Abstract
The residual symmetry of a (3+1)‐dimensional Korteweg‐de Vries (KdV)‐like equation is constructed using the truncated Painlevé expansion. Such residual symmetry can be localized and the (3+1)‐dimensional KdV‐like equation is extended into an enlarged system by introducing some new variables. By using Lie’s first theorem, the finite transformation is obtained for this localized residual symmetry. Further, the linear superposition of multiple residual symmetries is localized and the n‐th Bäcklund transformation in the form of the determinants is constructed for this equation. For illustration more detail, the first three multiple wave solutions‐the collisions of resonant solitons are depicted. Finally, with the aid of the link between the consistent tanh expansion (CTE) method and the truncated Painlevé expansion, the explicit soliton‐cnoidal wave interaction solution containing three kinds of Jacobian elliptic functions for this equation is derived.
Highlights
In scientific and engineering fields, nonlinear evolution equations have been studied in wide applications, such as in the nonlinear optics [1,2,3,4,5,6,7], plasma physics [8, 9], fluid mechanics [10, 11], textile engineering [12], and wave propagation phenomena [13,14,15,16]
Methodology, for finding nonlinear evolution equations having infinitely many symmetries or flows, Olver proposed a general method which preserve them and it was employed to the KdV, modified Korteweg-de Vries, Burgers’, and sine-Gordon equations [22, 23]
A consistent Riccati expansion (CRE) method, which can be considered as an extension of the usual Riccati equation method and the tanh function expansion method was proposed for some integrable systems [26]
Summary
In scientific and engineering fields, nonlinear evolution equations have been studied in wide applications, such as in the nonlinear optics [1,2,3,4,5,6,7], plasma physics [8, 9], fluid mechanics [10, 11], textile engineering [12], and wave propagation phenomena [13,14,15,16]. 2. Residual Symmetry and Finite Transformation of the (3+1)-Dimensional KdV-Like Equation. For the Lie point symmetry (23), the following nth Backlund transformation theorem can be summarized according to Lie’s first principle with the aid of its initial value problem. When the seed solution takes u = 0, V = 0, w = a for (2) and (3), it is not difficult to verify that the prolonged (3+1)-dimensional KdV-like system (2), (3), and (17)-(21) possesses the following soliton function fi: fi = 1 + ekix+liy+miz+nit, mi.
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