In this article, we study the time-fractional nonlinear Klein–Gordon equation in Caputo–Fabrizio’s sense and Atangana–Baleanu–Caputo’s sense. The modified double Laplace transform decomposition method is used to attain solutions in the form of series of the proposed model under aforesaid fractional operators. The suggested method is the composition of the double Laplace transform and decomposition method. The convergence of the considered method is demonstrated for the considered model. It is observed that the obtained solutions converge to the exact solution of the proposed model. For validity, we consider two particular examples with appropriate initial conditions and derived the series solution in the sense of both operators for the considered model. From numerical solutions, it is observed that the considered model admits pulse-shaped solitons. It is also observed that the wave amplitude enhances with variations in time, which infers the coefficient significantly increases the wave amplitude and affects the nonlinearity/dispersion effects, therefore may admit monotonic shocks. The physical behavior of the considered numerical examples is illustrated explicitly which reveals the evolution of localized shock excitations.

The potential Kadomtsev-Petviashvili (pKP) equation delineates the development of small-amplitude, nonlinear, long waves characterized by a gradual variation in the transverse coordinate. The B-type KP equation outlines the relationships among exponentially localized shapes and was employed as a representation for shallow water waves and plasma physics. In this paper, we consider the combined pKP-BKP integrable equation. We discuss the multiple solitons of a newly proposed (3+1)-dimensional combined pKP-BKP integrable equation. We use the Hirota bilinear (HB) form of the considered equation to deduce fission process in higher order solitons with different orders. Moreover, the breather dynamics and its interaction with other solitons are investigated via HB. The lump solution and its interaction with first order and fourth order kink soliton is studied.

Four-dimensional continuous chaotic models with Caputo fractional derivative are presented. Fixed point theory is used to investigate the existence and uniqueness of complex systems. The dynamical properties are studied, including the Lyapunov exponent, phase portrait, and time series analysis. The hyperchaotic nature of each system is revealed by the positive exponents. The numerical method is introduced to describe the influence of the order of the Caputo fractional derivative. The phase portraits are presented to investigate the behavior and effect of some key parameters and fractional orders on model dynamics. The systems approach fixed point attractors for fractional-order and increase the visibility of the attractor by decreasing fractional order. This means that a change in fractional order has a significant impact on the dynamics of the models. When the order of the derivative is equal to one, both systems oscillate frequently. However, as the fractional order is reduced, the system oscillations decrease as compared to the integer-order, and the system moves towards its fixed point, reveals the hidden attractors inherent in the system, and enabling it to develop to a stable state more efficiently.

In this manuscript, a new three-dimensional continuous chaotic model is presented based on the modification in the Lorenz system. The dynamical aspects of the complex system are investigated, covering equilibrium points and linear stability, dissipation, phase portraits, Poincaré mapping, Lyapunov exponent, attractor projection, bifurcations, time series analysis, and sensitivity. The model is also studied numerically using the Haar wavelet scheme with Caputo fractional derivative. The positive exponent reveals that the system is chaotic. The symmetric limit cycle and butterfly type attractors are observed because the trajectories of the model ultimately progress to a bounded region. The existence of the chaotic attractor is shown by Poincaré section. In the Poincaré section, the kindling is integrated and connected as a single attractor. From the sensitivity analysis of the system, it is noted the system is dependent on the initial conditions that show chaos in the system. The evolution of the attractor is depicted by fixing the first two parameters and varying the third. The theoretical and numerical studies exhibit that the model has complex dynamics with certain stimulating physical characteristics.