An integrable evolution equation for surface waves in deep water

R A Kraenkel; H Leblond; M A Manna

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An integrable evolution equation for surface waves in deep water

机译：深水表面波的可积演化方程

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In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative method, an asymptotic model for small wave steepness ratio is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical results is performed.

机译：为了描述深水中单色表面波的动力学，我们从无限深度的欧拉方程中推导了非线性和色散方程组，用于自由表面标高和自由表面速度。由此，并使用多尺度微扰方法，推导出了小波陡度比的渐近模型。该模型显示为完全可集成的。显示了松散对，第一个守恒量和对称性。理论和数值研究表明，它支持周期性的斯托克斯波，该波在有限的时间内达到峰值并破裂。进行了渐近模型的极限波解与经典结果的比较。

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《Journal of physics, A. Mathematical and theoretical》 |2014年第2期|共17页
作者
R A Kraenkel; H Leblond; M A Manna;
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正文语种 eng
中图分类物理学;
关键词
integrable systems; multi-scale methods; deep water; gravity waves;

机译：可积系统;多尺度方法;深水;重力波;

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1. 两类新的Lie代数和它们的Liouville可积演化方程族 [J] . 夏铁成, 李季上海大学学报（英文版） . 2008,第002期
2. Exact similarity and traveling wave solutions to an integrable evolution equation for surface waves in deep water [J] . Meng Qing, He Bin, Liu Wei Nonlinear dynamics . 2018,第3期

机译：深水中表面波的完全相似性和行进波解
3. Dynamics of optical solitons and conservation laws of a new (2+1)-dimensional integrable nonlinear evolution equation in deep water oceanic waves [J] . Modern Physics Letters, B. Condensed Matter Physics, Statistical Physics, Applied Physics . 2020,第5期

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4. Fourth-order nonlinear evolution equations for a capillary-gravity wave packet in the presence of another wave packet in deep water [J] . Debsarma S, Das KP Physics of fluids . 2007,第9期

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5. EXTENDING INTEGRABILITY OF NONLINEAR WATER WAVE EQUATIONS: NONLINEAR FOURIER ANALYSIS OF BREATHER PACKETS AND ROGUE WAVES AT HIGHER ORDER [C] . Alfred R. Osborne ASME International Conference on Ocean, Offshore and Arctic Engineering . 2019

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6. A pseudospectral implementation of Hamiltonian surface water wave equations for coastal wave simulation [D] . Schauf, Andrew Johnathan 2013

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7. Dynamical behaviour of travelling wave solutions to the conformable time-fractional modified Liouville and mRLW equations in water wave mechanics [O] . Abdulla - Al Mamun, Samsun Nahar Ananna, Tianqing An, 2021

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8. An integrable evolution equation for surface waves in deep water [O] . R.A. Kraenkel, H. Leblond, M.A. Manna 2014

机译：深水地表波的可积演化方程
9. Nonlinear Evolution Equations, Rescalings, Model Partial Differential Equations (PDES) and Their Integrability, 2 [R] . Calogero, F., Eckhaus, W. 1987

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An integrable evolution equation for surface waves in deep water

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