In this paper, we propose a generalised perturbation ($n, N − n$)-fold Darboux transformation (DT) of the modified Korteweg–de Vries (mKdV) equation using the Taylor expansion and a parameter limit procedure. We apply the generalised perturbation ($1, N − 1$)-fold DT to find the new explicit higher-order rational soliton (RS) solutions in terms of determinants of the mKdV equation. These higher-order RS solutions are different from those known soliton results in terms of hyperbolic functions which are obtained from the classical iterated DT. The dynamics behaviours of the first-, second-, third-, and fourth-order RS solutions are shown graphically. The wave propagation characteristics and stability are also discussed using numerical simulations. We find that the initial constant seed solution plays an important role on the wave propagation stability of RS. Through Miura transformation, we give some complex higher-order rational solutions of the Korteweg–de Vries (KdV) equation which are different from the known results. The relevant structures also are discussed using some figures. The method used can also be extended to seek explicit rational solutions of other nonlinear integrable equations.

In this paper, three nonlinear differential-difference equations (NDDEs) from the same hierarchy are investigated using the generalised perturbation $(n, N − n)$-fold Darboux transformation (DT) technique. The dark multisoliton solutions in terms of determinants for three equations are obtained by means of the discrete $N$-fold DT. Propagation and elastic interaction structures of such soliton solutions are shown graphically. The details of their evolutions are studied through numerical simulations. Numerical results show the accuracy of our numerical scheme and the stable evolutions of such dark multisolitons without a noise.We find that the solutions of lower-order NDDEs in the same hierarchy are more robust against a small noise than their corresponding higher-order NDDEs. The discrete generalised perturbation $(1, N − 1)$-fold DT is used to derive some discrete rational and semirational solutions of the first equation, and a few mathematical features are also discussed. Results in this paper might be helpful for understanding some physical phenomena.

The coupled Volterra lattice equation associated with $4 \times 4$ Lax pair is under investigation, which is an integrable discrete form of a coupled KdV equation applied widely in fluids, Bose–Einstein condensation and atmospheric dynamics. First, we explore the conditions for modulational instability (MI) of the constant seed background for this equation. Secondly, we present the discrete Darboux transformation (DT) and generalised DT based on the new $4 \times 4$ Lax pair. Through the resulting discrete DT, the bell-shaped and anti-$N$-shaped soliton solutions of the coupled Volterra lattice equation are derived. Moreover, we derive the $M$-shaped and $N$-shaped rational solitons and bell-shaped and $N$-shaped semirational soliton solutions of the coupled Volterra lattice equation via the discrete generalised DT. Finally, we numerically study the dynamical behaviours of such soliton solutions and find that the rational and semirational soliton solutions have better numerical stability than the usual soliton solution, although three types of solutions are robust against a small noise. The results may be helpful for understanding the two-layered fluid waves near ocean shores described by the coupled Korteweg–de Vries (KdV) equation.

In this paper, the higher-order nonlinear self-dual network equation is investigated. Firstly, an integrable lattice hierarchy associated with this equation is constructed from a discrete matrix spectral problem without the denominator. Secondly, the condition of modulational instability for this equation is given. Thirdly, the infinitely many conservation laws are constructed on the basis of its new Lax representation. Finally, the discrete generalised $(m, N − m)$-fold Darboux transformation (DT) with non-zero constant as seed solution is used to derive new rational soliton (RS) and mixed interaction solutions. As an application, RS solutions can be obtained by the discrete generalised $(1, N −1)$-fold DT (i.e., generalised $(m, N −m)$-fold DT with $m = 1$), and mixed interaction solutions of the usual sech-type soliton (US) and RS can be derived by the discrete generalised $(2, N −2)$-fold DT (i.e., generalised $(m, N − m)$-fold DT with $m = 2$). We also perform the numerical simulations for such soliton solutions to explore their dynamical behaviours. Results given in this paper may have some prospective applications for understanding the propagation of electrical signals.

In this paper, we investigate two extended higher-order KdV models (i.e., the extended Sawada–Kotera equation and the extended Lax equation), which can successfully describe propagation of dimly nonlinear long waves in fluids and ion-acoustic waves in harmonic sparklers. First, we present a general formula of multisoliton solutions of the two models. We then build the interaction solutions in terms of hyperbolic and sinusoidal functions by using multisoliton solutions with appropriate complex conjugate parameters controlling the phase shifts, propagation direction and energies of the waves. In particular, we present their collision solutions in the identical plane with different parametric constraints, which degenerate to the line rogue waves, x-shaped rogue waves, cnoidal periodic waves, interactions of rogue and bell waves, line breather and double breather waves. The dynamical characteristics of the wave solutions are shown graphically by choosing some special parameter values.

Belov–Chaltikian (BC) lattice equation, which is related to the research of lattice analogues of W-algebras, is under consideration in this work. This equation may be viewed as an extension of the Volterra lattice equation. Firstly, we correspond BC lattice equation to several continuous equations under the continuous limit. Secondly, based on the known 3 × 3 matrix form Lax pair of this discrete equation, we construct its discrete generalised (m, 3N −m)-fold Darboux transformation for the first time and successfully popularise this technique from 2×2 Lax pair to 3×3 Lax pair. Finally, by applying the resulting Darboux transformation, we get its higher order rational solutions and analyse their singular trajectories and dynamics using the graphics and limit-state analysis.