Inverse scattering transform for the Degasperis–Procesi equation

Constantin, Adrian; Ivanov, Rossen I.; Lenells, Jonatan

doi:10.1088/0951-7715/23/10/012

Cited by 141 publications

(92 citation statements)

References 50 publications

(75 reference statements)

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“…The next example, and the final one directly related to simple problems involving integrable equations with vorticity, is the CH equation (see [6,7,11,20]) as it appears in the water wave context. We start with Eqs.…”

Section: The Ch Equation With Vorticitymentioning

confidence: 99%

A Problem in the Classical Theory of Water Waves: Weakly Nonlinear Waves in the Presence of Vorticity

Johnson¹

2012

JNMP

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The classical water-wave problem is described, and two parameters (ε-amplitude; δ-long wave or shallow water) are introduced. We describe various nonlinear problems involving weak nonlinearity (ε → 0) associated with equations of integrable type ("soliton" equations), but with vorticity. The familiar problem of propagation governed by the Korteweg-de Vries (KdV) equation is introduced, but allowing for an arbitrary distribution of vorticity. The effect of the constant vorticity on the solitary wave is described. The corresponding problem for the Nonlinear Schrödinger (NLS) equation is briefly mentioned but not explored here. The problem of two-way propagation (admitting head-on collisions), as described by the Boussinesq equation, is examined next. This leads to a new equation: the Boussinesq-type equation valid for constant vorticity. However, this cannot be transformed into an integrable Boussinesq equation (as is possible for the corresponding KdV and NLS equations). The solitary-wave solution for this new equation is presented. A description of the Camassa-Holm equation for water waves, with constant vorticity, with its solitary-wave solution, is described. Finally, we outline the problem of propagation of small-amplitude, large-radius ring waves over a flow with vorticity (representing a background flow in one direction). Some properties of this flow, for constant vorticity, are described.

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Section: The Ch Equation With Vorticitymentioning

confidence: 99%

A Problem in the Classical Theory of Water Waves: Weakly Nonlinear Waves in the Presence of Vorticity

Johnson¹

2012

JNMP

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“…It is worth drawing attention to the fact that the DP equation is not merely bi-Hamiltonian, it is also integrable [13].…”

Section: Introductionmentioning

confidence: 99%

Analysis on the blow-up of solutions to a class of integrable peakon equations

Chen

Guo

Liu

et al. 2016

Journal of Functional Analysis

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We investigate the blow-up mechanism of solutions to a class of quasilinear integrable equations which could possess peakons. The dynamics of the blow-up quantity along the characteristics is established by the Riccati-type differential inequality which involves the interaction among three parts: a local nonlinearity, a nonlocal term, and a term stemming from the weak linear dispersion. To analyse the interplay among these quantities, we provide two different approaches. The first one is designed for the case when the equations do not exhibit a weak linear dispersion and hence focuses on the interplay between the first two parts. The method is based on a refined analysis on either evolution of the solution u and its gradient u x , that is, Cu ± u x or the growth rate of the relative ratio u x /u. The second one handles the general situation when all of three parts are present. The idea is to extract the "truly" blow-up component from the Riccati-type differential inequality and utilizes the Morawetz-type identity or higher order conservation laws to show that such a component blows up in finite time before the other component degenerates.

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“…Other exact solutions also have been investigated by many researchers, and some of powerful methods have been presented, such as the extended Jacobi elliptic function expansion method [10][11][12][13], inverse scattering transformation method [14,15], multiple exp-function method [16,17], extended F-expansion method [18], the Hirota method [19][20][21][22][23][24][25], G G -expansion method [26,27], the Weierstrass elliptic function method [28][29][30] and so on. Nakamura [31,32] proposed a convenient way to construct a kind of quasi-periodic solutions of nonlinear equations in his two serial papers.…”

Section: Introductionmentioning

confidence: 99%

Soliton and Riemann theta function quasi-periodic wave solutions for a $$(2+1)$$ ( 2 + 1 ) -dimensional generalized shallow water wave equation

2015

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In this paper, a (2 + 1)-dimensional generalized shallow water wave equation is investigated through bilinear Hirota method. Interestingly, the breather-type and lump-type soliton solutions are obtained. Furthermore, dynamic properties of the soliton waves are revealed by means of the asymptotic analysis. Based on Hirota bilinear method and Riemann theta function, we succeed in constructing quasi-periodic wave solutions with a generalized form. We also display the asymptotic properties of these quasi-periodic wave solutions and point out the relation between the quasiperiodic wave solutions and the soliton solutions.

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Inverse scattering transform for the Degasperis–Procesi equation

Cited by 141 publications

References 50 publications

A Problem in the Classical Theory of Water Waves: Weakly Nonlinear Waves in the Presence of Vorticity

A Problem in the Classical Theory of Water Waves: Weakly Nonlinear Waves in the Presence of Vorticity

Analysis on the blow-up of solutions to a class of integrable peakon equations

Soliton and Riemann theta function quasi-periodic wave solutions for a $$(2+1)$$ ( 2 + 1 ) -dimensional generalized shallow water wave equation

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