Articles written in Pramana – Journal of Physics
Volume 83 Issue 1 July 2014 pp 29-37
In this paper, a method that can be used to construct the infinitely-many conservation laws with the Lax pair is generalized from the (1+1)-dimensional nonlinear evolution equations (NLEEs) to the (2+1)-dimensional ones. Besides, we apply that method to the Kadomtsev– Petviashvili (KP) and Davey–Stewartson equations in fluids, and respectively obtain their infinitelymany conservation laws with symbolic computation. Based on that method, we can also construct the infinitely-many conservation laws for other multidimensional NLEEs possessing the Lax pairs, including the cylindrical KP, modified KP and (2+1)-dimensional Gardner equations, in fluids, plasmas, optical fibres and Bose–Einstein condensates.
Volume 90 Issue 4 April 2018 Article ID 0045 Research Article
In this paper, investigation is made on a Kadomtsev–Petviashvili-based system, which can be seen in fluid dynamics, biology and plasma physics. Based on the Hirota method, bilinear form and Bäcklund transformation(BT) are derived. $N$-soliton solutions in terms of the Wronskian are constructed, and it can be verified that the $N$ soliton solutions in terms of the Wronskian satisfy the bilinear form and Bäcklund transformation. Through the $N$-soliton solutions in terms of the Wronskian, we graphically obtain the kink-dark-like solitons and parallel solitons, which keep their shapes and velocities unchanged during the propagation.
Volume 91 Issue 3 September 2018 Article ID 0043 Research Article
In this paper, a (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation, which describes the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersionand weak perturbation in fluid mechanics, is investigated. Lump, lump–soliton and rouge–soliton solutions are obtained with the aid of symbolic computation. For the lump and soliton, amplitudes are related to the nonlinearity coefficient and dispersion coefficient, while velocities are related to the perturbation coefficients. Fusion and fission phenomena between the lump and soliton are observed, respectively. Graphic analysis shows that: (i) soliton’s amplitude becomes larger after the fusion interaction, and becomes smaller after the fission interaction; (ii) afterthe interaction, the soliton propagates along the opposite direction to before when any one of the perturbation coefficients is a time-dependent function. For the interactions between the rogue wave and two solitons, the roguewave splits from one soliton and merges into the other one, and the two solitons exchange the amplitudes through the energy transfer by the rogue wave.
Volume 92 Issue 1 January 2019 Article ID 0009 Research Article
In this paper, we investigate the three coupled variable-coefficient nonlinear Schrödinger equations, which describe the amplification or attenuation of the picosecond pulse propagation in the inhomogeneous multicomponent optical fibre with different frequencies or polarisations. Based on the Darboux dressing transformation, semirational rogue wave solutions are derived. Semirational rogue waves, Peregrine combs and Peregrine walls are obtained and demonstrated. Splitting behaviour of the semirational Peregrine combs and attenuating phenomenon of the semirational Peregrine wall are exhibited. Effects of the group velocity dispersion, nonlinearity and fibre gain/loss are discussed according to the different fibres.We find that the maximum amplitude of the hump of the semirational rogue wave is less than nine times the background height due to the interaction between the soliton part and rogue wave part. Further, there is a bent in the soliton part of the semirational rogue.
Volume 93 Issue 3 September 2019 Article ID 0038 Research Article
In this paper, the AB system with time-dependent coefficients for the ultrashort pulses in an inhomogeneous optical fibre or the marginally unstable baroclinic wave packets in an atmospheric or oceanic system is investigated via the Lie symmetry analysis. We obtain the Lie symmetries, reduced equations and groupinvariant solutions. The nonlinear self-adjointness of the AB system is proved, and the conservation laws associated with the Lie symmetries are constructed. For the amplitude of the electric field in the inhomogeneous optical fibre or the amplitude of the wave packet in the atmospheric or oceanic system, and for the quantity associated with the occupation number which gives a measure of the atomic inversion in the inhomogeneous optical fibre or the quantity measuring the correction of the basic flow in the atmospheric or oceanic system, we get some solitons through the Lie symmetry transformations, whose amplitudes,widths, velocities and backgrounds are different from those of the given ones and can be adjusted via the Lie group parameters. We find a family of the ultrashort pulses propagating in the inhomogeneous optical fibre or a family of the marginally unstable baroclinic wave packets propagating in the atmospheric or oceanic system.
Volume 95 All articles Published: 19 August 2021 Article ID 0137 Research Article
Two-layer fluids are seen in fluid mechanics, thermodynamics and medical sciences. Lattices are seen in solid-state physics. In a two-layer liquid or a lattice, a (3 + 1)-dimensional generalised Yu–Toda–Sasa–Fukuyama equation is hereby studied with symbolic computation. Via the Hirota method, bilinear form and bilinear auto-Bäcklund transformation under certain coefficient constraints are obtained. Breather solutions are worked out based on the Hirota method and extended homoclinic test approach. Considering that the periods of breather solutions tend to infinity, we derive the lump solutions under a limit procedure. We observe that the amplitudes of the breather and lump remain unchanged during the propagation. Furthermore, we graphically present the breathers and lumps under the influence of different coefficients in the equation.
Volume 96, 2022
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