Soliton molecules and the CRE method in the extended mKdV equation

The soliton molecules of the (1+1)-dimensional extended modified Korteweg–de Vries (mKdV) system are obtained by a new resonance condition, which is called velocity resonance. One soliton molecule and interaction between a soliton molecule and one-soliton are displayed by selecting suitable parameters. The soliton molecules including the bright and bright soliton, the dark and bright soliton, and the dark and dark soliton are exhibited in figures 1–3, respectively. Meanwhile, the nonlocal symmetry of the extended mKdV equation is derived by the truncated Painlevé method. The consistent Riccati expansion (CRE) method is applied to the extended mKdV equation. It demonstrates that the extended mKdV equation is a CRE solvable system. A nonauto-Bäcklund theorem and interaction between one-soliton and cnoidal waves are generated by the CRE method.