Exact and numerical investigation of soliton dynamics in nonlinear oscillatory system

Abstract

The generalized reaction Duffing model is an important nonlinear system that describes a variety of oscillatory phenomena arising in physics, engineering, and applied sciences. In this work, we investigate its rich soliton dynamics through a combined analytical and numerical framework. Exact traveling-wave solutions are constructed by employing the generalized Riccati equation mapping method (GREMM), whereby the governing partial differential equation is transformed into a nonlinear ordinary differential equation via a traveling-wave transformation. Under suitable parameter constraints, several families of exact solutions expressed in hyperbolic, trigonometric, and exponential functions are obtained, yielding dark, bright, singular, dark–bright, dark–singular, and periodic–singular soliton structures. The validity of all derived solutions is verified through direct substitution into the original governing equation. To complement the analytical findings, the differential transform method (DTM) is utilized to numerically approximate selected solution families, and an excellent agreement between the exact and differential transform method solutions confirms the accuracy and effectiveness of the proposed semi-analytical technique. A conserved-quantity-based stability criterion is further employed to examine the robustness of the obtained solitary waves, revealing that the considered solutions remain stable for representative parameter values. Moreover, comprehensive graphical illustrations, including three-dimensional absolute-value surfaces, density distributions, and two-dimensional wave profiles, provide physical insights into the localization, propagation, and interaction characteristics of the nonlinear wave structures. The present study demonstrates the capability of the combined generalized Riccati equation mapping method and differential transform method framework to analyze complex nonlinear dynamical systems and offers benchmark solutions that may be useful for future investigations of generalized Duffing-type models and nonlinear oscillatory media.

1. Introduction

Nonlinear partial differential equations (NLPDEs) are of great importance in describing a vast array of physical and engineering systems.^1–6 In contrast to linear equations, NLPDEs can model intricate interactions and non-trivial phenomena that take place in many natural processes.^7,8 NLPDEs appear in various areas of physics, including fluid mechanics, ⁹ optics, ¹⁰ plasma physics, ¹¹ nuclear physics, ¹² biophysics, ¹³ and quantum field theory. ¹⁴ As a result, the determination of exact solutions to NLPDEs has drawn significant interest, since these solutions give profound insight into the physical process under consideration and serve as benchmarks for numerical and approximate techniques.^15–24

One of the significant mathematical models in nonlinear dynamics is the generalized reaction Duffing model (GRDM), which models a broad class of oscillatory phenomena in nonlinear systems. The GRDM is given by the following equation ²⁵ :

$$S_{tt}+\delta S_{xx}+\omega S+p{S}^2+\kappa S^3=0,t\ge 0.$$ (1)

Equation (1) represents a generalized reaction Duffing-type wave model that captures the interplay between inertia, spatial dispersion, linear restoring forces, and nonlinear reaction effects in oscillatory media. The term S _tt denotes the inertial (temporal acceleration) response of the medium and governs wave propagation in time. The dispersive term δS _xx accounts for spatial coupling and diffusion/dispersion effects, which are responsible for the spreading or localization of wave packets depending on the sign and magnitude of δ. The linear term ωS represents the restoring force associated with linear stiffness or potential, determining the natural frequency and linear stability of small-amplitude oscillations. The quadratic nonlinearity pS² models asymmetric reaction or quadratic stiffness effects that arise in non-centrosymmetric or biased media, while the cubic nonlinearity κS³ captures higher-order nonlinear restoring forces responsible for hardening or softening behavior. The combined action of dispersion and nonlinear reaction terms enables the formation of localized nonlinear wave structures such as solitary waves and periodic patterns in a wide range of physical contexts, including nonlinear vibrations of elastic structures, wave propagation in dispersive media, and nonlinear oscillations in engineering and physical systems.

In the limit where p = 0, equation (1) simplifies to

$$S_{tt}+\delta S_{xx}+\omega S+\kappa S^3=0,t\ge 0,$$ (2)

which is a simpler nonlinear wave model with only cubic nonlinearity. Although the quadratic term is not present, the cubic nonlinearity still allows for complex behaviors of solitary waves, periodic waves, and other nonlinear waveforms. The Duffing equation, presented by George Duffing in 1918, is now a standard model in nonlinear dynamics and chaos theory. ²⁶ It is valuable because it can model nonlinear vibrations, finite-amplitude shallow water waves, and chaotic oscillations in mechanical systems. ²⁷ As such, it remains a basis for researching nonlinear wave propagation, soliton dynamics, and spectral energy transfer in various physical situations.

Exact solutions to these nonlinear evolution equations play essential roles in understanding intricate phenomena in science and engineering. For example, in nonlinear optics, it is necessary to use complex soliton solutions to describe the propagation of high-intensity laser pulses, self-phase modulation, and soliton self-focusing in order to ensure fiber optic communication over long distances.^28–32 A number of analytical methods have been formulated for the derivation of exact or approximate solutions to NLPDEs, such as Hirota’s bilinear method,^33,34 the Kudryashov method,^35,36 a novel technique of extra equations, ³⁷ extended sinh-Gordon expansion method, ³⁸ and modified Exp-function approaches.^39,40 Other significant techniques are the generalized Riccati equation mapping method (GREMM),^41–43 the q-HATM method, ⁴⁴ and the Sardar sub-equation method. ⁴⁵ The generalized Khater (GK) method with Atangana’s flexible partial derivative operator, ⁴⁶ auto-Backlund modification, ⁴⁷ Jacobi elliptic function method,^48,49 the modified exponential rational function method, ⁵⁰ the unknown coefficient method ⁵¹ and many others.^52–59

This work presents a unified analytical–numerical framework for investigating soliton dynamics in the generalized reaction Duffing model, which is relevant to a broad class of nonlinear oscillatory phenomena in physics and engineering. By employing the generalized Riccati equation mapping method (GREMM), we systematically derive multiple families of exact traveling-wave solutions, including bright, dark, singular, dark–bright, and periodic–singular solitons. ⁴¹ To complement the exact results, the Differential Transform Method (DTM) is utilized for numerical validation, and a stability analysis based on conserved quantities is carried out to assess the robustness of the obtained solutions under parametric variations. ⁶⁰ Comprehensive two-dimensional and three-dimensional graphs are given in order to highlight the characteristics of the solutions qualitatively. There are many benefits associated with the application of the methods used in the study of nonlinear evolution equations. The first benefit is offered by the GREMM, which provides a systematic procedure to construct a broad variety of exact solutions in the form of hyperbolic functions, trigonometric functions, and exponential functions. It allows the creation of several families of soliton solutions due to the variation of parameters in one structure. Another important advantage is offered by the differential transform method (DTM), which is a semi-analytical procedure that exhibits fast convergence without discretization and linearization. Thus, DTM can be considered a powerful and computationally efficient method. Finally, it is worth noting that DTM is a great tool to validate the exact solutions.

Novelty of this research is in using the combination of the two methods, namely, GREMM and DTM, for building, validating, and analyzing the soliton structures in the form of a wide range of solution classes of the generalized reaction Duffing equation in one unified way. In contrast to many other research papers that deal only with some particular type of solutions obtained by the same analytical method, this research offers a larger variety of solutions along with the analysis of their stability and validation through numerics. The solution form offers more general parametric solution families including a number of different wave forms as some limiting or particular cases discussed in previous literature.

Even though much effort has been put into the development of nonlinear Duffing-type models, many of the studies carried out so far have been either dealing with the derivation of specific exact solutions or with purely numerical simulations. The number of papers that consider the construction of analytical solutions, numerical verification, and stability analysis together is rather limited. Based on the above, this paper presents a new GREMM-DTM method used to construct different kinds of solitons, to validate them numerically, and to analyze their stability properties.

The paper structure is as follows: Section 2 describes the methodology and its application used in this research. Section 3 discusses the undetermined coefficient method, providing bright, dark, and singular soliton solutions. In Section 4, stability analysis is performed. Sections 5 present the Differential Transform Method (DTM) and its application to the first and second families of solutions derived using GREMM and the undetermined coefficient method. Section 6 depicts the results by means of detailed 2D, 3D, and density plots. The paper ends with an overview of conclusions and future work in the last Section 7.

2. Description of the generalized Riccati equation mapping method

In this section, we describe the procedure to solve the generalized reaction Duffing model using the Generalized Riccati Equation Mapping Method (GREMM). ⁴¹

Consider a general non-linear evolution equation for S (x, t):

$$U(S,S_x,S_t,S_{xx},S_{tt},\dots )=0,$$ (3)

• Step 1: Wave Transformation. We apply the wave transformation.

$$S(x,t)=\mathrm{\Psi }(\psi ),\psi =n_{0}x-kt,$$ (4)

where n₀ and k represent the wave number and wave velocity, respectively. Substituting equation (4) into equation (3) reduces the NLPDE to an ordinary differential equation (ODE) in terms of Ψ(ψ):

$$\mathrm{\Delta }(\mathrm{\Psi },{\mathrm{\Psi }}^{\prime },{\mathrm{\Psi }}^{″},\mathrm{\Psi }‴,{\mathrm{\Psi }}^{″″},\dots )=0.$$ (5)

• Step 2: Ansatz for the Solution. Assume that the solution of equation (5) takes the form

$$\mathrm{\Psi }(\psi )={\mu }_0+\sum _{i=1}^N{\mu }_{i}G{(\psi )}^i,$$ (6)

where μ _i (i = 0, 1, …, N) are constants to be determined and G(ψ) satisfies the first-order nonlinear ODE:

$$G^{\prime }(\psi )=b_1+b_{2}G(\psi )+b_{3}G{(\psi )}^2,$$ (7)

with b₁, b₂, b₃ being unknown parameters.

• Step 3: Determination of N. The integer N is determined by balancing the highest-order derivative term with the highest-order nonlinear term in equation (5). This ensures that the resulting polynomial has consistent powers for an exact solution.

• Step 4: Parameter Determination. Substituting equation (6) and its derivatives, obtained from equation (7), into equation (5) yields a polynomial in terms of G(ψ). Equating the coefficients of like powers of G(ψ) to zero produces a system of algebraic equations. Solving this system provides the values of the unknown parameters μ _i (i = 0, 1, …, N) and b _j (j = 1, 2, 3).

• Step 5: Construction of Exact Solutions. Finally, substituting the determined parameters back into equation (6) and using the wave transformation equation (4) yields multiple exact solutions of the original NLPDE in equation (3). These solutions include solitons, periodic waves, and other nonlinear structures, depending on the chosen parameters.

• Case 1: When $b_2^2-4b_1{b}_3>0$

$$G_1=-\frac{\sqrt{b_2^2-4b_3{b}_1}\tanh \left(\frac{1}{2}\sqrt{b_2^2-4b_3{b}_1}\psi \right)+b_2}{2b_3},$$

$$G_2=-\frac{\sqrt{b_2^2-4b_3{b}_1}\coth \left(\frac{1}{2}\sqrt{b_2^2-4b_3{b}_1}\psi \right)+b_2}{2b_3},$$

$$\begin{array}{cc}\hfill G_3& =-\frac{\sqrt{b_2^2-4b_3{b}_1}\left(\tanh \left(\sqrt{b_2^2-4b_3{b}_1}\psi \right)\right.}{2b_3}\hfill \\ \hfill & \left.+\text{sech}\left(\sqrt{b_2^2-4b_3{b}_1}\psi \right)\right)-\frac{b_2}{2b_3}.\hfill \end{array}$$

$$G_4=-\frac{\sqrt{b_2^2-4b_3{b}_1}\left(\coth \left(\sqrt{b_2^2-4b_3{b}_1}\psi \right)+\text{csch}\left(\sqrt{b_2^2-4b_3{b}_1}\psi \right)\right)+b_2}{2b_3},$$

$$G_5=-\frac{\sqrt{b_2^2-4b_3{b}_1}\left(\tanh \left(\frac{1}{4}\sqrt{b_2^2-4b_3{b}_1}\psi \right)+\coth \left(\frac{1}{4}\sqrt{b_2^2-4b_3{b}_1}\psi \right)\right)+2b_2}{4b_3},$$

$$G_6=\frac{\frac{\begin{array}{c}\hfill \sqrt{\left(b_2^2-4b_3{b}_1\right)\left(A^2+B^2\right)}\hfill \\ \hfill -A\sqrt{b_2^2-4b_3{b}_1}\cosh \left(\sqrt{b_2^2-4b_3{b}_1}\psi \right)\hfill \end{array}}{A\sinh \left(\sqrt{b_2^2-4b_3{b}_1}\psi \right)+B}-b_2}{2b_3},$$

$$G_7=\frac{-\frac{\begin{array}{c}\hfill \sqrt{\left(b_2^2-4b_3{b}_1\right)\left(A^2+B^2\right)}\hfill \\ \hfill +A\sqrt{b_2^2-4b_3{b}_1}\cosh \left(\sqrt{b_2^2-4b_3{b}_1}\psi \right)\hfill \end{array}}{A\sinh \left(\sqrt{b_2^2-4b_3{b}_1}\psi \right)+B}-b_2}{2b_3},$$

$$G_8=\frac{2b_3\cosh \left(\frac{1}{2}\sqrt{b_2^2-4b_3{b}_1}\psi \right)}{\sqrt{b_2^2-4b_3{b}_1}\sinh \left(\frac{1}{2}\sqrt{b_2^2-4b_3{b}_1}\psi \right)-b_2\cosh \left(\frac{1}{2}\sqrt{b_2^2-4b_3{b}_1}\psi \right)},$$

$$G_9=-\frac{2b_3\sinh \left(\frac{1}{2}\sqrt{b_2^2-4b_3{b}_1}\psi \right)}{b_2\sinh \left(\frac{1}{2}\sqrt{b_2^2-4b_3{b}_1}\psi \right)-\sqrt{b_2^2-4b_3{b}_1}\cosh \left(\frac{1}{2}\sqrt{b_2^2-4b_3{b}_1}\psi \right)},$$

$$G_{10}=\frac{2b_3\cosh \left(\frac{1}{2}\sqrt{b_2^2-4b_3{b}_1}\psi \right)}{\begin{array}{c}\hfill \sqrt{b_2^2-4b_3{b}_1}\sinh \left(\sqrt{b_2^2-4b_3{b}_1}\psi \right)-b_2\cosh \left(\sqrt{b_2^2-4b_3{b}_1}\psi \right)\hfill \\ \hfill +i\sqrt{b_2^2-4b_3{b}_1}\hfill \end{array}},$$

$$G_{11}=\frac{2b_3\sinh \left(\frac{1}{2}\sqrt{b_2^2-4b_3{b}_1}\psi \right)}{\begin{array}{c}\hfill -b_2\sinh \left(\sqrt{b_2^2-4b_3{b}_1}\psi \right)+\sqrt{b_2^2-4b_3{b}_1}\cosh \left(\sqrt{b_2^2-4b_3{b}_1}\psi \right)\hfill \\ \hfill +i\sqrt{b_2^2-4b_3{b}_1}\hfill \end{array}},$$

$$G_{12}=\frac{4b_3\sinh \left(\frac{1}{4}\sqrt{b_2^2-4b_3{b}_1}\psi \right)\cosh \left(\frac{1}{4}\sqrt{b_2^2-4b_3{b}_1}\psi \right)}{\begin{array}{c}\hfill 2\sqrt{b_2^2-4b_3{b}_1}{\text{Cosh}}^2\left(\frac{1}{4}\sqrt{b_2^2-4b_3{b}_1}\psi \right)-\sqrt{q_2^2-4b_3{b}_1}\hfill \\ \hfill -2b_2\sinh \left(\frac{1}{4}\sqrt{b_2^2-4b_3{b}_1}\psi \right)\cosh \left(\frac{1}{4}\sqrt{b_2^2-4b_3{b}_1}\psi \right)\hfill \end{array}},$$

• Case 2: When $b_2^2-4b_1{b}_3<0$

$$G_{13}=\frac{\sqrt{4b_3{b}_1-b_2^2}\tan \left(\frac{1}{2}\sqrt{4b_3{b}_1-b_2^2}\psi \right)-b_2}{2b_3},$$

$$G_{14}=-\frac{\sqrt{4b_3{b}_1-b_2^2}\cot \left(\frac{1}{2}\sqrt{4b_3{b}_1-b_2^2}\psi \right)+b_2}{2b_3},$$

$$G_{15}=\frac{\sqrt{4b_3{b}_1-b_2^2}\left(\tan \left(\sqrt{4b_3{b}_1-b_2^2}\psi \right)+\sec \left(\sqrt{4b_3{b}_1-b_2^2}\psi \right)\right)-b_2}{2b_3},$$

$$G_{16}=-\frac{\sqrt{4b_3{b}_1-b_2^2}\left(\cot \left(\sqrt{4b_3{b}_1-b_2^2}\psi \right)+\csc \left(\sqrt{4b_3{b}_1-b_2^2}\psi \right)\right)+b_2}{2b_3},$$

$$G_{17}=\frac{\sqrt{4b_3{b}_1-b_2^2}\left(\tan \left(\frac{1}{4}\sqrt{4b_3{b}_1-b_2^2}\psi \right)-\cot \left(\frac{1}{4}\sqrt{4b_3{b}_1-b_2^2}\psi \right)\right)-2b_2}{4b_3},$$

where A ≠ B represents random constants that satisfy the condition that A² > B²

$$G_{18}=\frac{\frac{\begin{array}{c}\hfill \sqrt{\left(4b_3{b}_1-b_2^2\right)\left(A^2-B^2\right)}\hfill \\ \hfill -A\sqrt{4b_3{b}_1-b_2^2}\cos \left(\sqrt{4b_3{b}_1-b_2^2}\psi \right)\hfill \end{array}}{A\sin \left(\sqrt{4b_3{b}_1-b_2^2}\psi \right)+B}-b_2}{2b_3},$$

$$G_{19}=\frac{-\frac{\begin{array}{c}\hfill \sqrt{\left(4b_3{b}_1-b_2^2\right)\left(A^2-B^2\right)}\hfill \\ \hfill +A\sqrt{4b_3{b}_1-b_2^2}\cos \left(\sqrt{4b_3{b}_1-b_2^2}\psi \right)\hfill \end{array}}{A\sin \left(\sqrt{4b_3{b}_1-b_2^2}\psi \right)+B}-b_2}{2b_3},$$

$$G_{20}=-\frac{2b_3\cos \left(\frac{1}{2}\sqrt{4b_3{b}_1-b_2^2}\psi \right)}{\sqrt{4b_3{b}_1-b_2^2}\sin \left(\frac{1}{2}\sqrt{4b_3{b}_1-b_2^2}\psi \right)+b_1\cos \left(\frac{1}{2}\sqrt{4b_3{b}_1-b_2^2}\psi \right)},$$

$$G_{21}=\frac{2b_3\sin \left(\frac{1}{2}\sqrt{4b_3{b}_1-b_2^2}\psi \right)}{b_1\sin \left(\frac{1}{2}\sqrt{4b_3{b}_1-b_2^2}\psi \right)+\sqrt{4b_3{b}_1-b_2^2}\cos \left(\frac{1}{2}\sqrt{4b_3{b}_1-b_2^2}\psi \right)},$$

$$G_{22}=-\frac{2b_3\cos \left(\frac{1}{2}\sqrt{4b_3{b}_1-b_2^2}\psi \right)}{\sqrt{4b_3{b}_1-b_2^2}\sin \left(\sqrt{4b_3{b}_1-b_2^2}\psi \right)+b_1\cos \left(\sqrt{4b_3{b}_1-b_2^2}\psi \right)+i\sqrt{4b_3{b}_1-b_2^2}},$$

$$\begin{array}{cc}\hfill G_{23}& =\frac{2b_3\sin \left(\frac{1}{2}\sqrt{4b_3{b}_1-b_2^2}\psi \right)}{-b_2\sin \left(\sqrt{4b_3{b}_1-b_2^2}\psi \right)}\hfill \\ \hfill & \times \frac{1}{-\sqrt{4b_3{b}_1-b_2^2}\cos \left(\sqrt{4b_3{b}_1-b_2^2}\psi \right)+i\sqrt{4b_3{b}_1-b_2^2}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill G_{24}& =\frac{4b_3\sin \left(\frac{\sqrt{4b_3{b}_1-b_2^2}}{4}\psi \right)\cos \left(\frac{\sqrt{4b_3{b}_1-b_2^2}}{4}\psi \right)}{-2b_2\sin \left(\frac{\sqrt{4b_3{b}_1-b_2^2}}{4}\psi \right)\cos \left(\frac{\sqrt{4b_3{b}_1-b_2^2}}{4}\psi \right)}\hfill \\ \hfill & \times \frac{1}{+2\sqrt{4b_3{b}_1-b_2^2}{\cos }^2\left(\frac{\sqrt{4b_3{b}_1-b_2^2}}{4}\psi \right)-\sqrt{4b_3{b}_1-b_2^2}}.\hfill \end{array}$$

• Case 3: When b₂ = 0, b₃b₁ ≠ 0

$$G_{25}=-\frac{b_{2}d}{b_1\left(-\sinh \left(b_2\psi \right)+\cosh \left(b_2\psi \right)+d\right)},$$

$$G_{26}=-\frac{b_2\left(\cosh \left(b_2\psi \right)-\sinh \left(b_2\psi \right)\right)}{b_1\left(-\sinh \left(b_2\psi \right)+\cosh \left(b_2\psi \right)+d\right)},$$

• Case 4: When b₃ = 0, b₂ = 0, b₁ ≠ 0

$$G_{27}=-\frac{1}{b_1\psi +c_1}.$$

2.1. Implementation of the technique

The traveling wave transformation is applied as follows:

$$S(x,t)=\mathrm{\Psi }(\psi ),\psi =n_{0}x-kt,$$ (8)

where n₀ and k represent the wave number and wave velocity, respectively. Substituting equation (8) into equation (2) reduces the NLPDE to the ordinary differential equation:

$$(k^2-\delta n_0^2){\mathrm{\Psi }}^{″}(\psi )+\kappa \mathrm{\Psi }{(\psi )}^3+\omega \mathrm{\Psi }(\psi )=0.$$ (9)

The parameter N denotes the highest power of the auxiliary function G(ψ) appearing in the solution ansatz. Its value is determined by employing the homogeneous balance principle, which equates the order of the highest derivative term with that of the highest-order nonlinear term in the reduced ordinary differential equation. For equation (9), balancing the derivative term Ψ″(ψ) with the nonlinear term Ψ³(ψ) yields

$$N+2=3N,$$

which gives

$$N=1.$$

Consequently, the solution ansatz in equation (6) reduces to:

$$\mathrm{\Psi }(\psi )={\mu }_0+{\mu }_{1}G(\psi ),$$ (10)

where μ₀ and μ₁ are constants to be determined, and G(ψ) satisfies the Riccati equation given in equation (7).

Substituting equations (7) and (10) in equation (9) and then equating the coefficients of each power of G(ψ) (i = 0, 1, 2, 3) to zero, this leads to a system of algebraic equations, whose solution determines the unknown parameters μ₀, μ₁, and b _j (j = 1, 2, 3), which in turn provide the exact soliton solutions of the original non-linear equation.

$$\begin{array}{lll}\hfill & \hfill & \hfill {(G(\psi ))}^0:b_1{b}_2{\mu }_1\left(k^2-\delta n_0^2\right)+\kappa {\mu }_0^3+{\mu }_0\omega =0,\\ \hfill & \hfill & \hfill {(G(\psi ))}^1:b_2^2{\mu }_1\left(k^2-\delta n_0^2\right)+2b_1{b}_3{\mu }_1\left(k^2-\delta n_0^2\right)\\ \hfill & \hfill & \hfill +3\kappa {\mu }_0^2{\mu }_1+{\mu }_1\omega =0,\\ \hfill & \hfill & \hfill {(G(\psi ))}^2:3b_2{b}_3{\mu }_1\left(k^2-\delta n_0^2\right)+3\kappa {\mu }_0{\mu }_1^2=0,\\ \hfill & \hfill & \hfill {(G(\psi ))}^3:2b_3^2{\mu }_1\left(k^2-\delta n_0^2\right)+\kappa {\mu }_1^3=0.\end{array}$$ (11)

To determine the unknown constants and obtain explicit results, the system of algebraic equations in equation (11) is solved using Mathematica, resulting in

$$\begin{array}{lll}\hfill {\mu }_0& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }},\\ \hfill {\mu }_1& =\hfill & \hfill \frac{\sqrt{2}{b}_3\sqrt{k^2-\delta n_0^2}}{\sqrt{-\kappa }},\\ \hfill b_1& =\hfill & \hfill \frac{b_2^2(k^2-\delta n_0^2)-2\omega }{4b_3(k^2-\delta n_0^2)}.\end{array}$$ (12)

Substituting these constants back into the ansatz in equation (10), we obtain the explicit soliton solutions of the nonlinear system as follows:

• Case 1: If $b_2^2-4b_1{b}_3>0$

$$\begin{array}{lll}\hfill S_1(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill -\frac{\sqrt{k^2-\delta n_0^2}\left(\sqrt{\chi }\tanh \left(\frac{1}{2}\psi \sqrt{\chi }\right)+b_2\right)}{\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_2(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill -\frac{\sqrt{k^2-\delta n_0^2}\left(\sqrt{\chi }\coth \left(\frac{1}{2}\psi \sqrt{\chi }\right)+b_2\right)}{\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_3(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill -\frac{\sqrt{k^2-\delta n_0^2}\left(\sqrt{\chi }\left(\coth \left(\psi \sqrt{\chi }\right)+\text{csch}\left(\psi \sqrt{\chi }\right)\right)+b_2\right)}{\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_4(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill -\frac{\sqrt{k^2-\delta n_0^2}\left(\sqrt{\chi }\left(\tanh \left(\psi \sqrt{\chi }\right)+\text{sech}\left(\psi \sqrt{\chi }\right)\right)+b_2\right)}{\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_5(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill -\frac{\sqrt{k^2-\delta n_0^2}\left(\sqrt{\chi }\left(\tanh \left(\frac{1}{4}\psi \sqrt{\chi }\right)+\coth \left(\frac{1}{4}\psi \sqrt{\chi }\right)\right)+2b_2\right)}{2\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_6(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{\sqrt{k^2-\delta n_0^2}\left(\frac{\sqrt{\left(A^2+B^2\right)\left(\chi \right)}-A\sqrt{\chi }\cosh \left(\psi \sqrt{\chi }\right)}{A\sinh \left(\psi \sqrt{\chi }\right)+B}-b_2\right)}{\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_7(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{\sqrt{k^2-\delta n_0^2}\left(-\frac{\sqrt{\left(A^2+B^2\right)\left(\chi \right)}+A\sqrt{\chi }\cosh \left(\psi \sqrt{\chi }\right)}{A\sinh \left(\psi \sqrt{\chi }\right)+B}-b_2\right)}{\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_8(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{2\sqrt{2}{b}_3^2\sqrt{k^2-\delta n_0^2}\cosh \left(\frac{1}{2}\psi \sqrt{\chi }\right)}{\sqrt{-\kappa }\left(\sqrt{\chi }\sinh \left(\frac{1}{2}\psi \sqrt{\chi }\right)-b_2\cosh \left(\frac{1}{2}\psi \sqrt{\chi }\right)\right)},\end{array}$$

$$\begin{array}{lll}\hfill S_9(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill -\frac{2\sqrt{2}{b}_3^2\sqrt{k^2-\delta n_0^2}\sinh \left(\frac{1}{2}\psi \sqrt{\chi }\right)}{\sqrt{-\kappa }\left(b_2\sinh \left(\frac{1}{2}\psi \sqrt{\chi }\right)-\sqrt{\chi }\cosh \left(\frac{1}{2}\psi \sqrt{\chi }\right)\right)},\end{array}$$

$$\begin{array}{lll}\hfill S_{10}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{2\sqrt{2}{b}_3^2\sqrt{k^2-\delta n_0^2}\cosh \left(\frac{1}{2}\psi \sqrt{\chi }\right)}{\sqrt{-\kappa }\left(\sqrt{\chi }\sinh \left(\psi \sqrt{\chi }\right)-b_2\cosh \left(\psi \sqrt{\chi }\right)+i\sqrt{\chi }\right)},\end{array}$$

$$\begin{array}{lll}\hfill S_{11}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{2\sqrt{2}{b}_3^2\sqrt{k^2-\delta n_0^2}\sinh \left(\frac{1}{2}\psi \sqrt{\chi }\right)}{\sqrt{-\kappa }\left(-b_2\sinh \left(\psi \sqrt{\chi }\right)+\sqrt{\chi }\cosh \left(\psi \sqrt{\chi }\right)+i\sqrt{\chi }\right)},\end{array}$$

$$\begin{array}{lll}\hfill S_{12}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{4\sqrt{2}{b}_3^2\sqrt{k^2-\delta n_0^2}\sinh \left(\frac{1}{4}\psi \sqrt{\chi }\right)\cosh \left(\frac{1}{4}\psi \sqrt{\chi }\right)}{\sqrt{-\kappa }\left(2\sqrt{\chi }{\text{Cosh}}^2\left(\frac{1}{4}\psi \sqrt{\chi }\right)-2b_2\sinh \left(\frac{1}{4}\psi \sqrt{\chi }\right)\cosh \left(\frac{1}{4}\psi \sqrt{\chi }\right)-\sqrt{\chi }\right)},\end{array}$$

where $\chi =b_2^2-b_2^2{k}^2-b_2^2\delta n_0^2-2\omega /k^2-\delta n_0^2$ .

• Case 2: When $b_2^2-4b_3{b}_1<0$

$$\begin{array}{lll}\hfill S_{13}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{\sqrt{k^2-\delta n_0^2}\left(\sqrt{\chi }\tan \left(\frac{1}{2}\psi \sqrt{\chi }\right)-b_2\right)}{\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_{14}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill -\frac{\sqrt{k^2-\delta n_0^2}\left(\sqrt{\chi }\cot \left(\frac{1}{2}\psi \sqrt{\chi }\right)+b_2\right)}{\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_{15}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{\sqrt{k^2-\delta n_0^2}\left(\sqrt{\chi }\left(\tan \left(\psi \sqrt{\chi }\right)+\sec \left(\psi \sqrt{\chi }\right)\right)-b_2\right)}{\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_{16}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill -\frac{\sqrt{k^2-\delta n_0^2}\left(\sqrt{\chi }\left(\cot \left(\psi \sqrt{\chi }\right)+\csc \left(\psi \sqrt{\chi }\right)\right)+b_2\right)}{\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_{17}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{\sqrt{k^2-\delta n_0^2}\left(\sqrt{\chi }\left(\tan \left(\frac{1}{4}\psi \sqrt{\chi }\right)-\cot \left(\frac{1}{4}\psi \sqrt{\chi }\right)\right)-2b_2\right)}{2\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_{18}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{\sqrt{k^2-\delta n_0^2}\left(\frac{\sqrt{\left(A^2-B^2\right)\left(\chi \right)}-A\sqrt{\chi }\cos \left(\psi \sqrt{\chi }\right)}{A\sin \left(\psi \sqrt{\chi }\right)+B}-b_2\right)}{\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_{19}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{\sqrt{k^2-\delta n_0^2}\left(-\frac{\sqrt{\left(A^2-B^2\right)\left(\chi \right)}+A\sqrt{\chi }\cos \left(\psi \sqrt{\chi }\right)}{A\sin \left(\psi \sqrt{\chi }\right)+B}-b_2\right)}{\sqrt{2}\sqrt{-\kappa }},\end{array}$$

$$\begin{array}{lll}\hfill S_{20}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill -\frac{2\sqrt{2}{b}_3^2\sqrt{k^2-\delta n_0^2}\cos \left(\frac{1}{2}\psi \sqrt{\chi }\right)}{\sqrt{-\kappa }\left(\sqrt{\chi }\sin \left(\frac{1}{2}\psi \sqrt{\chi }\right)+\frac{\left(b_2^2{k}^2-b_2^2\delta n_0^2-2\omega \right)\cos \left(\frac{1}{2}\psi \sqrt{\chi }\right)}{4b_3\left(k^2-\delta n_0^2\right)}\right)},\end{array}$$

$$\begin{array}{lll}\hfill S_{21}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{2\sqrt{2}{b}_3^2\sqrt{k^2-\delta n_0^2}\sin \left(\frac{1}{2}\psi \sqrt{\chi }\right)}{\sqrt{-\kappa }\left(\frac{\left(b_2^2{k}^2-b_2^2\delta n_0^2-2\omega \right)\sin \left(\frac{1}{2}\psi \sqrt{\chi }\right)}{4b_3\left(k^2-\delta n_0^2\right)}+\sqrt{\chi }\cos \left(\frac{1}{2}\psi \sqrt{\chi }\right)\right)},\end{array}$$

$$\begin{array}{lll}\hfill S_{22}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill -\frac{2\sqrt{2}{b}_3^2\sqrt{k^2-\delta n_0^2}\cos \left(\frac{1}{2}\psi \sqrt{\chi }\right)}{\sqrt{-\kappa }\left(\sqrt{\chi }\sin \left(\psi \sqrt{\chi }\right)+\frac{\left(b_2^2{k}^2-b_2^2\delta n_0^2-2\omega \right)\cos \left(\psi \sqrt{\chi }\right)}{4b_3\left(k^2-\delta n_0^2\right)}+i\sqrt{\chi }\right)},\end{array}$$

$$\begin{array}{lll}\hfill S_{23}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{2\sqrt{2}{b}_3^2\sqrt{k^2-\delta n_0^2}\sin \left(\frac{1}{2}\psi \sqrt{\chi }\right)}{\sqrt{-\kappa }\left(-b_2\sin \left(\psi \sqrt{\chi }\right)-\sqrt{\chi }\cos \left(\psi \sqrt{\chi }\right)+i\sqrt{\chi }\right)},\end{array}$$

$$\begin{array}{lll}\hfill S_{24}(x,t)& =\hfill & \hfill \frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}\\ \hfill & \hfill & \hfill +\frac{4\sqrt{2}{b}_3^2\sqrt{k^2-\delta n_0^2}\sin \left(\frac{1}{4}\psi \sqrt{\chi }\right)\cos \left(\frac{1}{4}\psi \sqrt{\chi }\right)}{\sqrt{-\kappa }\left(2\sqrt{\chi }{\text{Cos}}^2\left(\frac{1}{4}\psi \sqrt{\chi }\right)-2b_2\sin \left(\frac{1}{4}\psi \sqrt{\chi }\right)\cos \left(\frac{1}{4}\psi \sqrt{\chi }\right)-\sqrt{\chi }\right)},\end{array}$$

where $\chi =b_2^2-b_2^2{k}^2-b_2^2\delta n_0^2-2\omega /k^2-\delta n_0^2$

3. Undetermined coefficient approach

3.1. Bright soliton solution

To derive the bright soliton solution, we apply the soliton ansatz of the form:

$$\mathrm{\Psi }(\psi )=M{\text{sech}}^W(N\psi ).$$ (13)

Differentiating equation (13) with respect to ψ, we obtain:

$${\mathrm{\Psi }}^{″}(\psi )=M{N}^2{W}^2{\text{sech}}^W(N\psi )-M{N}^{2}W(W+1){\text{sech}}^{W+2}(N\psi ),$$ (14)

and

$${\mathrm{\Psi }}^3(\psi )=M^3{\text{sech}}^{3W}(N\psi ).$$ (15)

Substituting equations (13)–(15) into equation (9), we obtain:

$$\begin{array}{ll}\hfill & (k^2-\delta n_0^2)\left[M{N}^2{W}^2{\text{sech}}^W(N\psi )-M{N}^{2}W(W+1){\text{sech}}^{W+2}(N\psi )\right]\hfill \\ \hfill & +\omega M{\text{sech}}^W(N\psi )+\kappa M^3{\text{sech}}^{3W}(N\psi )=0.\hfill \end{array}$$ (16)

Balancing the highest powers of the hyperbolic function, i.e., W + 2 = 3W, yields W = 1. Substituting W = 1 into equation (16) leads to:

$$\begin{array}{ll}\hfill & (k^2-\delta n_0^2)\left[M{N}^2\text{sech}(N\psi )-2M{N}^2{\text{sech}}^3(N\psi )\right]\hfill \\ \hfill & +\omega M\text{sech}(N\psi )+\kappa M^3{\text{sech}}^3(N\psi )=0.\hfill \end{array}$$ (17)

Equating the coefficients of sech (Nψ) and sech³(Nψ) gives:

$$M=\frac{\sqrt{2\omega }}{\sqrt{-\kappa }},N=\frac{\sqrt{-\omega }}{\sqrt{k^2-\delta n_0^2}}.$$ (18)

Substituting equation (18) into equation (13), the bright soliton solution is obtained as:

$$S(\psi )=\frac{\sqrt{2\omega }}{\sqrt{-\kappa }}\text{sech}\left(\frac{\psi \sqrt{-\omega }}{\sqrt{k^2-\delta n_0^2}}\right).$$ (19)

3.2. Dark soliton solution

To obtain the dark soliton profile, we consider the following ansatz:

$$\mathrm{\Psi }(\psi )=M{\tanh }^W(N\psi ).$$ (20)

Differentiating equation (20) twice with respect to ψ, we obtain:

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}^{″}(\psi )=& M{N}^{2}W(W+1){\tanh }^{W+2}(N\psi )\hfill \\ \hfill & +M{N}^{2}W(W-1){\tanh }^{W-2}(N\psi )\hfill \\ \hfill & -2M{N}^2{W}^2{\tanh }^W(N\psi ).\hfill \end{array}$$ (21)

and

$${\mathrm{\Psi }}^3(\psi )=M^3{\tanh }^{3W}(N\psi ).$$ (22)

Substituting equations (20)–(22) into equation (9) yields:

$$\begin{array}{ll}\hfill & (k^2-\delta n_0^2)\left[M{N}^{2}W(W+1){\tanh }^{W+2}(N\psi )+M{N}^2\right.\hfill \\ \hfill & \left.W(W-1){\tanh }^{W-2}(N\psi )-2M{N}^2{W}^2{\tanh }^W(N\psi )\right]\hfill \\ \hfill & +\omega M{\tanh }^W(N\psi )+\kappa M^3{\tanh }^{3W}(N\psi )=0.\hfill \end{array}$$ (23)

Balancing the dominant exponents in the derivative and nonlinear terms (W + 2 = 3W) gives W = 1. Substituting W = 1 into equation (23), we get:

$$\begin{array}{l}\hfill (k^2-\delta n_0^2)\left[2M{N}^2{\tanh }^3(N\psi )-2M{N}^2\tanh (N\psi )\right]\end{array}$$ (24)

$$\begin{array}{l}\hfill +\omega M\tanh (N\psi )+\kappa M^3{\tanh }^3(N\psi )=0.\end{array}$$ (25)

Equating the coefficients of tanh (Nψ) and tanh³(Nψ), we obtain:

$$M=\sqrt{\frac{-\omega }{\kappa }},N=\sqrt{\frac{\omega }{2(k^2-\delta n_0^2)}}.$$ (26)

Substituting equation (26) into the ansatz equation (20), the dark soliton solution is obtained as:

$$\mathrm{\Psi }(\psi )=\frac{\sqrt{-\omega }}{\sqrt{\kappa }}\tanh \left(\frac{\psi \sqrt{\omega }}{\sqrt{2(k^2-\delta n_0^2)}}\right).$$ (27)

3.2. Singular soliton solution

To construct the singular soliton profile, we consider the ansatz:

$$\mathrm{\Psi }(\psi )=M{\coth }^W(N\psi ).$$ (28)

The second derivative of equation (28) with respect to ψ is calculated as:

$$\begin{array}{lll}\hfill & \hfill & \hfill {\mathrm{\Psi }}^{″}(\psi )=M{N}^{2}W(W+1){\coth }^{W+2}(N\psi )\\ \hfill & \hfill & \hfill -2M{N}^2{W}^2{\coth }^W(N\psi )+2M{N}^{2}W(W-1)\\ \hfill & \hfill & \hfill {\coth }^{W-2}(N\psi ),\end{array}$$

and the nonlinear term is given by:

$${\mathrm{\Psi }}^3(\psi )=M^3{\coth }^{3W}(N\psi ).$$ (29)

Substituting equations (28)–(29) into equation (9) yields:

$$\begin{array}{ll}\hfill & (k^2-\delta n_0^2)\left[M{N}^{2}W(W+1){\coth }^{W+2}(N\psi )-2M{N}^2{W}^2\right.\hfill \\ \hfill & \left.{\coth }^W(N\psi )+2M{N}^{2}W(W-1){\coth }^{W-2}(N\psi )\right]\hfill \\ \hfill & +\omega M{\coth }^W(N\psi )+\kappa M^3{\coth }^{3W}(N\psi )=0.\hfill \end{array}$$ (30)

Balancing the dominant powers of coth (Nψ) from the derivative and nonlinear terms, i.e., W + 2 = 3W, gives W = 1. Substituting W = 1 into equation (30) leads to:

$$\begin{array}{lll}\hfill & \hfill & \hfill (k^2-\delta n_0^2)\left[2M{N}^2{\coth }^3(N\psi )-2M{N}^2\coth (N\psi )\right]+\\ \hfill & \hfill & \hfill \omega M\coth (N\psi )+\kappa M^3{\coth }^3(N\psi )=0.\end{array}$$ (31)

Comparing the coefficients of coth (Nψ) and coth³ (Nψ) yields:

$$M=\frac{\sqrt{-2\omega }}{\sqrt{\kappa }},N=\frac{\sqrt{\omega }}{\sqrt{2(k^2-\delta n_0^2)}}.$$ (32)

Substituting equation (32) into equation (28), we obtain the singular soliton solution as:

$$\mathrm{\Psi }(\psi )=\frac{\sqrt{2}\sqrt{-\omega }}{\sqrt{\kappa }}\coth \left(\frac{\psi \sqrt{\omega }}{\sqrt{2(k^2-\delta n_0^2)}}\right).$$ (33)

4. Stability analysis

Stability analysis constitutes a central component in the qualitative assessment of nonlinear dynamical systems and has been investigated using a variety of analytical frameworks in the literature. For wave-type and soliton-bearing equations, energy-based criteria, spectral stability, modulational instability, and perturbation approaches are commonly employed to assess the robustness of localized structures under parametric variations and external disturbances. In parallel, for fractional-order and compartmental dynamical models arising in applied sciences, Lyapunov-based methods and fractional stability theories have been widely used to characterize equilibrium stability and long-term dynamical behavior. These diverse stability frameworks reflect the model-dependent nature of stability concepts and provide complementary perspectives on system robustness. The present work aligns with this body of literature by adopting a conserved-quantity-based stability criterion tailored to traveling-wave solutions of nonlinear dispersive systems, thereby offering a physically interpretable stability assessment consistent with established approaches for soliton dynamics. ⁶¹ The solitary wave solution S₁ (x, t) is considered, and the corresponding conserved quantity is defined as follows:

$$M=\frac{1}{2}{\int }_{-\infty }^{\infty }{S}_1^2(x,t)dx,$$ (34)

which represents the energy (or mass) of the soliton. For the numerical computations, the infinite integration domain is approximated by the finite interval x ∈ [−10, 10], giving

$$M=\frac{1}{2}{\int }_{-10}^{10}{\left[\frac{b_2\sqrt{{\kappa }^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }}-\frac{\sqrt{{\kappa }^2-\delta n_0^2}\left(b_2+\sqrt{\chi }\tanh \left(\frac{1}{2}(-kt+n_{0}x)\sqrt{\chi }\right)\right)}{\sqrt{2}\sqrt{-\kappa }}\right]}^{2}dx.$$ (35)

To evaluate the stability of the soliton, we calculate the derivative of M with respect to the propagation parameter k. The stability criterion is given by

$$\frac{\partial M}{\partial k}>0\Rightarrow \text{stable},\frac{\partial M}{\partial k}<0\Rightarrow \text{unstable}.$$

Carrying out the differentiation symbolically in Mathematica, we obtain the following expression for ∂M/∂k:

$$\begin{array}{cc}\hfill \frac{\partial M}{\partial k}=& \frac{\left(-n_0^2\delta +{\kappa }^2\right)\sqrt{\chi }}{2n_0\kappa }\left[-\frac{1}{2}t\sqrt{\chi }{\text{sech}}^2\left(\frac{(10n_0-kt)\sqrt{\chi }}{2}\right)\right.\hfill \\ \hfill & \left.+\frac{1}{2}t\sqrt{\chi }{\text{sech}}^2\left(\frac{(10n_0+kt)\sqrt{\chi }}{2}\right)\right].\hfill \end{array}$$ (36)

For the soliton solution S₁(x, t), stability is determined by the sign of ∂M/∂k. The solution is considered stable when ∂M/∂k > 0. For n₀ = 1.0, δ = 0.2, κ = −1.0, χ = 0.8, t = 1.0, k = 0.5.

Substituting these values into the analytical expression for ∂M/∂k gives

$$\frac{\partial M}{\partial k}\approx 7.7\times 10^{-6}>0,$$

which indicates that the soliton is stable. For another set of parameters n₀ = 1.2, δ = 0.15, κ = −1.5, χ = 1.0, t = 1.0, k = 0.6, we obtain

$$\frac{\partial M}{\partial k}\approx 8.84\times 10^{-6}>0,$$

so the soliton is again stable.

In both examples, the derivative ∂M/∂k remains positive, confirming the stability of the soliton solution for these parameter choices.

5. Differential transform method

Almost all nonlinear issues are usually difficult to solve analytically and produce accurate answers in the fields of chemistry, physics, and engineering. In this situation, the semi-analytical and numerical methods so emerge as a powerful alternative tactic. The effectiveness of the differential transform technique (DTM) in handling nonlinear has been established. In order to solve both linear and nonlinear initial value problems in the study of electric circuits, Zhou first introduced the idea of the differential transform. The definition of the hth derivative of the differential transform of the function y(x) is

$$Y(h)=\frac{1}{h!}\left(\frac{d^{h}y(x)}{d{x}^h}\right)$$ (37)

where the initial purpose is y(x) and the transformed function is Y(h). The definition of the differential inverse transform is Y(h) is

$$y(x)=\sum _{h=0}^{\infty }Y(h)x^h\approx y_N(x)=\sum _{h=0}^{N}Y(h)x^h.$$ (38)

When we replace equation (37) in equation (38), we obtain

$$y(x)=\sum _{h=0}^{\infty }\frac{x^h}{h!}\frac{d^{h}y(x)}{d{x}^h}{|}_{x=0}.$$ (39)

It demonstrates how the differential transform idea is derived via Taylor series expansion. Although applicable to any given real number, x = x₀, the earlier version only considers the situation in which x = 0. Equations (38)–(41) determine the main theorems that can be deduced from equations (37) and (39).

5.1. DTM validation of the first family of GREMM solutions

This section contrasts the numerical treatment of the wave solutions of equation (9) with the exact solution of the same problem using the first family of GREMM of the equation.

Making use of the values

$$\begin{array}{lll}\hfill & \hfill & \hfill b_3=5,b_2=-4,\delta =2,\kappa =-2,n_0=-1,\\ \hfill & \hfill & \hfill \omega =-2,{\mu }_0=\frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }},{\mu }_1=\frac{\sqrt{2}{b}_3\sqrt{k^2-\delta n_0^2}}{\sqrt{-\kappa }},\\ \hfill & \hfill & \hfill b_1=\frac{b_2^2{k}^2-b_2^2\delta n_0^2-2\omega }{4b_3\left(k^2-\delta n_0^2\right)}.\end{array}$$ (40)

Equation (9) coverts in

$${\mathrm{\Psi }}^{″}-{\mathrm{\Psi }}^3-\mathrm{\Psi }=0,$$ (41)

the precise solution of equation (41), and these are the prerequisites:

$$\begin{array}{lll}\hfill & \hfill & \hfill \mathrm{\Psi }(\psi )=\tan \left(\frac{\psi +1}{\sqrt{2}}\right),\psi =x+t,\\ \hfill & \hfill & \hfill \mathrm{\Psi }(0)=\tan \left(\frac{1}{\sqrt{2}}\right),{\mathrm{\Psi }}^{\prime }(0)=\frac{{\sec }^2\left(\frac{1}{\sqrt{2}}\right)}{\sqrt{2}}.\end{array}$$ (42)

When applying the differential transform technique to the equations (40) and (41) system, the outcomes are as follows:

$$(h+1)(h+2)Y(h+2)-F(h)-Y(h)=0,$$ (43)

where Y(h) is the function that has been converted to match the original function Ψ(ψ), and F(h) is the change for the following nonlinear term Ψ³.,

$$F(h)=\left\{\begin{array}{cc}{\left(Y(0)\right)}^3,\hfill & \text{if\hspace{0.17em}}h=0,\hfill \\ \frac{1}{Y(0)}\sum _{r=1}^h\frac{\left((m+1)r-h\right)Y(r)F(h-r)}{h},\hfill & \text{if\hspace{0.17em}}h\ge 1.\hfill \end{array}\right.$$ (44)

From equation (43), we learn about their current relationships:

$$\begin{array}{lll}\hfill & \hfill & \hfill Y(h+2)=\frac{F(h)+Y(h)}{(h+1)(h+2)};Y(0)=\mathrm{\Psi }(0)=\tan \left(\frac{1}{\sqrt{2}}\right),\\ \hfill & \hfill & \hfill Y(1)={\mathrm{\Psi }}^{\prime }(0)=\frac{{\sec }^2\left(\frac{1}{\sqrt{2}}\right)}{\sqrt{2}}.\end{array}$$ (45)

When we enter h = 0 in equation (45) we obtain

$$Y(2)=\frac{F(0)+Y(0)}{(1)(2)}=0.739232.$$

Similarly, we can obtain the coefficients Y(h) as

$$Y(2)=0.739232,Y(3)=0.650571.$$

The DTM solution of equation (41) with initial conditions equation (42) is then

$$\begin{array}{lll}\hfill & \hfill & \hfill {\mathrm{\Psi }}_{\mathit{DTM}}(\psi )=\sum _{h=0}^{\infty }Y(h){\psi }^h\Rightarrow {\mathrm{\Psi }}_N(\psi )\cong \sum _{h=0}^{N}Y(h){\psi }^h=\\ \hfill & \hfill & \hfill \sum _{h=0}^{10}Y(h){\psi }^h=0.85451+1.22343\psi +\\ \hfill & \hfill & \hfill 0.739232{\psi }^2+0.650571{\psi }^3+0.5163{\psi }^4+0.427189{\psi }^5\\ \hfill & \hfill & \hfill +0.348277{\psi }^6+0.285576{\psi }^7+0.233664{\psi }^8\\ \hfill & \hfill & \hfill +0.191344{\psi }^9+0.156641{\psi }^{10}.\end{array}$$ (46)

The expansion of the exact solution in the Maclaurin-Taylor series is very close to this.

$$\begin{array}{lll}\hfill & \hfill & \hfill {\mathrm{\Psi }}_{\mathit{Exact}}(\psi )=0.85451+1.22343\psi +0.739232{\psi }^2\\ \hfill & \hfill & \hfill +0.650571{\psi }^3+0.5163{\psi }^4+0.427189{\psi }^5+\\ \hfill & \hfill & \hfill 0.348277{\psi }^6+0.285576{\psi }^7+0.233664{\psi }^8\\ \hfill & \hfill & \hfill +0.191344{\psi }^9+0.156641{\psi }^{10}.\end{array}$$ (47)

Similarly, Figure 7 illustrates the DTM’s complete discrepancy between the exact answer and the numerical solution.

$$Error=|Exactsolution-DTMsolution|=\left[\begin{array}{c}\hfill 0.\hfill \\ \hfill 1.11022\ast 10^{-16}.\hfill \\ \hfill 2.22045\ast 10^{-16}.\hfill \\ \hfill 0.\hfill \\ \hfill 2.22045\ast 10^{-16}.\hfill \\ \hfill 2.22045\ast 10^{-16}.\hfill \\ \hfill -4.44089\ast 10^{-16}.\hfill \\ \hfill 4.44089\ast 10^{-16}.\hfill \\ \hfill 4.44089\ast 10^{-16}.\hfill \\ \hfill -8.88178\ast 10^{-16}.\hfill \\ \hfill 0.\hfill \end{array}\right]$$ (48)

Figure 7.

Pictorial 2D display of Ψ (ψ).

5.2. DTM validation of the second family of GREMM solutions

In this example, the numerical solution of equation (9) obtained via the Differential Transform Method (DTM) is compared with the exact solution derived from the first family (GREMM) of the equation.

Using the parameter values:

$$\begin{array}{ll}\hfill & b_3=5,b_2=-4,\delta =2,\kappa =-2,n_0=-1,\omega =2,\hfill \\ \hfill & {\mu }_0=\frac{b_2\sqrt{k^2-\delta n_0^2}}{\sqrt{2}\sqrt{-\kappa }},{\mu }_1=\frac{\sqrt{2}{b}_3\sqrt{k^2-\delta n_0^2}}{\sqrt{-\kappa }},\hfill \\ \hfill & b_1=\frac{b_2^2{k}^2-b_2^2\delta n_0^2-2\omega }{4b_3(k^2-\delta n_0^2)},\hfill \end{array}$$ (49)

Equation (9) reduces to the following nonlinear ordinary differential equation:

$${\mathrm{\Psi }}^{″}(\psi )-{\mathrm{\Psi }}^3(\psi )+\mathrm{\Psi }(\psi )=0.$$ (50)

The exact solution of equation (50), subject to the initial conditions, is given by:

$$\begin{array}{lll}\hfill & \hfill & \hfill \mathrm{\Psi }(\psi )=-\tanh \left(\frac{\psi +1}{\sqrt{2}}\right),\psi =x+t,\\ \hfill & \hfill & \hfill \mathrm{\Psi }(0)=-\tanh \left(\frac{1}{\sqrt{2}}\right),{\mathrm{\Psi }}^{\prime }(0)=-\frac{{\text{sech}}^2\left(\frac{1}{\sqrt{2}}\right)}{\sqrt{2}}.\end{array}$$ (51)

Applying the Differential Transform Method to equations (50) and (51) yields:

$$(h+1)(h+2)Y(h+2)-F(h)+Y(h)=0.$$ (52)

From equation (52), the recurrence relation is obtained as:

$$\begin{array}{cc}\hfill Y(h+2)& =\frac{F(h)-Y(h)}{(h+1)(h+2)},\hfill \\ \hfill Y(0)& =\mathrm{\Psi }(0)=-\tanh \left(\frac{1}{\sqrt{2}}\right),\hfill \\ \hfill Y(1)& ={\mathrm{\Psi }}^{\prime }(0)=-\frac{{\text{sech}}^2\left(\frac{1}{\sqrt{2}}\right)}{\sqrt{2}}.\hfill \end{array}$$ (53)

Substituting h = 0 into equation (53), we obtain:

$$Y(2)=\frac{F(0)-Y(0)}{2}=0.191575.$$

Continuing in this manner, the subsequent coefficients are found as:

$$Y(2)=0.191575,Y(3)=-0.00831579.$$

Thus, the approximate DTM solution of equation (50) up to order N = 10 is:

$$\begin{array}{ll}\hfill {\mathrm{\Psi }}_{\mathit{DTM}}(\psi )& =\sum _{h=0}^{N}Y(h){\psi }^h\hfill \\ \hfill & =-0.608859-0.444975\psi +0.191575{\psi }^2\hfill \\ \hfill & -0.00831579{\psi }^3-0.0283489{\psi }^4\hfill \\ \hfill & +0.0111189{\psi }^5+0.00100212{\psi }^6\hfill \\ \hfill & -0.00221307{\psi }^7+0.000577596{\psi }^8\hfill \\ \hfill & +0.00017826{\psi }^9-0.000157411{\psi }^{10}.\hfill \end{array}$$ (54)

The Maclaurin–Taylor series expansion of the exact solution equation (51) is found to be identical to the above truncated series:

$$\begin{array}{ll}\hfill {\mathrm{\Psi }}_{\text{Exact}}(\psi )& =-0.608859-0.444975\psi +0.191575{\psi }^2-\hfill \\ \hfill & 0.00831579{\psi }^3-0.0283489{\psi }^4+0.0111189{\psi }^5\hfill \\ \hfill & +0.00100212{\psi }^6-0.00221307{\psi }^7\hfill \\ \hfill & +0.000577596{\psi }^8+0.00017826{\psi }^9\hfill \\ \hfill & -0.000157411{\psi }^{10}.\hfill \end{array}$$ (55)

Similarly, Figure 8 illustrates the DTM’s complete discrepancy between the exact answer and the numerical solution.

$$Error=|Exactsolution-DTMsolution|=\left[\begin{array}{c}\hfill 0.\hfill \\ \hfill 0.\hfill \\ \hfill 0.\hfill \\ \hfill 0.\hfill \\ \hfill 1.11022\ast 10^{-16}.\hfill \\ \hfill 0.\hfill \\ \hfill 0.\hfill \\ \hfill 1.11022\ast 10^{-16}.\hfill \\ \hfill 1.11022\ast 10^{-16}.\hfill \\ \hfill 1.11022\ast 10^{-16}.\hfill \\ \hfill 0.\hfill \end{array}\right]$$ (56)

Figure 8.

Pictorial 2D display of Ψ (ψ).

5.3. DTM validation of the soliton solution obtained by the undetermined coefficient method

This section contrasts the numerical treatment of the wave solutions of equation (9) with the precise resolution of the same issue utilizing the equation’s UCM family.

Making use of the values

$$\delta =0,\kappa =1,n_0=1,W=1,\omega =-\frac{1}{2}$$ (57)

Equation (9) turns into

$${\mathrm{\Psi }}^{″}+{\mathrm{\Psi }}^3+\frac{\mathrm{\Psi }}{2}=0$$ (58)

Equation (13), the precise solution of equation (58), and the preconditions are:

$$\begin{array}{cc}\hfill \mathrm{\Psi }(\psi )& =\text{sech}\left(\frac{\psi }{\sqrt{2}}\right),\psi =x+t,\hfill \\ \hfill \mathrm{\Psi }(0)& =\text{sech}\left(\frac{1}{\sqrt{2}}\right),\hfill \\ \hfill {\mathrm{\Psi }}^{\prime }(0)& =-\frac{\tanh \left(\frac{1}{\sqrt{2}}\right)\text{sech}\left(\frac{1}{\sqrt{2}}\right)}{\sqrt{2}}.\hfill \end{array}$$ (59)

When the differential transform method is used on equations (57) and (58), the following results are obtained:

$$(h+1)(h+2)Y(h+2)+F(h)+\frac{Y(h)}{2}=0,$$ (60)

where Y(h) is the converted function that corresponds to the original function Ψ(ψ), and F(h) is the transformation for the subsequent on linear term Ψ³., From equation (60),we get

$$\begin{array}{cc}\hfill Y(h+2)& =\frac{-2F(h)-Y(h)}{2(h+1)(h+2)},\hfill \\ \hfill Y(0)& =\mathrm{\Psi }(0)=\text{sech}\left(\frac{1}{\sqrt{2}}\right),\hfill \\ \hfill Y(1)& ={\mathrm{\Psi }}^{\prime }(0)=-\frac{\tanh \left(\frac{1}{\sqrt{2}}\right)\text{sech}\left(\frac{1}{\sqrt{2}}\right)}{\sqrt{2}}.\hfill \end{array}$$ (61)

When we enter h = 0 in equation (61) we obtain

$$Y(2)=\frac{-2F(0)-Y(0)}{2(1)(2)}=-0.0512816.$$

Similarly, we can obtain the coefficients Y (h) as

$$Y(2)=-0.0512816,Y(3)=0.0789997.$$

The DTM solution of equation (58) with initial conditions equation (59) is then

$$\begin{array}{ll}\hfill {\mathrm{\Psi }}_{\mathit{DTM}}(\psi )& =\sum _{h=0}^{\infty }Y(h){\psi }^h\Rightarrow \hfill \\ \hfill & {\mathrm{\Psi }}_N(\psi )\cong \sum _{h=0}^{N}Y(h){\psi }^h=\sum _{h=0}^{10}Y(h){\psi }^h\hfill \\ \hfill & =0.793278-0.341529\psi -0.0512816{\psi }^2\hfill \\ \hfill & +0.0789997{\psi }^3-0.0172014{\psi }^4\hfill \\ \hfill & -0.00765831{\psi }^5+0.00546595{\psi }^6\hfill \\ \hfill & -0.000547626{\psi }^7-0.000736316{\psi }^8\hfill \\ \hfill & +0.000340313{\psi }^9+9.38474\times 10^{-6}{\psi }^{10}.\hfill \end{array}$$ (62)

The expansion of the exact solution in the Maclaurin-Taylor series is very close to this.

$$\begin{array}{lll}\hfill & \hfill & \hfill {\mathrm{\Psi }}_{\mathit{Exact}}(\psi )=0.793278-0.341529\psi -0.0512816{\psi }^2\\ \hfill & \hfill & \hfill +0.0789997{\psi }^3-0.0172014{\psi }^4-0.00765831{\psi }^5\\ \hfill & \hfill & \hfill +0.00546595{\psi }^6-0.000547626{\psi }^7-0.000736316{\psi }^8\\ \hfill & \hfill & \hfill +0.000340313{\psi }^9+9.38474\ast 10^{-6}{\psi }^{10}.\end{array}$$ (63)

Similarly, Figure 9 illustrates the DTM’s complete discrepancy between the exact answer and the numerical solution.:

$$Error=|Exactsolution-DTMsolution|=\left[\begin{array}{c}\hfill 0.\hfill \\ \hfill 0.\hfill \\ \hfill 0.\hfill \\ \hfill 0.\hfill \\ \hfill 0.\hfill \\ \hfill 0.\hfill \\ \hfill 0.\hfill \\ \hfill 0.\hfill \\ \hfill 0.\hfill \\ \hfill 5.55112\ast 10^{-17}.\hfill \\ \hfill 5.55112\ast 10^{-17}.\hfill \end{array}\right]$$ (64)

Figure 9.

Pictorial 2D display of Ψ (ψ).

6. Results and discussion

The present study provides a comprehensive analytical–numerical framework for investigating soliton dynamics in the generalized reaction Duffing model, which extends several classical Duffing-type formulations reported in the literature. In contrast to earlier works that typically focus on a limited class of solitary wave solutions obtained via a single analytical technique, this work employs the generalized Riccati equation mapping method (GREMM) to derive a broad spectrum of exact solutions, including bright, dark, singular, dark–bright, and periodic–singular solitons. Unlike many existing analytical approaches that yield a restricted set of solutions and numerical schemes that may introduce discretization errors, the combined GREMM–DTM framework enables the construction of broader solution families and provides independent numerical verification with high accuracy and fast convergence. In addition, the analytical results are consistently supported by the use of numerical analysis through the differential transformation method (DTM) and a stability analysis utilizing conserved quantities, as well as 2D and 3D illustrations. This holistic approach to the study of the nonlinear waves in the generalized reaction Duffing equation provides a more comprehensive understanding of the obtained results and demonstrates their novelty and general applicability.

A comparative study of the current analysis with other relevant studies on Duffing-type and associated nonlinear oscillatory equations brings out several unique features. In Ref. 25, a mapping technique is used for the classical Duffing equation to obtain bell-shape and singular solutions without any stability or numerical validation. Likewise, in Ref. 30, a Duffing-type nonlinear system of equations is examined and modified simple equation and expansion techniques are used to extract periodic and solitary wave solutions but without stability analysis. In Ref 62, a nonlinear oscillatory wave equation is analyzed through the use of Modified Exp-function and Kudryashov approaches to get bright and dark soliton solutions; however, no graphical validation is provided. Contrary to the above investigations, in the current work, the generalized reaction Duffing equation is considered and an analytic-numerical scheme combining GREMM and DTM is applied. This results in a variety of solutions including bright, dark, singular, dark-bright and periodic-singular solitons. In addition, the analysis includes a thorough stability analysis along with numerical validation using DTM and 2D/3D plots. The results of the numerical simulations using the DTM match the exact soliton solutions of the proposed model equations derived with the generalized Riccati equation mapping method. The values of the system parameters used in the simulations are in accordance with the constraints of the exact solutions. The graphs for various time instances of the system show the stable propagation of the solitons with almost the same amplitude and shape, experiencing mainly a translation along the propagation direction. These results confirm the traveling character of the waves and their robust dynamics. Also, the changes in the values of system parameters such as the dispersion parameter δ and the nonlinear coefficients influence the width, amplitude, and localization of the wave profiles. Therefore, the good agreement between the exact and numerical results as well as the obtained temporal behavior provide sufficient evidence of the efficiency and validity of the suggested approach. The general reaction Duffing equation can be formulated based on the properties of a non-uniform nonlinear material that possesses inertial effect, spatial dispersion, and non-linear elasticity forces. Parameter δ accounts for the influence of spatial dispersion, ω denotes the stiffness constant, and p and κ indicate the presence of quadratic and cubic non-linearity. The interaction between these physical processes defines whether localized bright solitons, singular structures, and periodic waves are possible in the system. All the exact and numerical examples in this paper contain explicitly stated values of the parameters involved, which are clearly indicated in the captions of figures and numerical examples used in this paper. The values of the parameters have been chosen according to the condition of existence of the solutions, thus ensuring the mathematical and physical acceptability of the solutions. Choosing particular parameter values makes it possible to visualize qualitatively the wave dynamics, whereas deviation from the accepted parameter values proves the stability of the soliton solutions in changing parameters.

The main aim of the current study is to perform graphical analysis of the generalized reaction Duffing equation with the aid of Mathematica. Several graphs, such as 3D graphs, 2D graphs, and density graphs, are applied to study the evolution of wave propagation in time for various sets of parameters and configurations of solitons. The calculations are carried out for the space and time interval −20 ≤ x ≤ 20 and 0 ≤ t ≤ 20, respectively, with different physically relevant values of the parameters.

Even though some of the wave forms obtained show similarities with those found in previous works, the current graphs are plotted based on newly found families of solutions by the application of GREMM-DTM method. In fact, several sets of parameters yield dark, bright, singular, dark-bright, dark-singular, and periodic-singular waves, which have not been discussed before in the literature for the generalized reaction Duffing equation.

Figure 1(a)–(c) show the 3D plot for the Abs surface as well as the respective density plot for the dark soliton state. It is clear from the 3D plot how the notch appears. The density plot, on the other hand, shows the depleted region due to the soliton core. The 2D plots show the evolution of the amplitude and phase profiles as well, with an obvious dip.

Figure 1.

The 3D and 2D graphical behavior of the dark soliton S₁ (x, t) is illustrated through its absolute value and density profiles for the parameters $b_3=5,b_2=-4,\delta =-2,\kappa =-1,n_0=-1,\omega =2,{\mu }_0=-2\sqrt{6},{\mu }_1=5\sqrt{6},b_1=11/15$ .

The 3D Abs surface and density plot in Figure 2(a)–(c) describe the singular soliton solution. From the 3D Abs plot, one can see the presence of the singular peak and, on the other hand, the density plot highlights the localisation nature around the singular region.

Figure 2.

The 3D and 2D graphical behavior of the singular soliton S₃ (x, t) is depicted through its absolute value and density profiles for the parameters $b_3=5,b_2=-4,\delta =-2,\kappa =-1,n_0=-1,\omega =2,{\mu }_0=-2\sqrt{6},{\mu }_1=5\sqrt{6},b_1=11/15$ .

Figure 3(a)–(c) gives the dark-bright soliton structure. From the 3D Abs surface, we can see the co-existence of bright localized peaks in the dark background. The density graph also helps us visualize the spatial distribution of intensity. In addition, we can see the 2D graphs where the amplitude and phase vary simultaneously to prove the co-existence of bright and dark soliton structures.

Figure 3.

The 3D and 2D graphical behavior of the dark-bright soliton S₄ (x, t) is illustrated through its absolute value and density profiles for the parameters $b_3=5,b_2=-4,\delta =-2,\kappa =-1,n_0=-1,\omega =2,{\mu }_0=-2\sqrt{6},{\mu }_1=5\sqrt{6},b_1=11/15.$

Figure 4(a)–(c) shows the profile of the singular dark soliton. In the 3D Abs surface, we can find the singularity along with the dark soliton. From the density graph, we can find the region of the localization around the singular core. Also, the 2D graphs show the variation of the amplitude and phase at different orders.

Figure 4.

The 3D and 2D graphical behavior of the dark-singular soliton S₅ (x, t) is illustrated through its absolute value and density profiles for the parameters $b_3=5,b_2=-4,\delta =-2,\kappa =-1,n_0=-1,\omega =2,{\mu }_0=-2\sqrt{6},{\mu }_1=5\sqrt{6},b_1=11/15.$

Figure 5(a)–(c) correspond to the periodic-singular soliton setup. In the 3D Abs plot, one can see periodically occurring singular peaks, and the density plot shows the recurrent high-energy spots along the propagation direction. The 2D plots prove that the amplitude and phase are modulated periodically in space and time.

Figure 5.

The 3D and 2D graphical behavior of the periodic-singular soliton S₁₄ (x, t) is shown through its absolute value and density profiles for the parameters b₃ = 1, b₂ = 5, δ = 2, κ = −2, n₀ = −1, $\omega =-2,{\mu }_0=5/\sqrt{2},{\mu }_1=\sqrt{2}$ , b₁ = 27/4.

Figure 6(a)–(c) present the bright soliton solution setup. The 3D Abs plot is characterized by the presence of strongly localized high-amplitude peaks typical for bright solitons. The density plot demonstrates the localization of energy near the soliton core, and the 2D plots prove the stability of the solution evolution.

Figure 6.

Graphical representation of the absolute value and density profile of the bright soliton Ψ(ψ) for δ = −1, κ = 1, n₀ = 1, W = 1, and $\omega =-\frac{1}{2}$ .

Lastly, Figures 7–9 illustrate the comparison between the DTM and the exact solutions. The excellent match between the graphs in space and time domains illustrates the highly accurate nature of the DTM in modeling wave profiles. The insignificant discrepancies that appear in some domains are due to the truncation error associated with numerical methods.

From the foregoing graphs, the variety of the soliton structures formed through the generalized reaction Duffing equation is quite evident, such as dark, bright, singular, dark-bright, and periodic singular solitons. The application of 3D graphs, density graphs, and amplitude-phase graphs provides a comprehensive depiction of the behavior of wave propagation and highlights the effect of primary parameters on the solitons.

7. Conclusion

For this research, the generalized reaction Duffing equation has been analyzed via an integrated analysis of numerical and analytical techniques to discover nonlinear waves behaviors. Generalized Riccati equation mapping technique and undetermined coefficient method have been used to acquire a range of types of solutions such as bright soliton solution, dark soliton solution, singular soliton solution, dark-bright soliton solution, dark-singular soliton solution, and periodic singular soliton solution. All the solutions obtained have been validated by substitution in the corresponding governing equation and comparison with differential transform method. Graphical depictions provided in Figures 1–6 give the complete visual representation of the achieved wave forms. 3D absolute value surfaces show the evolution and propagation of the solitons in space and time, whereas density plots indicate localization and concentration properties of them. 2D profiles of these waves at different time moments show that the wave shape and amplitude are conserved, which gives physical insights about the propagation of dark, singular, dark-bright, dark-singular, bright, and periodic-singular solitons. In addition, the conserved quantity based stability criteria have been used, where the positivity of ∂M/∂k showed that the solitons were stable for some chosen parameters. In conclusion, it can be stated that the presented analytical-numerical technique increases the solution set of the generalized reaction Duffing model, which gives reliable numerical results. The future works on generalized Duffing type equations may include higher order dispersive and nonlinear derivatives terms, another stability criteria, and so on.

ORCID iDs

Salah Mahmoud Boulaaras https://orcid.org/0000-0003-1308-2159

Mohammad Safi Ullah https://orcid.org/0000-0002-8889-7865

Data Availability Statement

The article contains all the data that substantiates the results of this study.*