Fission and fusion interaction phenomena of the discrete kink multi-soliton solutions for the Chen–Lee–Liu lattice equation

Chen–Lee–Liu (CLL) lattice equation is an integrable discretization of the CLL equation which can be used to model the evolution of the self-steepening optical pulses without self-phase modulation. In this paper, the discrete N-fold Darboux transformation (DT) is used to derive the discrete kink multi-soliton solutions in terms of determinant for CLL lattice equation. Soliton fission and fusion interaction structures of such solutions are shown graphically. The details of their evolution are investigated by using numerical simulations, showing that a small noise with amplitude less than or equal to 0.01 produces a strong oscillation and instability of these kink soliton solutions. The discrete generalized perturbation [Formula: see text]-fold DT is constructed to express some rational solutions in terms of the determinants of CLL lattice equation by modifying the discrete N-fold DT. Infinitely many conservation laws for CLL lattice equation are constructed based on its Lax representation. Results in this paper might be helpful for understanding the propagation of optical pulses.

Dynamics and elastic interactions of the discrete multi-dark soliton solutions for the Kaup–Newell lattice equation

Under consideration in this paper is the Kaup–Newell (KN) lattice equation which is an integrable discretization of the KN equation. Infinitely, many conservation laws and discrete N-fold Darboux transformation (DT) for this system are constructed and established based on its Lax representation. Via the resulting N-fold DT, the discrete multi-dark soliton solutions in terms of determinants are derived from non-vanishing background. Propagation and elastic interaction structures of such solitons are shown graphically. Overtaking interaction phenomena between/among the two, three and four solitons are discussed. Numerical simulations are used to explore their dynamical behaviors of such multi-dark solitons. Numerical results show that their evolutions are stable against a small noise. Results in this paper might be helpful for understanding the propagation of nonlinear Alfvén waves in plasmas.

Integrable discretization of hodograph-type systems, hyperelliptic integrals and Whitham equations

Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In particular, we present discrete counterparts of (generalized) hodograph equations, hyperelliptic integrals and associated cycles, characteristic speeds of Whitham-type and (implicitly) the corresponding Whitham equations. By construction, the intimate relationship with integrable system theory is maintained in the discrete setting.