_0 a_2>a_1^2 ,~ a_1<0. \end{aligned}$$

(42)

Fig. 7

The solutions (41) and (42) are defined by assigning suitable parameter values as follows: (A) shows a lump Kink type solitary wave with CP and its 2-D representation in (B), and (C) depicts a kink soliton with CP and its 2-D representation in (D), respectively.

Stability analysis

Numerous higher-order nonlinear evolution systems exhibit instability, prompting the investigation of steady-state modulation arising from the interplay between nonlinear and dispersive effects. Applying a standard linear stability analysis^39,46,47, we examine the MI of model (9). The steady-state solution for model (9) is given by:

$$\begin{aligned} u(x,t)= \sqrt{S} + \mho (x,t)e^{\Gamma (t)}, ~\Gamma (t)= t S \rho \epsilon , \end{aligned}$$

(43)

here, S represents the normalized optical power, a measure of the steady energy level in the system. $\mho (x,t)$ is a small perturbation, i.e. $\mho< < \sqrt{S}$ introduced to analyze how the system behaves under slight deviations from the steady state. $\Gamma (t)$ is an exponential growth term associated with nonlinear gain/loss mechanisms. By performing linearization and substituting Eq. (43) into Eq. (9), we obtain:

$$\begin{aligned} \mho _t + S \rho \epsilon \mho + 252 S^{\frac{3}{2}} \mho _x +126 S \mho _{3x} +21 \sqrt{S} \mho _{5x} + \mho _{7x}= 0. \end{aligned}$$

(44)

To study the temporal and spatial evolution of the perturbation, considering the solution of Eq. (44) has as

$$\begin{aligned} \mho (x,t) = \eta e^{\lambda x -\tau t}, \end{aligned}$$

(45)

here, $\eta$ is the amplitude of the perturbation, $\lambda$ denote the wave number representing spatial frequency and $\tau$ is the growth rate (normalized frequency) indicating how the perturbation evolves over time. Substituting Eq. (45) into Eq. (44) yields the following relation:

$$\begin{aligned} \tau = \lambda ^7+252 \lambda S^{3/2}+21 \lambda ^5 \sqrt{S}+126 \lambda ^3 S+\rho S \epsilon . \end{aligned}$$

(46)

The dispersion relation in Eq. (46) indicates that the wave number, self-phase modulation, and stimulated Raman scattering influence the stability of the steady state. For all wave numbers $\lambda$, if $\tau$ in Eq. (46) remains real, the steady state remains stable under small perturbations.

Fig. 8

The relation in (46) between frequency($\tau$) and wave number ($\lambda$) is shown.

Discussion and physical interpretation of results

The soliton solutions obtained through the current methodology exhibit diverse forms, differing from those derived by other authors using alternative techniques. By precisely assigning parameter values in (5) and (10), various families of solutions have been achieved. The current techniques have proven effective in achieving novel soliton solutions. The range of solutions obtained includes peakon periodic solitons, multi-peak solitons, dark solitons, bright solitons, multipeak periodic solitons, singular periodic solitons, multipeakon solitons, kink solitons, anti-kink solitons, and lump kink-type solitons. The advantages of the current techniques effectively generates a wide range of novel and diverse soliton solutions, many of which are not found in existing literature. It outperforms previous techniques by offering exact, flexible, and previously unreported solution forms. These diverse solutions are distinct and introduce novel aspects not commonly found in the existing literature. Notably, some of the solutions discovered in this study belong to new categories that are unfamiliar in current research. The authors in⁵³ used the sub-equation method to investigate this dynamical model. The researchers in⁵⁴ investigated this model via the $\left( G'/G \right)$-expansion method and obtained exact closed-form solutions. Approximate solutions of this model were constructed by the authors in⁵⁷ using the homotopy perturbation method and the ZZ-transform. The residual series method⁵⁸ was used to obtain the approximate solution of this model in fractional form. The authors in⁵⁹ used the Bell-polynomial approach to construct wave solutions of this model. In⁵⁵, the authors employed the exp-function method, the $\left( G'/G \right)$-expansion method, and the ansatz method to construct exact solutions of this dynamical model. Therefore, this study includes solutions that have not been previously constructed using the aforementioned methods. As a result, it has produced numerous groundbreaking outcomes that are not documented in prior research.

In Fig. 1, by adjusting parameters to appropriate values, peakon periodic solitons and multi-peak solitons are obtained. In Fig. 2, the resulting solutions are bright and dark solitons. In Fig. 3, the outcome is multipeak periodic solitons. In Fig. 4, the results are multipeakon solitons. In Fig. 5, singular periodic solitary waves and periodic solitary waves are presented. In Fig. 6, dark solitons and anti-kink solitons are obtained. In Fig. 7, lump kink-type solitons and anti-kink solitons are shown. Fig. 8 depicts a plot of the dispersion relation (between frequency $\left( \tau \right)$ and wave number $\left( \lambda \right)$, demonstrating that for a wide range of wave numbers, $\tau$ remains real and smooth, which verifies the robustness of the steady-state wave profile under linear disturbances.

Conclusion

In this work, we have successfully employed the $\phi ^6$-model expansion method and the extended simplest equation method to analyze the seventh-order Sawada-Kotera-Ito equation. As a result, we have obtained novel analytical solutions in various forms, including trigonometric, hyperbolic, exponential, and rational functions. This NLPDE is fundamental in modeling complex wave phenomena in fluid dynamics, plasma physics, and other physical systems. The derived solutions have potential applications in engineering, nonlinear physics, and fiber optics. They offer valuable insights into wave propagation in dispersive optical media and provide a foundational understanding of more complex phenomena in such systems. By assigning appropriate parameter values, the obtained solutions can represent a wide range of wave structures, such as kink-type periodic waves, bright-dark periodic wave trains, multipeak solitons, and breather-type waves. This diversity enhances the understanding of the model’s dynamic complexity. MI analysis is conducted to investigate the stability of the model and its solutions. The computational results confirm that the proposed methods are simple, efficient, and effective. Overall, these findings contribute to the deeper understanding of higher-order nonlinear wave equations and expand their potential applications in mathematical physics and engineering. Furthermore, the techniques presented here can be extended to solve other nonlinear wave equations encountered in applied sciences.