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Residual symmetry, n th Bäcklund transformation, and soliton‐cnoidal wave interaction solution for the combined modified KdV–negative‐order modified KdV equation
Author(s) -
Cheng Wenguang,
Qiu Deqin,
Xu Tianzhou
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5935
Subject(s) - korteweg–de vries equation , mathematics , homogeneous space , soliton , symmetry (geometry) , mathematical physics , integrable system , transformation (genetics) , cnoidal wave , residual , lax pair , order (exchange) , mathematical analysis , conjecture , pure mathematics , wave equation , physics , nonlinear system , quantum mechanics , geometry , finance , algorithm , economics , gene , biochemistry , chemistry
This paper is concerned with the n th Bäcklund transformation (BT) related to multiple residual symmetries and soliton‐cnoidal wave interaction solution for the combined modified KdV–negative‐order modified KdV (mKdV‐nmKdV) equation. The residual symmetry derived from the truncated Painlevé expansion can be extended to the multiple residual symmetries, which can be localized to Lie point symmetries by prolonging the combined mKdV‐nmKdV equation to a larger system. The corresponding finite symmetry transformation, ie, n th BT, is presented in determinant form. As a result, new multiple singular soliton solutions can be obtained from known ones. We prove that the combined mKdV‐nmKdV equation is integrable, possessing the second‐order Lax pair and consistent Riccati expansion (CRE) property. Furthermore, we derive the exact soliton and soliton‐cnoidal wave interaction solutions by applying the nonauto‐BT obtained from the CRE method.