Xiangpeng Xin

To address the limitations of the standard Estevez–Mansfield–Clarkson (EMC) equation in describing practical physical scenarios-such as inhomogeneous media and multi-dimensional flow fields-this paper also aims to accurately capture the dynamic evolution characteristics of non-autonomous systems. Building on the basic framework of the standard equation, it proposes the generalized multi-dimensional EMC equation with time-varying coefficients (TVC-GM-EMC) through dimensional extension and time-varying coefficient modification. This paper analyzes the TVC-GM-EMC. It investigates the equation’s analytical solutions, including soliton solutions, interaction solutions, and lump solutions. It also explores the learning and prediction of these solutions using physics-informed neural network (PINN) and augmented graph PINN (AGPINN). First, the Hirota bilinear method is used to successfully construct soliton solutions for the equation. The regulatory mechanisms of TVC on soliton amplitude, propagation velocity and waveform stability are analyzed in depth. Second, lump solutions of the equation are derived via the long-wave limit method. Finally, PINN and AGPINN are respectively applied to learn the analytical solutions of the equation. A comparative analysis is conducted on the prediction accuracy of the two networks in handling solutions to multi-dimensional equations with TVC. The research findings not only deepen the theoretical understanding of the solution space structure of the TVC-GM-EMC. They also verify the scientific validity and practicality of the equation’s extension. The equation demonstrates broad application prospects in cutting-edge fields, including multi-dimensional wave propagation in plasma physics, multi-scale coupling mechanisms in fluid turbulence and multi-dimensional pulse control in nonlinear optics.

This paper proposes a dual framework combining analytical and numerical methods to study high-dimensional nonlocal integrable systems. In terms of the analytical method, a high order nonlocal Ablowitz-Kaup-Newell-Segur(AKNS) system was constructed through symmetric reduction, and the exact solutions(including singular solutions) were derived by using the gauge transformation. In terms of numerical methods, combined with physical laws and boundary conditions, and due to the influence of singular solutions, an improved multi-step physics informed neural network(MS-PINN) is adopted for region segmentation. By using activation functions(Tanh, ReLU, Sigmoid) and the Adam optimizer, this framework effectively captures complex nonlinear behaviors and is analyzed and illustrated through comparative images of the two regions. This study demonstrates the effectiveness of Darboux transformation for nonlocal systems and MS-PINN for high-dimensional nonlocal problems, providing insights into nonlinear partial differential equations in fluid dynamics and optics.

Physics-informed neural networks (PINNs), built upon general neural networks, integrate physical information into the loss function, achieving remarkable results in solving partial differential equations (PDEs). Meanwhile, PINNs face challenges in solving complex interface problems, such as the Navier–Stokes/Darcy interface model considered in this paper. To address these challenges, we extend the general PINNs to propose subregion–identified physics-informed neural networks (SI-PINNs), which consist of three different neural networks for different subregions (Navier–Stokes, Darcy, and interface) with separated datasets. By identifying and classifying the regional features of the neural networks and datasets, the complexity of the physical information that SI-PINNs need to process is effectively split, while the networks also take all physical information into account during the training. Additionally, this splitting feature substantially reduces the training costs while enhancing or maintaining the accuracy of the general PINNs for the target model. Numerical experiments are performed to validate the effectiveness of the SI-PINN method.

To analyze the generalized Broer-Kaup (GBK) system, we have employed the methodology suggested by Lou et al, which involves constructing high-dimensional equations from conservation laws associated with low-dimensional partial differential equations. For the first time, we have derived the (2+1)-dimensional integrable GBK equations by leveraging the conservation law of the (1+1)-dimensional GBK equations. Through the imposition of constraints within the (2+1)-dimensional GBK equations, we have obtained a novel (1+1)-dimensional GBK equations. Subsequently, we pioneered the use of a nonlocal approach to analyze this new equations. Specifically, we constructed a closed system incorporating nonlocal symmetries. This was achieved by introducing the Lax pairs associated with potential functions and GBK equations, utilizing conservation laws. By applying the nonlocal symmetries to this closed system, we were able to deduce the generating element. The exact solution of the equation is achieved by combining finite symmetry transformations with a symmetry reduction technique that involves approximations. The dynamic behavior of the equations is studied by means of figures of the exact solutions.

In this paper, the (2+1)-dimensional integrable long water wave equations (LWWs) are constructed for the first time using the conservation law of the (1+1)-dimensional LWWs. The new (1+1)-dimensional LWWs can be obtained by introducing a constraint to the (2+1)-dimensional LWWs. This new (1+1)-dimensional LWWs are studied by using nonlocal symmetry methods for the first time. The closed system corresponding to nonlocal symmetry is established by the lax pairs of equations and the potential function determined using conservation laws. Exact solutions of the equations are obtained by finite symmetry transformation and symmetry approximation of this closed system. The dynamic behavior of the equations is studied by means of figures of the exact solutions.