Analytical and Numerical Computations of Multi-Solitons in the Korteweg-de Vries (KdV) Equation

Orapine, Hycienth O.; Ayankop-Andi, E.; Ibeh, G. J.

doi:10.4236/am.2020.117037

Cited by 6 publications

(3 citation statements)

References 9 publications

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“…Due to the cancellation of nonlinearity and dispersion effects, this waveform maintains its original characteristics after pairwise collision. Although this characteristic can be observed through numerical simulation [1], it is found that it does not completely return to the original shape after collision after one-step analysis of the numerical results [2]. Therefore, the study of collision is inseparable from the exact solution of the nonlinear evolution equation.…”

Section: Introductionmentioning

confidence: 95%

Families of exact solutions of a Generalized (2+1)-dimensional Boussinesq type equation

Chen,

2024

Wave Motion

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Section: Introductionmentioning

confidence: 95%

Families of exact solutions of a Generalized (2+1)-dimensional Boussinesq type equation

Chen,

2024

Wave Motion

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“…It has since been used to study a variety of physical systems such as ionic acoustic waves in plasma, lattice acoustic waves, long inner waves in densely layered oceans, and weakly interacting shallow water waves. Over the years, various techniques and methods have been developed to explore the KdV equation [7][8][9][10][11]. Nonlinear evolution equations with variable coefficients are known to be better than those with constant coefficients in describing the heterogeneity of medium and boundary.…”

Section: Introductionmentioning

confidence: 99%

Exact soliton solutions and soliton diffusion of two kinds of stochastic KdV equations with variable coefficients

Wang,

Li,

Wang

et al. 2023

Phys. Scr.

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In this paper, we consider the exact solutions and soliton diffusion phenomenon of two kinds of stochastic KdV equations with variable coefficients. Firstly, according to symmetric reduction, stochastic KdV equations with variable coefficients are transformed into a coupled system of a deterministic KdV-type equation with variable coefficients and a solvable stochastic ordinary differential equation. Then, by the generalized wave transformation and the Clarkson-Kruskal direct method, we obtain the exact solutions of the deterministic KdV-type equation with variable coefficients. By coupling with the exact solutions of the stochastic ordinary differential equation, the exact solutions of stochastic KdV equations with variable coefficients are obtained. Compared to Wick-type stochastic KdV equations, our research work does not require additional inverse transformations, but solves stochastic partial differential equations more concisely, systematically, and directly. Secondly, two examples are given to verify the correctness of the theoretical analysis, and the soliton diffusion phenomenon of the system is discussed. Finally, by the Zabusky-Kruskal finite difference scheme, numerical simulations are provided to demonstrate the effectiveness of the analytic methods. The results indicate that the soliton diffusion phenomenon is subject to noise influence. In particular, the wave speed accelerates the soliton diffusion over time in the multiplicative noise background, and the wave speed slows down the soliton diffusion over time in the additive noise background.

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“…But the detailed analysis of the numerical results exposed the existence of some ripples after a collision meaning that the original identity is not completely recovered [4]. Therefore, it is directive to explore exact solutions of NLEEs admitting soliton solutions for proper scrutiny of collisions.…”

Section: Introductionmentioning

confidence: 99%

Novel localized waves and interaction solutions for a dimensionally reduced (2 + 1)-dimensional Boussinesq equation from N-soliton solutions

et al. 2022

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The Boussinesq equation (BqE) has been of considerable interest in coastal and ocean engineering models for simulating surface water waves in shallow seas and harbors, tsunami wave propagation, wave over-topping, inundation, and near-shore wave process in which nonlinearity and dispersion effects are taken into consideration. The study deals with the dynamics of localized waves and their interaction solutions to a dimensionally reduced (2+1)dimensional BqE from N-soliton solutions with the use of Hirota's bilinear method (HBM).Taking the long-wave limit approach in coordination with some constraint parameters in the N-soliton solutions, the localized waves (i.e., soliton, breather, lump, and rogue waves) and their interaction solutions are constructed. The interaction solutions can be obtained among localized waves, such as (i) one breather or one lump from the two solitons, (ii) one stripe and one breather, and one stripe and one lump from the three solitons, and (iii) two stripes and one breather, one lump and one periodic breather, two stripes and one lump, two breathers, and two lumps from the four solitons. It is to be found that all interactions among the solitons are elastic.The energy, phase shift, shape, and propagation direction of these localized waves and their interaction solutions can be influenced and controlled by the involved constraint parameters.The dynamical characteristics of these localized waves and their interaction solutions are demonstrated through some 3D and density graphs. The outcomes achieved in this study can be used to illustrate the wave interaction phenomena in shallow water.

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Analytical and Numerical Computations of Multi-Solitons in the Korteweg-de Vries (KdV) Equation

Cited by 6 publications

References 9 publications

Families of exact solutions of a Generalized (2+1)-dimensional Boussinesq type equation

Families of exact solutions of a Generalized (2+1)-dimensional Boussinesq type equation

Exact soliton solutions and soliton diffusion of two kinds of stochastic KdV equations with variable coefficients

Novel localized waves and interaction solutions for a dimensionally reduced (2 + 1)-dimensional Boussinesq equation from N-soliton solutions

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