This paper investigates the exact solutions and linear stability of a class of high-order non-linear evolution equations with fractional non-linearity. We mainly consider the case where the exponent takes p=2/n or p=4/n with n ∈ N, including representative values (p ∈ 1,2,1/2,2/3,2/5). By employing the Jacobi elliptic function expansion method, various new localized exact solutions are derived, and their properties are analyzed in detail, providing a theoretical basis for practical applications. The dynamical behaviors and evolution characteristics of these solutions are visualized through graphical representations. Furthermore, the linear stability of constant solutions is discussed to reveal the effects of fractional non-linearity. Finally, Physics-Informed Neural Networks (PINNs) are utilized to approximate the analytical solutions. The numerical results demonstrate excellent agreement with the exact solutions, verifying the effectiveness of the proposed approach.
This work focuses on the unstable non-linear Schrödinger equation, which is used to elaborate the propagation of ultra-short signals in fiber optics. In this work, the variational principle of the unstable non-linear Schrödinger equation is established using the semi-inverse method. By employing the Galilean transformation, the planar dynamical system of the unstable non-linear Schrödinger equation is constructed, and its Hamiltonian function, bifurcation analysis, chaotic behavior and sensitivity analysis are presented. Moreover, some new soliton solutions and periodic solutions of the unstable non-linear Schrödinger equation are derived using two excellent mathematical approaches, which are called (H/K) -expansion method and improved e^ - Θ( ζ) -expansion method. These solutions hold practical significance in the respective physical fields. The internal structure and evolution are represented in three-dimensional, two-dimensional, and contour graphs. These graphs are generated with appropriate choice of parameters to emphasize some important features of waves.
The present study addresses the relativistic Volterra lattice system, denoted as RVL_-(α ) system, a key member of the relativistic Volterra lattice hierarchy, serving as a prototypical discrete integrable model for describing nonlinear wave dynamics and energy transfer in lattice structures. The present study focuses on four main aspects: (1) Our first focus is on the continuum limit and conservation laws of RVL_-(α ) system; (2) Employing the Lax pair, we construct the discrete generalized (m, 2N-m) -fold Darboux transformation (DT), which yields multi-soliton and higher-order rational solutions. Asymptotic analysis confirms that the two-soliton interaction is elastic, characterized by well-defined phase shifts, whereas the rational counterparts feature distinct singularity structures and trace patterns; (3) For more accurate soliton prediction, we extend our previously proposed symmetric difference enhanced physics-informed neural networks (SDE-PINNs) to Lax pair-constrained SDE-PINNs (Lax-SDE-PINNs), embedding both spatial and temporal Lax pair equations into the loss function, supplemented by symmetric difference data augmentation, RAR sampling and NTK-based weight balancing. Numerical simulations of one- and two-soliton solutions demonstrate low reconstruction errors and stable extrapolation, confirming the effectiveness of combining Lax pair-constraints with the SDE-PINNs enhancements. (4) For the inverse problem, we focus on recovering the parameter α . Comparative numerical tests demonstrate that, under the same experimental settings, the Lax-SDE-PINNs outperforms the traditional PINNs and SDE-PINNs, yielding more accurate predictions and better robustness. These findings contribute to a deeper understanding of the physical phenomena described by RVL_-(α ) system, and may inform future studies on related integrable lattice models.
We study fluctuations of the accumulated activity h(x, t) in the Bak–Sneppen model of evolution. When the worst fitness occurs at x at time t , the activity becomes one, and h(x, t) increases by one unit. The accumulated activity resembles a height profile as observed in the discrete models of kinetic rough interfaces. We employ the tools developed for roughness analysis and examine the structure factor, since it can reveal anomalous features. A systematic finite-time scaling yields the roughness exponent χ =1.94(7) and the dynamical exponent z=2.44(8) . We find the spectral roughness exponent χ _s = 0.72(3) < χ , indicating that the interface shows intrinsic anomalous behavior. The anisotropic version of the BS model also exhibits intrinsic anomalous behavior with different scaling exponents. We also examine the accumulated activity in a class of random walks that display qualitatively similar behavior. The structure factor S(k,t) of the accumulated activity in the BS model. S(k, t) exhibits two different regimes separated by a cutoff wavenumber, varying as ∼ t^-1/z , where z is the dynamical exponent. In the low-wavenumber regime, the structure factor remains constant, but increases with time. On the other hand, S(k, t) decays in power law along with a time-dependent variation in the high-wavenumber regime.
Physics-informed neural networks (PINNs) with a self-adaptive mechanism are employed to accurately solve the high-order Sawada–Kotera equation, achieving relative L_2 -norm errors on the order of 10^-5 . This represents a one-to-two order-of-magnitude improvement in accuracy over previously reported results for various soliton molecule solutions of the Sawada–Kotera equation. Moreover, the simulation domains are enlarged by two to three orders of magnitude while maintaining high prediction accuracy for the SK equation. Despite the substantially larger domains, the computational cost remains comparable, benefiting from the self-adaptive weighting mechanism and the mesh-free framework enabled by automatic differentiation. The proposed model also accurately solves genuine rational soliton solutions of the derivative Schrödinger equation, achieving relative L_2 -norm errors on the order of 10^-5 . This corresponds to a two-to-three order-of-magnitude improvement in accuracy over previous results, despite the considerably larger simulation domains.
