In this paper, we study the Fokas–Lenells (F-L) equation in the presence of perturbation terms and we construct exact solutions. The modified extended direct algebraic method (MEDAM) is utilized, and soliton solutions, solitary wave solutions and elliptic function solutions are achieved. The physical meaning of the geometrical shapes for some of the obtained results is investigated for various choices of the free parameters that appear in the results. The stability of the model is investigated by using the standard linear stability analysis, which verifies that the exact solutions are stable. The achieved solutions and computational work show that the current method is powerful and influential.

We investigate the recently proposed holographic dark energy models with the apparent horizon as the IR cutoff by assuming Kaniadakis and generalized Tsallis entropies in the fractal universe. The implications of these models are discussed for both the interacting ( Γ = 3 H b 2 ρ m ) and noninteracting ( b 2 = 0 ) cases through different cosmological parameters. Accelerated expansion of the universe is justified for both models through deceleration parameter q . In this way, the equation of state parameter ω d describes the phantom and quintessence phases of the universe. However, the coincidence parameter r ~ = Ω m / Ω d shows the dark energy- and dark matter-dominated eras for different values of parameters. It is also mentioned here that the squared speed of sound gives the stability of the model except for the interacting case of the generalized Tsallis holographic dark energy model. It is mentioned here that the current dark energy models at the apparent horizon give consistent results with recent observations.

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In this work, we study the cosmological implications of Kaniadakis holographic dark energy model in the context of Einstein and deformed Hořava-Lifshitz theories of gravity. We examine the various cosmological parameters to analyze the evolution of the universe. The Hubble parameter [Formula: see text] is discussed as it is evidence for accelerating universe and source to examine some other cosmological parameters. We probe coincidence parameter [Formula: see text] to check the matter- and energy-dominated eras of the universe. The deceleration parameter [Formula: see text] is explored for the expansion behavior of the universe while the jerk parameter [Formula: see text] is examined to compare the different dark energy models with [Formula: see text]CDM [Formula: see text] model. We examine the equation of state parameter [Formula: see text] to look into different phases of the universe. We calculate the perturbed parameter squared speed of sound [Formula: see text] to assess the stable behavior of the model. We also investigate the behavior of rate of change of entropy to discuss the validity of generalized second law of thermodynamics.