(2021) Supersolidity of Cnoidal Waves in an Ultracold Bose

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• Waves on Deep Water, I
Lecture 14: Waves on deep water, I Lecturer: Harvey Segur. Write-up: Adrienne Traxler June 23, 2009 1 Introduction In this lecture we address the question of whether there are stable wave patterns that propagate with permanent (or nearly permanent) form on deep water. The primary tool for this investigation is the nonlinear Schr¨odinger equation (NLS). Below we sketch the derivation of the NLS for deep water waves, and review earlier work on the existence and stability of 1D surface patterns for these waves. The next lecture continues to more recent work on 2D surface patterns and the eﬀect of adding small damping. 2 Derivation of NLS for deep water waves The nonlinear Schr¨odinger equation (NLS) describes the slow evolution of a train or packet of waves with the following assumptions: • the system is conservative (no dissipation) • then the waves are dispersive (wave speed depends on wavenumber) Now examine the subset of these waves with • only small or moderate amplitudes • traveling in nearly the same direction • with nearly the same frequency The derivation sketch follows the by now normal procedure of beginning with the water wave equations, identifying the limit of interest, rescaling the equations to better show that limit, then solving order-by-order. We begin by considering the case of only gravity waves (neglecting surface tension), in the deep water limit (kh → ∞). Here h is the distance between the equilibrium surface height and the (ﬂat) bottom; a is the wave amplitude; η is the displacement of the water surface from the equilibrium level; and φ is the velocity potential, u = ∇φ.
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• 21 Non-Linear Wave Equations and Solitons
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• Soliton Molecules, Rational Positon Solution and Rogue Waves for the Extended Complex Modi Ed Kdv Equation
Soliton Molecules, Rational Positon Solution and Rogue Waves for the Extended Complex Modied KdV Equation Lin Huang Hangzhou Dianzi University Nannan Lv ( [email protected] ) Hangzhou Dianzi University School of Science Research Article Keywords: Darboux transformation, Soliton molecule, Positon solution, Rational positon solution, Rogue waves solution Posted Date: May 11th, 2021 DOI: https://doi.org/10.21203/rs.3.rs-434604/v1 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Version of Record: A version of this preprint was published at Nonlinear Dynamics on August 5th, 2021. See the published version at https://doi.org/10.1007/s11071-021-06764-x. Soliton molecules, rational positons and rogue waves for the extended complex modified KdV equation Lin Huang ∗and Nannan Lv† School of Science, Hangzhou Dianzi University, 310018 Zhejiang, China. May 6, 2021 Abstract We consider the integrable extended complex modified Korteweg–de Vries equation, which is generalized modified KdV equation. The first part of the article considers the construction of solutions via the Darboux transformation. We obtain some exact solutions, such as soliton molecules, positon solutions, rational positon solutions, and rogue waves solutions. The second part of the article analyzes the dynamics of rogue waves. By means of the numerical analysis, under the standard decomposition, we divide the rogue waves into three patterns: fundamental pattern, triangular pattern and ring pattern. For the fundamental pattern, we define the length and width of the rogue waves and discuss the effect of different parameters on rogue waves. Keywords: Darboux transformation, Soliton molecule, Positon solution, Rational positon so- lution, Rogue waves solution.
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