Soliton molecules and the CRE method in the extended mKdV equation

Abstract

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 –, 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.

Residual Symmetry, Bäcklund Transformation, and Soliton Solutions of the Higher-Order Broer-Kaup System

“…In reference [26], Lou start from the truncated Painlevé expansion to propose the definition of consistent Riccati expansion (CRE) solvable. Inspired by this reference, we consider the special case of CRE-consistent Tanh expansion (CTE), which is a more generalized but much simpler method to find interaction solutions between solitons and other nonlinear excitations, such as soliton-resonant solutions, soliton and condial wave, and soliton and sin-cosine wave [38][39][40].…”

Painlevé Analysis, Soliton Molecule, and Lump Solution of the Higher-Order Boussinesq Equation

“…The high-order dispersive terms play a key role in the velocity resonance mechanism [10]. The velocity resonance mechanism is developed to some integrable systems, the (2+1)-dimensional fifth-order Korteweg-de Vries (KdV) equation [11], the complex modified KdV equation [12], the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation [13], and so on [14][15][16]. Combining the Darboux transformation and the variable separation approach, some interactions between soliton molecules and breather solutions and between soliton molecules and dromions are explored [11][12][13][14][15]17].…”

“…The velocity resonance mechanism is developed to some integrable systems, the (2+1)-dimensional fifth-order Korteweg-de Vries (KdV) equation [11], the complex modified KdV equation [12], the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation [13], and so on [14][15][16]. Combining the Darboux transformation and the variable separation approach, some interactions between soliton molecules and breather solutions and between soliton molecules and dromions are explored [11][12][13][14][15]17]. In addition to the soliton molecule, lump solutions are a kind of rational function solutions which have become a hot field in nonlinear systems [18][19][20][21][22].…”

Characteristics of the Soliton Molecule and Lump Solution in the 2 + 1 -Dimensional Higher-Order Boussinesq Equation

“…To balance the nonlinear effects, the high-order dispersive terms may play a key role in the velocity resonance mechanism [21]. The soliton molecule of a variety of integrable systems has been verified with the velocity resonance mechanism: the fifth-order Korteweg-de Vries (KdV) equation [22,23], the modified KdV equation [24,25], the ð3 + 1Þ -dimensional Boiti-Leon-Manna-Pempinelli equation [26], and so on [27]. The dynamics between soliton molecules and breather solutions and between soliton molecules and dromions are presented by the velocity resonance mechanism, the Darboux transformation, and the variable separation approach [25][26][27][28].…”