The main purpose of this work is to discuss an extended KdV equation, which can provide some physically significant integrable evolution equations to model the propagation of two-dimensional nonlinear solitary waves in various science fields. Based on the bilinear Bäcklund transformation, a Lax system is constructed, which guarantees the integrability of the introduced equation. The linear superposition principle is applied to homogeneous linear differential equation systems, which plays a key role in presenting linear superposition solutions composed of exponential functions. Moreover, some special linear superposition solutions are also derived by extending the involved parameters to the complex field. Finally, a set of sufficient conditions on Wronskian solutions is given associated with the bilinear Bäcklund transformation. The Wronskian identities of the bilinear KP hierarchy provide a direct and concise way for proving the Wronskian determinant solution. The resulting Wronskian structure generates $ N $-soliton solutions and a few of special Wronskian interaction solutions, which enrich the solution structure of the introduced equation.
Citation: Li Cheng, Yi Zhang, Ying-Wu Hu. Linear superposition and interaction of Wronskian solutions to an extended (2+1)-dimensional KdV equation[J]. AIMS Mathematics, 2023, 8(7): 16906-16925. doi: 10.3934/math.2023864
The main purpose of this work is to discuss an extended KdV equation, which can provide some physically significant integrable evolution equations to model the propagation of two-dimensional nonlinear solitary waves in various science fields. Based on the bilinear Bäcklund transformation, a Lax system is constructed, which guarantees the integrability of the introduced equation. The linear superposition principle is applied to homogeneous linear differential equation systems, which plays a key role in presenting linear superposition solutions composed of exponential functions. Moreover, some special linear superposition solutions are also derived by extending the involved parameters to the complex field. Finally, a set of sufficient conditions on Wronskian solutions is given associated with the bilinear Bäcklund transformation. The Wronskian identities of the bilinear KP hierarchy provide a direct and concise way for proving the Wronskian determinant solution. The resulting Wronskian structure generates $ N $-soliton solutions and a few of special Wronskian interaction solutions, which enrich the solution structure of the introduced equation.
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Li Cheng, Yi Zhang, Ying-Wu Hu. Linear superposition and interaction of Wronskian solutions to an extended (2+1)-dimensional KdV equation[J]. AIMS Mathematics, 2023, 8(7): 16906-16925. doi: 10.3934/math.2023864
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