Soliton Molecules, Asymmetric Solitons and Hybrid Solutions for (2+1)-Dimensional Fifth-Order KdV Equation*

Abstract

Soliton molecules were first discovered in optical systems and are currently a hot topic of research. We obtain soliton molecules of the (2+1)-dimensional fifth-order KdV system under a new resonance condition called velocity resonance in theory. On the basis of soliton molecules, asymmetric solitons can be obtained by selecting appropriate parameters. Based on the N-soliton solution, we obtain hybrid solutions consisting of soliton molecules, lump waves and breather waves by partial velocity resonance and partial long wave limits. Soliton molecules, and some types of special soliton resonance solutions, are stable under the meaning that the interactions among soliton molecules are elastic. Both soliton molecules and asymmetric solitons obtained may be observed in fluid systems because the fifth-order KdV equation describes the ion-acoustic waves in plasmas, shallow water waves in channels and oceans.

Resonance Y-shape solitons and mixed solutions for a (2+1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics

“…Through long wave limit approach, one can gain lump solution [9,10]. By combining the above methods, one can also get a hybrid solution consisting of soliton molecules, breather solutions and lump solutions [11][12][13][14][15][16].…”

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

“…The linear superposition principle can be used for constructing the resonance solutions [27][28][29]. The hybrid solutions consisting of soliton molecules and lump waves were investigated by partial velocity resonance and partial long wave limits [30,31]. It is interesting that Li et al considered a more generalized constraint of the parameters in N -solitons to construct the resonance Y -type soliton solutions and the hybrid solutions among the resonance Y -type solitons, breathers, soliton molecules and lumps [32,33].…”