Fig 4 - uploaded by Evgueni Dinvay
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The upper part of the solution diagram C 5 is plotted for (2.15) with h = π/5. Three secondary branches bifurcating from C 5 at μ ≈ 0.23484 and w ∞ ≈ 0.10444 are denoted by C 51 , C 52 and C 53 . The upper bound μ/2 mentioned prior to (1.7) is included.
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A new operator equation for periodic gravity waves on water of finite depth is derived and investigated; it is equivalent to Babenko’s equation considered in Kuznetsov and Dinvay (Water Waves, 1, 2019). Both operators in the proposed equation are nonlinear and depend on the parameter equal to the mean depth of water, whereas each solution defines a...
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Context 1
... (0, 0.35686). The last value is approximately equal to the L ∞ -norm of the solution corresponding to the wave of extreme form. The single maximum of this curve approximately equal to 0.54543 is attained at w ∞ ≈ 0.33433. The latter value is slightly less than the norm corresponding to the turning point solution; namely, ≈ 0.34553. Now we turn to Fig. 4, where the upper part of the solution diagram C 5 is plotted. This curve bifurcates from the zero solution at μ ≈ 0.19925, whereas there are three secondary branches (denoted by C 51 , C 52 and C 53 ), which, within the accuracy of our computations, bifurcate from C 5 at the point with μ ≈ 0.23484 and w ∞ ≈ 0.10444. Unfortunately, the ...
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One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution...
Citations
... Gravity water waves are known to play a crucial role in coastal, energy, and hydraulic engineering and attract much current interest [1][2][3][4][5][6][7]. A series of nonlinear partial differential equations are proposed to analyze the characteristics of gravity water waves [8][9][10][11][12]; a typical example is the ZK-mZK-BBM equation, which describes gravity water waves in a fluid [13]. ...
The ZK–mZK–BBM equation plays a crucial role in actually depicting the gravity water waves with the long wave region. In this article, the bilinear forms of the (2 + 1)-dimensional ZK–mZK–BBM equation were derived using variable transformation. Then, the multiple soliton solutions of the ZK–mZK–BBM equation are obtained by bilinear forms and symbolic computation. Under complex conjugate transformations, quasi-soliton solutions and mixed solutions composed of one-soliton and one-quasi-soliton are derived from soliton solutions. These solutions are further studied graphically to observe the propagation characteristics of gravity water waves. The results enrich the research of gravity water wave in fluid mechanics.
This document is an announcement and preview of a memoir whose full version is available on the Open Math Notes repository of the American Mathematical Society (OMN:202109.111309). In this memoir, I try to provide a fairly comprehensive picture of (mostly shallow water) asymptotic models for water waves. The work and presentation is heavily inspired by the book of D. Lannes, yet extends the discussion into several directions, notably high order and fully dispersive models, and internal/interfacial waves.
One hundred years ago, Nekrasov published the widely cited paper (Nekrasov in Izvestia Ivanovo-Voznesensk Politekhn Inst 3:52–65, 1921), in which he derived the first of his two integral equations describing steady periodic waves on the free surface of water. We examine how Nekrasov arrived at these equations and his approach to investigating their solutions. In connection with this, Nekrasov’s life after 1917 is briefly outlined, in particular, how he became a prisoner in Stalin’s Gulag. Further results concerning Nekrasov’s equations and related topics are surveyed.
