Abstract In this work, we mainly address two new integrable (2+1)- and (3+1)-dimensional sinh-Gordon equations, which naturally appear in surface theory and fluid dynamics. The first equation includes constant coefficients, while the other is characterized with time-dependent coefficients. It is of further value to investigate the integrability of each model. This study puts forward a Painleve test to reveal the Painleve integrability. We show that the first equation passes the Painleve test to confirm its integrability. However, the compatibility conditions of the second model with time-dependent coefficients provides the relation between these coefficients to ensure its integrability. We show that the dispersion relations of the two equations are distinct, whereas the phase shifts are identical. We apply the simplified Hirota's method where four sets of multiple soliton are derived for these equations.

中文翻译：

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新的可积 (2+1) 和 (3+1) 维 sinh-Gordon 方程，具有常数和时间相关系数

摘要 在这项工作中，我们主要解决了两个新的可积 (2+1) 维和 (3+1) 维 sinh-Gordon 方程，它们自然出现在表面理论和流体动力学中。第一个方程包括常数系数，而另一个方程的特征是时间相关系数。研究每个模型的可集成性具有进一步的价值。本研究提出了 Painleve 检验来揭示 Painleve 可积性。我们表明第一个方程通过了 Painleve 检验以确认其可积性。然而，第二个模型与时间相关系数的兼容性条件提供了这些系数之间的关系，以确保其可积分性。我们表明这两个方程的色散关系是不同的，而相移是相同的。我们应用简化的 Hirota'