Soliton solutions to the Novikov equation and a negative flow of the Novikov hierarchy

Wu, Lihua; Li, Chao; Li, Nianhua

doi:10.1016/j.aml.2018.07.036

Cited by 22 publications

(12 citation statements)

References 15 publications

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“…It is easy to see that by integrating the second equation of (2.15) and subtituting it into (2.16), we reach the results appeared in [15,37].…”

Section: Reductions Of the Reciprocal Transformationsupporting

confidence: 59%

“…They also related this equation to a negative flow of the Sawada-Kotera hierarchy via a reciprocal transformation. Smooth multi-soliton solutions of the Novikov equation have been presented via several approaches such as the Hirota bilinear method, Riemann-Hilbert method and DT [31,2,37]. Furthermore, multipeakons of the Novikov equation may be computed by inverse spectral method [14].…”

Section: Introductionmentioning

confidence: 99%

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Smooth multisoliton solutions of the Geng-Xue equation

Li¹,

Liu²

2022

Preprint

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We present a reciprocal transformation which links the Geng-Xue equation to a particular reduction of the first negative flow of the Boussinesq hierarchy. We discuss two reductions of the reciprocal transformation for the Degasperis-Procesi and Novikov equations, respectively. With the aid of the Darboux transformation and the reciprocal transformation, we obtain a compact parametric representation for the smooth soliton solutions such as multi-kink solutions of the Geng-Xue equation.

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“…It is easy to see that by integrating the second equation of (2.15) and subtituting it into (2.16), we reach the results appeared in [15,37].…”

Section: Reductions Of the Reciprocal Transformationsupporting

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Smooth multisoliton solutions of the Geng-Xue equation

Li¹,

Liu²

2022

Preprint

Self Cite

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“…Recently, Boutet de Monvel et al developed the inverse scattering transform method for the Novikov equation (1.1) [11] with nonzero constant background. And Wu et al obtained a parametric representation for N-soliton solutions to the Novikov equation and the negative flow of the Novikov hierarchy through Darboux transformations [10].…”

Section: Introductionmentioning

confidence: 99%

Long-time asymptotic behavior for the Novikov equation in solitonic regions of space time

Yang¹,

Fan²

2021

Preprint

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In this paper, we consider the long time asymptotic behavior for the initial value problem of the Novikov equationwhere u 0 (x) is assumed to decay to a nonzero constant: u(x) → κ > 0,x → ±∞. The study makes crucial use of the inverse scattering transform as well as of the ∂ generalization of Deift-Zhou steepest descent method for oscillatory Riemann-Hilbert (RH) problems. Based on the spectral analysis of the Lax pair associated with the Novikov equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a RH problem. In different space-time solitonic region of x/t, we further compute the different long time asymptotic expansion of the solution u(x, t), which implies soliton resolution conjecture and can be characterized with an N (Λ)-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the region. And it has diverse residual error order from ∂

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“…However, it is trivial to derive from equations (2.6a) and (2.8) in [11] by introducing a potential (writing W = −F τ , U = 3F y ) followed by a rescaling A. G. Rasin, J. Schiff / A simple-looking relative of the Novikov, Hirota-Satsuma and Sawada-Kotera equations (and changes of names of the variables). See also [12], equation (9). Matsuno derived a different scalar equation from equations (2.6a) and (2.8) in [11].…”

Section: Introductionmentioning

confidence: 99%

A simple-looking relative of the Novikov, Hirota-Satsuma and Sawada-Kotera equations

Rasin

Schiff

2019

JNMP

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We study the simple-looking scalar integrable equation f xxt − 3( f x f t − 1) = 0, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a Bäcklund transformation, soliton and merging soliton solutions (some exhibiting instabilities), two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries, a Painlevé series, a scaling reduction to a third order ODE and its Painlevé series, and the Hirota form (giving further multisoliton solutions).

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Soliton solutions to the Novikov equation and a negative flow of the Novikov hierarchy

Cited by 22 publications

References 15 publications

Smooth multisoliton solutions of the Geng-Xue equation

Smooth multisoliton solutions of the Geng-Xue equation

Long-time asymptotic behavior for the Novikov equation in solitonic regions of space time

A simple-looking relative of the Novikov, Hirota-Satsuma and Sawada-Kotera equations

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