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<p>In this paper, a new integrable Camass–Holm type equation with cubic nonlinearity is studied. The Lie symmetries and group-invariant solutions of the equation are presented. Geometrically, this equation describes a non-trivial one-parameter family of pseudo-spherical surfaces and is the integrability condition of an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathfrak {sl}(2, \mathbb {R})\)</EquationSource> </InlineEquation>-valued linear problem. Moreover, its quadratic pseudo-potentials and infinite number of conservation laws are directly constructed. In addition, the equation is proved to possesses the property of admitting local isometric immersions in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {E}^3\)</EquationSource> </InlineEquation> with the “universal" coefficients of the second fundamental form, which depend only on <i>x</i> and <i>t</i>.</p>

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On a Generalized Camassa–Holm Equation Describing Pseudospherical Surfaces

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月6日

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  • Mingyue Guo,
  • Zhenhua Shi

摘要

In this paper, a new integrable Camass–Holm type equation with cubic nonlinearity is studied. The Lie symmetries and group-invariant solutions of the equation are presented. Geometrically, this equation describes a non-trivial one-parameter family of pseudo-spherical surfaces and is the integrability condition of an \(\mathfrak {sl}(2, \mathbb {R})\) -valued linear problem. Moreover, its quadratic pseudo-potentials and infinite number of conservation laws are directly constructed. In addition, the equation is proved to possesses the property of admitting local isometric immersions in \(\mathbb {E}^3\) with the “universal" coefficients of the second fundamental form, which depend only on x and t.