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This study investigates the (1+1) dimensional dissipative nonlinear Schrödinger equation, which has applications in modeling the evolution of swell in the ocean. Two different and impressive techniques are used for the solutions, which are the Jacobi elliptic function expansion method and the sine-Gorgon expansion method. The suggested techniques are designed to derive the solutions for many other the nonlinear partial differential equations that arising in various branches of sciences. By applying this methods, we obtain novel soliton solutions, which are expressed in terms dark, bright, singular and combo optical solitons. The efficient optical soliton solutions discovered through this methods have the potential to find use in a variety of fields, inclusive electronics, ocean waves, and optical fibers. The modulation instability criterion for governing model is obtained based on the standard linear stability analysis. Additionally, we will also present graphs in 2D and 3D form with constraints conditions. [ABSTRACT FROM AUTHOR]

In the intricate realms of quantum field theory, cosmology, and condensed matter physics, the Phi-4 model emerges as a potent catalyst, shaping mathematical landscapes with profound implications. Our research has uncovered a set of soliton solutions that were previously unreported in the existing literature. This transformative journey has allowed us to unravel a rich tapestry of new insights in the field. Our findings will contribute significantly to advancing knowledge in this area of study. Through innovative analytical approaches, we breathe life into these solitons, delving into their dynamic behaviors and unveiling the nuanced influence of factors such as wave velocity (ω) and diverse free parameters. The elegance of our methodologies lies in their simplicity, offering accessible pathways for exploration. As we stand on the threshold of discovery, the solutions and intricate wave profiles we present broaden our current understanding and beckon further investigation, resonating with the Phi-4 model’s enduring melody that harmonizes across scientific frontiers.

In this work, we investigate the solutions of the (2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili (BKP) equation by three powerful analytical methods: the exp a function method, the (G ′ G) -expansion method, and the Sine-Gordon expansion method. This equation describes the nonlinear wave propagation in many applications like waves of evolutionary shallow water, electrical networks, and engineering devices. Moreover, we study the solutions numerically via the finite difference method. We analyze the bifurcation of dynamical system resulting from the BKP equation. Finally, the majority of our solutions are displayed graphically to present the strength of imposed methods. [ABSTRACT FROM AUTHOR]

The Sine-Gordon expansion (SGE) method is used to develop exact traveling wave solutions of a family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations. The governing equations are reduced to standard ordinary differential equations via companionable wave transformation. The order of the projected polynomial-type solution is figured out using the homogeneous stability approach based on the renowned Sine-Gordon equation. This solution's replacement corresponds to the earlier stage. A system of algebraic equations (SAE) is formed by comparing the coefficients of the powers of the projected solution. A robust coefficient scheme supplies the necessary relationships between parameters and coefficients to develop solutions. Some solutions are modeled for various parameter combinations. Several hyperbolic function solutions are created using the mentioned method. The MATLAB program displays the answers' graphical exemplifications in three-dimensional (3D) surface plots, contour plots, and 2D line plots. The resulting solutions to the equation, with the appropriate parameters, are used to depict the absolute wave configurations in all displays. Additionally, it can conclude that the discovered solutions and their physical characteristics could aid in understanding how shallow water waves propagate in nonlinear dynamics.

In this paper, we employ the sine-Gordon expansion method to shallow water wave models which are Kadomtsev-Petviashvili-Benjamin-Bona-Mahony and the Benney-Luke equations. We construct many new complex combined dark-bright soliton, anti-kink soliton solutions for the governing models. The 2D, 3D and contour plots are given under the suitable coefficients. The obtained results show that the approach proposed for these completely integrable equations can be used effectively.

In the field of quantum mechanics and fluid physics, especially in the study of nonlinear geometric optics and superconductivity, the Landau-Ginzburg-Higgs (LGH) and the (2 + 1)-dimensional Novikov-Veselov (NV) equations are two significant models. In this article, we have affirmed that wave profile changes with the change of the free parameters associated with them and are mainly dominated by linear effects. The effects of nonlinearity and wave speed on the wave contours have also been analyzed. On account of this, wave solutions are computed concerning rational, hyperbolic, and trigonometric structures balancing the exponents of linear and nonlinear terms of the highest order, from which scores of typical wave profiles including kink, bell-shape soliton, lump, and periodic waves have been extracted. The wave solutions are designated through extending the typical concept of the sine-Gordon expansion (SGE) method from the lower dimensional to the higher dimensional nonlinear evolution equations. This study establishes the capability of the stated method in solving both lower and higher dimensional nonlinear evolution equations. The solutions are analyzed by sketching figures for different values of related variables, and it is observed that the attributes of these solutions are pivotal in the selection of parameters.

This study investigates the Gilson–Pickering equation by using the sine-Gordon expansion method. Sine-Gordon expansion method is one of the most powerful methods for solving the nonlinear partial differential equations. We successfully construct various exact solitary wave solutions to the governing equation, such as shock wave, topological, non-topological, compound topological, and non-topological soliton wave solutions. In addition, the stability of the studied nonlinear equation is analyzed via the linear stability analysis. The 2D, 3D, and contour surfaces are also plotted for all obtained solutions. [ABSTRACT FROM AUTHOR]

The coupled Higgs field equation is significant for describing a system of conserved scalar nucleons interacting with the neutral scalar meson. The improved Bernoulli sub-equation function method and the sine–Gordon expansion method have been put in used in the current article, to interpret the dynamism of the particles through analytical solutions modelled by the coupled Higgs field equation. The interaction among the electrons and mesons has been found out and it is observed that the particle moves along the smooth path from one state to another. It is also vital to mark that the wave profile of the waves are velocity dependent and the incident is shown in the figures. Different types of soliton shape like, bell shape soliton, kink soliton, consolidated bell shape; anti-bell shape soliton, compacton, singular soliton, flat kink soliton etc. are ascertained by assigning individual quantities of the unidentified parameters whose propagations are portrayed in 3D and 2D surfaces. The solutions established in this study are further comprehensive, broad–ranging and some are new which have not been found in earlier research. The executed schemes are effective, compatible to computer algebra and deliver ample wide–ranging wave solutions. [ABSTRACT FROM AUTHOR]

This study aims to obtain travelling wave solutions of the doubly dispersive equation in nonlinear dynamic elasticity by the sine-Gordon expansion method. We give physical explanation of the presented solutions under suitable parameters via the 3D, 2D and contour simulations. [ABSTRACT FROM AUTHOR]

In this paper, a powerful sine-Gordon expansion method (SGEM) with aid of a computational program is used in constructing a new hyperbolic function solutions to one of the popular nonlinear evolution equations that arises in the field of mathematical physics, namely; longren-wave equation. We also give the 3D and 2D graphics of all the obtained solutions which are explaining new properties of model considered in this paper. Finally, we submit a comprehensive conclusion at the end of this paper. [ABSTRACT FROM AUTHOR]

Researching different solutions of nonlinear models has been interesting in different fields of science and application. In this study, we investigated different solutions of fourth-order nonlinear Ablowitz-Kaup-Newell-Segur wave equation. We have used the sine-Gordon expansion method (SGEM) during this research. We have given the 2D, 3D, and contour graphs acquired from the values of the solutions obtained using strong SGEM. [ABSTRACT FROM AUTHOR]

In this paper, we apply the sine-Gordon expansion method which is one of the powerful methods to the generalized-Zakharov equation with complex structure. This algorithm yields new complex hyperbolic function solutions to the generalized-Zakharov equation with complex structure. Wolfram Mathematica 9 has been used throughout the paper for plotting two- and three-dimensional surface of travelling wave solutions obtained.

This paper applies the sine-Gordon expansion method to the extended nonlinear (2+1)-dimensional Boussinesq equation. Many new dark, complex and mixed dark-bright soliton solutions of the governing model are derived. Moreover, for better understanding of the results, 2D, 3D and contour graphs under the strain conditions and the suitable values of parameters are also plotted.

In this manuscript, we investigate the analytical solutions of the new coupled Konno-Oono equation by using the sine-Gordon expansion method. We successfully obtained some new soliton solutions to this model. We plot the 2D and 3D surfaces of all the found soliton solutions in this study. The Wolfram Mathematica 9 is used in performing all the computations in this study. [ABSTRACT FROM AUTHOR]

In this article, we employ the sine-Gordon expansion method to construct a new solitary and optical wave structures to the (1 + 1)-dimensional combined KdV–mKdV equation. Sine-Gordon expansion method is developed based on the travelling wave transformation u ( x , y ) = U ( ξ ), ξ = μ ( x − ct ) and the sine-Gordon equation u xx − u tt = m 2 sin( u ). We find the travelling wave solutions to (1 + 1)-dimensional combined KdV–mKdV equation in the form of complex, hyperbolic, trigonometric and rational function. Our results show that all the travelling wave solutions we obtain are solitary and optical wave structures solutions. All the computations and graphics plot in this article are carried out with the help of Wolfram Mathematica 9. [ABSTRACT FROM AUTHOR]

This manuscript investigates the new acoustic wave behaviors of the Davey-Stewartson equation with power-law nonlinearity with the help of sine-Gordon expansion method. This technique yields many new acoustic wave behaviors such as hyperbolic, exponential and complex function structures to the problem considered. Wolfram Mathematica 9 has been used throughout the paper for mathematical calculations along with obtaining two- and three-dimensional surfaces of results. [ABSTRACT FROM AUTHOR]

In this paper, we apply the sine-Gordon expansion method which is one of the powerful methods to the generalized-Zakharov equation with complex structure. This algorithm yields new complex hyperbolic function solutions to the generalized-Zakharov equation with complex structure. Wolfram Mathematica 9 has been used throughout the paper for plotting two- and three-dimensional surface of travelling wave solutions obtained. [ABSTRACT FROM AUTHOR]

• The new 3D fractional WBMM equations are studied. • Soliton, lump and kink wave solutions are constructed. • Single and multiple soliton wave solutions are obtained. • The structure of the obtained solutions is reported graphically. The Sine-Gordon expansion (SGE) method is used to develop exact traveling wave solutions of a family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations. The governing equations are reduced to standard ordinary differential equations via companionable wave transformation. The order of the projected polynomial-type solution is figured out using the homogeneous stability approach based on the renowned Sine-Gordon equation. This solution's replacement corresponds to the earlier stage. A system of algebraic equations (SAE) is formed by comparing the coefficients of the powers of the projected solution. A robust coefficient scheme supplies the necessary relationships between parameters and coefficients to develop solutions. Some solutions are modeled for various parameter combinations. Several hyperbolic function solutions are created using the mentioned method. The MATLAB program displays the answers' graphical exemplifications in three-dimensional (3D) surface plots, contour plots, and 2D line plots. The resulting solutions to the equation, with the appropriate parameters, are used to depict the absolute wave configurations in all displays. Additionally, it can conclude that the discovered solutions and their physical characteristics could aid in understanding how shallow water waves propagate in nonlinear dynamics. [ABSTRACT FROM AUTHOR]

• Stable solitary solutions have been developed for NLEEs related to optics and superconductivity. • The effects of linearity and nonlinearity on the dynamics of wave profiles have been interpreted. • Standard wave profiles including kink, bell-shape soliton, and periodic solitons have been found. • The 3D and contour plots are depicted to illustrate profile of the phenomena. In the field of quantum mechanics and fluid physics, especially in the study of nonlinear geometric optics and superconductivity, the Landau-Ginzburg-Higgs (LGH) and the (2 + 1)-dimensional Novikov-Veselov (NV) equations are two significant models. In this article, we have affirmed that wave profile changes with the change of the free parameters associated with them and are mainly dominated by linear effects. The effects of nonlinearity and wave speed on the wave contours have also been analyzed. On account of this, wave solutions are computed concerning rational, hyperbolic, and trigonometric structures balancing the exponents of linear and nonlinear terms of the highest order, from which scores of typical wave profiles including kink, bell-shape soliton, lump, and periodic waves have been extracted. The wave solutions are designated through extending the typical concept of the sine-Gordon expansion (SGE) method from the lower dimensional to the higher dimensional nonlinear evolution equations. This study establishes the capability of the stated method in solving both lower and higher dimensional nonlinear evolution equations. The solutions are analyzed by sketching figures for different values of related variables, and it is observed that the attributes of these solutions are pivotal in the selection of parameters. [ABSTRACT FROM AUTHOR]

In the current study, we investigate the (3＋1)-dimensional extended quantum Zakharov–Kuznetsov equation through two different methods namely the sine–Gordon expansion method and the (1 ∕ G ′ )-expansion method. The given model represent the nonlinear propagation of the quantum ion-acoustic wave. Traveling wave transformation has been used, in order to convert the governing equation into a non-linear ordinary differential equation. As a result, we successfully construct various exact wave solutions, such as shock, topological, non-topological solutions, periodic wave solutions, and singular solutions. In addition, 2D, 3D and contour surfaces are plotted according to the appropriate parameters values. [ABSTRACT FROM AUTHOR]

This paper applies the sine-Gordon expansion method to the extended nonlinear (2+1)-dimensional Boussinesq equation. Many new dark, complex and mixed dark-bright soliton solutions of the governing model are derived. Moreover, for better understanding of the results, 2D, 3D and contour graphs under the strain conditions and the suitable values of parameters are also plotted. [ABSTRACT FROM AUTHOR]