We consider the evolution of an initial disturbance described by the modified Korteweg–de Vries equation with a positive coefficient of the cubic nonlinear term, so that it can support solitons. Our primary aim is to determine the circumstances which can lead to the formation of solitons and/or breathers. We use the associated scattering problem and determine the discrete spectrum, where real eigenvalues describe solitons and complex eigenvalues describe breathers. For analytical convenience we consider various piecewise-constant initial conditions. We show how complex eigenvalues may be generated by bifurcation from either the real axis, or the imaginary axis; in the former case the bifurcation occurs as the unfolding of a double real eigenvalue. A bifurcation from the real axis describes the transition of a soliton pair with opposite polarities into a breather, while the bifurcation from the imaginary axis describes the generation of a breather from the continuous spectrum. Within the class of initial conditions we consider, a disturbance of one polarity, either positive or negative, will only generate solitons, and the number of solitons depends on the total mass. On the other hand, an initial disturbance with both polarities and very small mass will favor the generation of breathers, and the number of breathers then depends on the total energy. Direct numerical simulations of the modified Korteweg–de Vries equation confirms the analytical results, and show in detail the formation of solitons, breathers, and quasistationary coupled soliton pairs. Being based on spectral theory, our analytical results apply to the entire hierarchy of evolution equations connected with the same eigenvalue problem.
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June 2000
Research Article|
June 01 2000
On the generation of solitons and breathers in the modified Korteweg–de Vries equation
Simon Clarke;
Simon Clarke
Department of Mathematics and Statistics, Monash University, Clayton, Victoria, Australia
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Roger Grimshaw;
Roger Grimshaw
Department of Mathematics and Statistics, Monash University, Clayton, Victoria, Australia
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Peter Miller;
Peter Miller
Department of Mathematics and Statistics, Monash University, Clayton, Victoria, Australia
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Efim Pelinovsky;
Efim Pelinovsky
Institute of Applied Physics and Nizhny Novgorod Technical University, Nizhny Novgorod, Russia
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Tatiana Talipova
Tatiana Talipova
Institute of Applied Physics and Nizhny Novgorod Technical University, Nizhny Novgorod, Russia
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Chaos 10, 383–392 (2000)
Article history
Received:
October 19 1999
Accepted:
March 28 2000
Citation
Simon Clarke, Roger Grimshaw, Peter Miller, Efim Pelinovsky, Tatiana Talipova; On the generation of solitons and breathers in the modified Korteweg–de Vries equation. Chaos 1 June 2000; 10 (2): 383–392. https://doi.org/10.1063/1.166505
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