https://doi.org/10.1140/epjp/s13360-026-07453-7
Regular Article
(2+1)-dimensional variable-coefficient nonlinear wave equation: exact solutions, lump solutions, and the fitting solution of the R-b PINN
School of Mathematical Sciences, Liaocheng University, 252059, Liaocheng, People’s Republic of China
a
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Received:
20
September
2025
Accepted:
17
February
2026
Published online:
3
March
2026
Abstract
In cases involving non-uniform media and non-uniform boundary conditions, variable-coefficient equations can effectively describe the occurrence and evolution processes of certain complex physical phenomena. This paper investigates a (2+1)-dimensional variable-coefficient nonlinear wave equation that describes the nonlinear localized wave solutions of the nonlinear partial differential equations and their interaction phenomena, which are crucial in the field of nonlinear research. Firstly, the Hirota bilinear method is employed successfully to derive the bilinear form of the (2+1)-dimensional variable-coefficient nonlinear wave equation, thereby obtaining some exact solutions, including soliton solutions and interaction solutions. Moreover, the first-order lump solution is obtained by using the long-wave limit method. By selecting specific parameter values, the different dynamic behaviors and effects exhibited by these exact solutions in the propagation of shallow water waves are systematically analyzed. Secondly, to address the complex problem of boundary features such as nonlinear changes at corners, low proportion of boundary samples, easy dilution of boundary features by internal ones, and weak gradient propagation at boundaries, this study proposes an enhanced residual boundary physical information neural network method based on the residual networks to obtain more accurate solutions. The core novelties of this method are threefold. First, it amplifies the learning weights of boundary features through a gating mechanism and adopts boundary-enhanced residual blocks and additional sampling at corners to strengthen the gradient propagation of boundary features. Second, it adopts a staged learning approach that first learns global features to lay a foundation and then specifically enhances boundary features, preventing boundary features from being diluted by internal features and reducing the risk of overfitting. Third, it introduces dynamic weight adjustment in the loss function, which automatically increases the weight when the boundary error is large to prioritize reducing boundary errors. A systematic comparison is conducted between the predictions provided by the traditional physical information neural network and the two-soliton solutions of the (2+1)-dimensional variable-coefficient nonlinear wave equation evaluated along the 
boundary at 
, 1, and 
. The residual-boundary physics-informed neural network exhibits a significantly higher fidelity in boundary reconstruction, hence validating the improved accuracy.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
