• 대한수학회보
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• 제56권3호
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• pp.703-728
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• 2019

Considered herein is the orbital stability in the energy space $H^1({\mathbb{R}})$ of a decoupled sum of N peakons for a Camassa-Holm-type equation with quartic nonlinearity, which admits single peakon and multi-peakons. Based on our obtained result of the stability of a single peakon, then combining modulation argument with monotonicity of local energy $H^1$-norm, we get the stability of the sum of N peakons.

• 대한수학회보
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• 제55권1호
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• pp.267-296
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• 2018

This paper studies the Cauchy problem and blow-up phenomena for a new generalized Camassa-Holm equation with cubic nonlinearity in the nonhomogeneous Besov spaces. First, by means of the Littlewood-Paley decomposition theory, we investigate the local well-posedness of the equation in $B^s_{p,r}$ with s > $max\{{\frac{1}{p}},\;{\frac{1}{2}},\;1-{\frac{1}{p}}\},\;p,\;r{\in}[0,{\infty}]$. Second, we prove that the equation is locally well-posed in $B^s_{2,r}$ with the critical index $s={\frac{1}{2}}$ by virtue of the logarithmic interpolation inequality and the Osgood's Lemma, and it is shown that the data-to-solution mapping is $H{\ddot{o}}lder$ continuous. Finally, we derive two kinds of blow-up criteria for the strong solution by using induction and the conservative property of m along the characteristics.

• Journal of applied mathematics & informatics
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• 제33권3_4호
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• pp.261-273
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• 2015

We consider the hyperelastic rod equation describing nonlinear dispersive waves in compressible hyperelastic rods. We investigate the existence of certain traveling wave solutions to this equation. We also determine whether two other equations(the b-family equation and the modified Camassa-Holm equation) have our solution type.