A modified (2+1)-dimensional nonlinear evolution equation is investigated to reveal the nonlinear superposition mechanisms between lump waves and other localized structures. Starting from the N-soliton solutions, a class of hybrid solutions is constructed via the long-wave limit approach under appropriate parameter constraints. These results include interaction patterns involving a single lump wave with solitons, as well as configurations composed of multiple lump waves and solitons. By further introducing complex-conjugate parameter conditions, hybrid states describing the coexistence of lump waves and breather structures are derived. In addition, the velocity resonance mechanism is employed to generate various bound states, such as lump–soliton and lump–breather molecules, together with interaction solutions between composite wave structures and individual solitons. Under additional constraints, a novel coupling form between lump waves and soliton molecules is also obtained. The dynamical properties of these solutions are illustrated through graphical analysis, demonstrating diverse interaction behaviors and structural features.
This paper employs multiple methods to conduct an in-depth study of the Lossy Nonlinear Electrical Transmission Line Model (LNETLM). First, we use the modified generalized Riccati equation mapping (mGREM) method to solve for soliton solutions of the LNETLM, discover various forms of solitons, including bright solitons, bright-dark alternating waves, singular solitons, and breather solitons. We also investigate the impact of fractional-order changes on the model. Second, by removing certain special restrictions on the parameters, we transform the LNETLM equation into a dynamical system and study its planar phase portraits. This approach is significantly different from the research in the reference. In terms of research methods, we introduce the Lyapunov exponent and Poincaré section, which are not mentioned in references. Additionally, we compare the effects of different types of fractional-order derivatives on the LNETLM and study the impact of relevant physical parameters. In the numerical computation aspect, we use the classical fourth-order explicit Runge–Kutta method to solve the equations. The numerical solutions achieve an accuracy level of 10^-6 , demonstrating the high precision of this method and its effectiveness in suppressing error propagation. Overall, this paper presents innovative contributions in soliton solution derivation, dynamical analysis, research methods, and numerical computation, offering new perspectives and approaches to the study of the LNETLM.
In this study, the conformable coupled nonlinear Maccari system (CCNMS) is analyzed by employing a modified ( G'G^2) -expansion technique. This method offers a systematic way to derive exact analytical traveling-wave solutions of the system. A wide class of closed-form solutions is obtained, encompassing trigonometric, hyperbolic, rational, and kink-type structures that reveal diverse nonlinear phenomena such as solitons and hybrid waves. The solutions, presented in compact analytical expressions, cover various parameter regimes and exhibit both mathematical simplicity and rich structural features. Their qualitative behaviors are illustrated using symbolic computations performed in Mathematica and MATLAB. Furthermore, a bifurcation analysis of the reduced ordinary differential equation is carried out to determine the conditions leading to periodic, solitary, and homoclinic wave formations.The results contribute to a deeper theoretical understanding of nonlinear wave propagation and demonstrate the relevance of CCNMS in areas such as fluid dynamics, nonlinear optics, magnetization, and Bose–Einstein condensates. The proposed framework can also be adapted to other types of conformable fractional partial differential equations.
The Kaup–Kupershmidt equation is used to study nonlinear wave propagation associated with ocean currents, capturing the combined effects of nonlinearity and dispersion. Due to the many solitons exhibited by the fractional Kaup–Kupershmidt (K–K) equation, which represent the many solitary waves observed in ocean currents, plasma physics, and optics, several analytical methods have been developed and consistently applied to study these phenomena. One such method is the homogeneous balance method (HBM), which invariably relies on the embedded Riccati equation. Thus, the workability of the HBM largely depends on the embedded Riccati equation, which by and large determines the number and type of solitons yielded when the HBM is applied to the fractional K–K equation. Undeniably, several Riccati equations appear in the literature, each with unique properties. Therefore, if the HBM is embedded with a particular Riccati equation for solving the fractional K–K equation, the underlying mechanisms of the solitary waves in ocean current, plasma physics and optics are unveiled. The HBM and its modified methods explored only by the researchers have not been exhaustively observed a fractional Riccati equation with a quadratic term as the dependent variable, where coefficient of the second term is one and coefficient of the linear term is zero, d^γϕ/dξ ^γ = p + ϕ ^2(ξ ), quadratic term as the dependent variable and applied this modified homogeneous balance method (MHBM) to obtain the exact solitons of the fractional K–K equation. Applying the MHBM in solving the fractional K–K equation yielded five solitons: a bright soliton with a flat top, a gray soliton, a dark soliton with a flat bottom, an L-tilted soliton, and a halfway dark soliton. Three of the five solitons, a bright soliton with a flat bottom, a dark soliton with a flat bottom, and an L-tilted soliton, were new solitons that have not been observed in literature. These three new solitons of the fractional K–K equation have been observed in this paper as a result of a new embedded Riccati equation in HBM. The soliton properties, such as the kinetic and potential energies of the gray soliton and L-tilted soliton, are similar, whereas in the bright soliton and in the flat top and dark soliton with flat bottom, the potential and kinetic energies are in opposite directions. Lastly, the kinetic energy of the halfway dark soliton increases exponentially.
In shallow water, solitary waves typically arise as a result of the combined effect of nonlinearity and dispersion. The Boussinesq equation plays a pivotal role in study nonlinear solitary wave due to existing dispersive and nonlinear terms. This study aims to conduct a comprehensive investigation on Boussinesq equation through Lie symmetry method. By determining the complete Lie algebra of infinitesimal generators and constructing one dimensional optimal system of subalgebras, the test equation is transformed to ordinary differential equations (ODEs). The ODEs enable to derive a rich variety of exact solutions, which include bright and dark solitons, double soliton configurations, and many others. To examine the qualitative behavior and structural transitions of these solutions, bifurcation theory is applied, offering a robust framework to study the effects of critical parameters on the dynamics and revealing shifts between distinct solution regimes. Stability is analyzed through bifurcation diagrams and phase portraits, capturing influence of perturbations or parameter variations to evolution of wave structures. Additionally, sensitivity and chaotic analysis are introduced to assess the effect of variations in initial conditions and parameters to persistence and robustness of the derived periodic and quasi-periodic solutions, thereby highlighting critical thresholds and transition points. Furthermore, the adjoint equation and conserved vectors are obtained in connection with the identified symmetries within the framework of the Lagrangian formulation. By synthesizing Lie symmetry reductions, bifurcation and stability analysis, sensitivity examination, and conservation law derivation, this work offers a unified framework for probing the intricate dynamics of the Boussinesq equation, advancing the theoretical understanding and practical applicability of nonlinear wave models.
