Articles written in Pramana – Journal of Physics
Volume 91 Issue 2 August 2018 Article ID 0023 Research Article
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.
Volume 92 Issue 1 January 2019 Article ID 0010 Research Article
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.
Volume 93 Issue 2 August 2019 Article ID 0023 Research Article
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.
Volume 94, 2020
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