A new integrable equation with no smooth solitons

Zhijun Qiao; Liping Liu

doi:10.1016/j.chaos.2007.11.034

A new integrable equation with no smooth solitons

Zhijun Qiao
, Liping Liu

Mathematics and Statistics

University of Texas Pan American

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

Abstract

In this paper, we propose a new completely integrable equation:mt = frac(1, 2) fenced(frac(1, m2))xxx - frac(1, 2) fenced(frac(1, m2))x,which has no smooth solitons. This equation is shown to have bi-Hamiltonian structure and Lax pair, which imply integrability of the equation. Studying this new equation, we develop two new kinds of soliton solutions under the inhomogeneous boundary condition lim| x | → ∞ m = B where B is nonzero constant. One is continuous and piecewise smooth "W/M"-shape-peaks solitary solution and the other one-single-peak soliton. The two new kinds of peaked solitons can not be written as the regular type peakon: c e- | x - ct |, where c is a constant. We will provide graphs to show those new kinds of peaked solitons. © 2008 Elsevier Ltd. All rights reserved.

Original language	English
Pages (from-to)	587-593
Number of pages	7
Journal	Chaos, Solitons and Fractals
Volume	41
Issue number	2
DOIs	https://doi.org/10.1016/j.chaos.2007.11.034
State	Published - Jul 30 2009

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10.1016/j.chaos.2007.11.034

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T1 - A new integrable equation with no smooth solitons

AU - Qiao, Zhijun

AU - Liu, Liping

PY - 2009/7/30

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N2 - In this paper, we propose a new completely integrable equation:mt = frac(1, 2) fenced(frac(1, m2))xxx - frac(1, 2) fenced(frac(1, m2))x,which has no smooth solitons. This equation is shown to have bi-Hamiltonian structure and Lax pair, which imply integrability of the equation. Studying this new equation, we develop two new kinds of soliton solutions under the inhomogeneous boundary condition lim| x | → ∞ m = B where B is nonzero constant. One is continuous and piecewise smooth "W/M"-shape-peaks solitary solution and the other one-single-peak soliton. The two new kinds of peaked solitons can not be written as the regular type peakon: c e- | x - ct |, where c is a constant. We will provide graphs to show those new kinds of peaked solitons. © 2008 Elsevier Ltd. All rights reserved.

AB - In this paper, we propose a new completely integrable equation:mt = frac(1, 2) fenced(frac(1, m2))xxx - frac(1, 2) fenced(frac(1, m2))x,which has no smooth solitons. This equation is shown to have bi-Hamiltonian structure and Lax pair, which imply integrability of the equation. Studying this new equation, we develop two new kinds of soliton solutions under the inhomogeneous boundary condition lim| x | → ∞ m = B where B is nonzero constant. One is continuous and piecewise smooth "W/M"-shape-peaks solitary solution and the other one-single-peak soliton. The two new kinds of peaked solitons can not be written as the regular type peakon: c e- | x - ct |, where c is a constant. We will provide graphs to show those new kinds of peaked solitons. © 2008 Elsevier Ltd. All rights reserved.

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JO - Chaos, Solitons and Fractals

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A new integrable equation with no smooth solitons

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