Abstract
Under investigation is the discrete modified Korteweg-de Vries (mKdV) equation, which is an integrable discretization of the continuous mKdV equation that can describe some physical phenomena such as dynamics of anharmonic lattices, solitary waves in dusty plasmas, and fluctuations in nonlinear optics. Through constructing the discrete generalized m , N − m -fold Darboux transformation for this discrete system, the various discrete soliton solutions such as the usual soliton, rational soliton, and their mixed soliton solutions are derived. The elastic interaction phenomena and physical characteristics are discussed and illustrated graphically. The limit states of diverse soliton solutions are analyzed via the asymptotic analysis technique. Numerical simulations are used to display the dynamical behaviors of some soliton solutions. The results given in this paper might be helpful for better understanding the physical phenomena in plasma and nonlinear optics.
Highlights
Nonlinear partial differential equations (NPDEs) such as Burgers equation, KdV equation, modified Korteweg-de Vries (mKdV) equation, and nonlinear Schrödinger (NLS) equation can describe many important physical phenomena in nonlinear optics, acoustic in the nonharmonized lattice, deep water waves, plasma environments, and so on
The point here is that when we perform asymptotic analysis to the usual two-soliton and three-soliton solutions, the final two and three solitons keep their shape before and after the collisions, and for asymptotic analysis to second-order and third-order RS solutions, we find that there is one first-order RS left at infinity for second-order RS solutions and the other first-order RS disappears at infinity, while there are two first-order RSs left at infinity for third-order RS solutions and the third RS disappears
For higher-order RS solutions, there is always a RS solution that disappears at infinity, which is a little like rogue wave solutions that are asymptotic to the background wave at infinity, and we find that these new phenomena about RS are interesting and quite different from those of the usual soliton (US), which is worthy of further investigation
Summary
Nonlinear partial differential equations (NPDEs) such as Burgers equation, KdV equation, mKdV equation, and nonlinear Schrödinger (NLS) equation can describe many important physical phenomena in nonlinear optics, acoustic in the nonharmonized lattice, deep water waves, plasma environments, and so on (see [1,2,3,4,5] and the references therein). Based on what we know, the US solutions, RS solutions, their mixed interaction soliton solutions via the discrete generalized ðm, N − mÞ-fold DT, and associated soliton limit states via the asymptotic analysis technique for equation (1) have not been investigated before. We will use the generalized ðm, N − mÞ-fold DT to get various soliton solutions, like US solutions, RS solutions, and their mixed soliton solutions and analyze the elastic interaction and limit state of such solitons by using asymptotic analysis
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