Soliton molecules, nonlocal symmetry and CRE method of the KdV equation with higher-order corrections

Abstract

The soliton molecules of the Korteweg–de Vries (KdV) equation with higher-order corrections are studied by using the velocity resonance mechanism and the multi-soliton solution. The interaction between a soliton molecule and one-soliton of the KdV equation with higher-order corrections is elastic by means of analytical and graphical ways. The nonlocal symmetry of the KdV equation with higher-order corrections is derived by the truncate Painlevé analysis. An nonauto-Bäcklund theorem is established by solving the initial value problem of the Lie’s first principle of the nonlocal symmetry. In the meanwhile, the KdV equation with higher-order corrections is proved to be a consistent Riccati expansion (CRE) solvable system by exploiting the CRE method.

Soliton Molecules and Full Symmetry Groups to the KdV-Sawada-Kotera-Ramani Equation

“…In 2019, Lou [11] introduced a velocity resonant mechanism to form soliton molecules and asymmetric solitons for three-fifth order systems. Very recently, soliton molecules and some hybrid solutions involving Lump, breather, and positon have been investigated for some (1 + 1)-dimensional and (2 + 1)-dimensional equations by Hirota bilinear method and Darboux transformation [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].…”

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].…”

Resonance Y-type soliton, hybrid and quasi-periodic wave solutions of a generalized (2+1)-dimensional nonlinear wave equation

“…Soliton molecules are the bound states of solitons which have been experimentally discovered on optical systems [21][22][23]. Soliton molecule is essentially a velocity resonance soliton [24][25][26]. The linear superposition principle can be used for constructing the resonance solutions [27][28][29].…”