Two-Soliton Interaction Within the Framework of the Modified Korteweg–de Vries Equation

Pelinovsky, Efim; Shurgalina, Ekaterina

doi:10.1007/s11141-015-9560-y

Cited by 33 publications

(14 citation statements)

References 22 publications

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“…′ . The powers of the coefficients obviously correlate with the powers in Equations (17)- (19). For the fixed q (q = 0.5), dependences I2(z), I3(z), and I4(z) are presented in Figure 10.…”

Section: Of 12mentioning

confidence: 88%

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Breather’s Properties within the Framework of the Modified Korteweg–de Vries Equation

Didenkulova

Pelinovsky

2020

Symmetry

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We study a breather’s properties within the framework of the modified Korteweg–de Vries (mKdV) model, where cubic nonlinearity is essential. Extrema, moments, and invariants of a breather with different parameters have been analyzed. The conditions in which a breather moves in one direction or another has been determined. Two limiting cases have been considered: when a breather has an N-wave shape and can be interpreted as two solitons with different polarities, and when a breather contains many oscillations and can be interpreted as an envelope soliton of the nonlinear Schrödinger equation (NLS).

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Section: Of 12mentioning

confidence: 88%

“…Optimal focusing conditions for transforming an arbitrarily large number of solitons to the single freak wave have been found analytically in [16,17]. The fundamental role of two-soliton interactions in complex multi-soliton dynamics has been shown in [18][19][20][21].…”

Section: Introductionmentioning

confidence: 96%

Breather’s Properties within the Framework of the Modified Korteweg–de Vries Equation

Didenkulova

Pelinovsky

2020

Symmetry

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“…Hence, the quantity |B| does not depend on the choice of the seed function and corresponds to the amplitude of the breather. Note that (45) may be written in the form B = ∓ (A + -A -), where A + and Aare the formal amplitudes of the positive and negative solitons with the complex parameter obtained according to (7). The breather amplitude |B| is smaller than the difference between the heights of solitons of different polarities with parameters Re .…”

Section: Collisions Of Breathers and Solitonsmentioning

confidence: 99%

“…Then much information about the gas characteristics may be achieved based on the detailed picture of the two-soliton solution. [7][8][9] However, collisions of many soliton-like structures can cause such exciting wave phenomenon as so-called super-rogue waves, which occur as a result of the modulational instability and describe amazing local enhancement of waves (they may be modeled by high-order rational solutions of the nonlinear Schrödinger equation [NLS] 10,11 ). Similar effects may be utilized to amplify the wave field many times, and are routinely used in the practice of generation of intense optical pulses.…”

Section: Introductionmentioning

confidence: 99%

On the optimal focusing of solitons and breathers in long‐wave models

Slunyaev

2019

Stud Appl Math

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Conditions of optimal (synchronized) collisions of any number of solitons and breathers are studied within the framework of the Gardner equation (GE) with positive cubic nonlinearity, which in the limits of small and large amplitudes tends to other long-wave models, the classic and the modified Korteweg-de Vries equations. The local solution for an isolated soliton or breather within the GE is obtained. The wave amplitude in the focal point is calculated exactly. It exhibits a linear superposition of partial amplitudes of the solitons and breathers. The crucial role of the choice of proper soliton polarities and breather phases on the cumulative wave amplitude in the focal point is demonstrated. Solitons are most synchronized when they have alternating polarities. The straightforward link to the problem of synchronization of envelope solitons and breathers in the focusing nonlinear Schrödinger equation is discussed (then breathers correspond to envelope solitons propagating above a condensate). K E Y W O R D S amplitude in the focal point, Gardner equation, Korteweg-de Vries equation, modified Korteweg-de Vries equation, nonlinear Schrödinger equation, optimal focusing of solitons and breathers Stud Appl Math. 2019;142:385-413.wileyonlinelibrary.com/journal/sapm

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“…In the context of the Riccati-type method there have been shown that the deformed SG, KdV and NLS models [8][9][10], respectively, possess linear system formulations and that they exhibit infinite towers of exact non-local conservation laws. The NLS-type, KdV-type and SG-type models share the same importance due to their potential applications, since they are ubiquitous in all areas of nonlinear physics, such as Bose-Einsten condensation and superconductivity [12][13][14], soliton gas and soliton turbulence in fluid dynamics [15][16][17][18][19][20], the Alice-Bob physics [21,22] and the understanding of a kind of triality among the gauge theories, integrable models and gravity theories [23].…”

Section: Introductionmentioning

confidence: 99%

Deformed Sine-Gordon Models, Solitons and Anomalous Charges

Blas¹,

Callisaya²,

Campos³

et al. 2021

Recent Developments in the Solution of Nonlinear Differential Equations

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We study certain deformations of the integrable sine-Gordon model (DSG). It is found analytically and numerically several towers of infinite number of anomalous charges for soliton solutions possessing a special space–time symmetry. Moreover, it is uncovered exact conserved charges associated to two-solitons with a definite parity under space-reflection symmetry, i.e. kink-kink (odd parity) and kink-antikink (even parity) scatterings with equal and opposite velocities. Moreover, we provide a linear formulation of the modified SG model and a related tower of infinite number of exact non-local conservation laws. We back up our results with extensive numerical simulations for kink-kink, kink-antikink and breather configurations of the Bazeia et al. potential V q w = 64 q 2 tan 2 w 2 1 − sin w 2 q 2 , q ∈ R , which contains the usual SG potential V 2 w = 2 1 − cos 2 w .

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Two-Soliton Interaction Within the Framework of the Modified Korteweg–de Vries Equation

Cited by 33 publications

References 22 publications

Breather’s Properties within the Framework of the Modified Korteweg–de Vries Equation

Breather’s Properties within the Framework of the Modified Korteweg–de Vries Equation

On the optimal focusing of solitons and breathers in long‐wave models

Deformed Sine-Gordon Models, Solitons and Anomalous Charges

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