A fractal variational theory of the Broer-Kaup system in shallow water waves

The Broer-Kaup equation is one of many equations describing some phenomena of shallow water wave. There are many errors in scientific research because of the existence of the non-smooth boundaries. In this paper, we generalize the Broer-Kaup equation to the fractal space and establish fractal variational formulations through the semi-inverse method. The acquired fractal variational formulation reveals conservation laws in an energy form in the fractal space and suggests possible solution structures of the morphology of the solitary waves.

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Variational principle of the Whitham-Broer-Kaup equation in shallow water wave with fractal derivatives

Thermal Science ◽

10.2298/tsci200301019l ◽

2021 ◽

pp. 19-19

Author(s):

Wei-Wei Ling ◽

Pin-Xia Wu

Keyword(s):

Variational Principle ◽

Shallow Water ◽

Solitary Waves ◽

Water Waves ◽

Water Wave ◽

Variational Formulation ◽

Inverse Method ◽

Solution Structure ◽

Shallow Water Waves ◽

Fractal Space

The Whitham-Broer-Kaup equation exists widely in shallow water waves, but unsmooth boundary seriously affects the properties of solitary waves and has certain deviations in scientific research. The aim of this paper is to introduce its modification with fractal derivatives in a fractal space and to establish a fractal variational formulation by the semi-inverse method. The obtained fractal variational principle shows conservation laws in an energy form in the fractal space and also hints its possible solution structure.

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New multi-soliton solutions of Whitham-Broer-Kaup shallow-water-wave equations

Thermal Science ◽

10.2298/tsci17s1137z ◽

2017 ◽

Vol 21 (suppl. 1) ◽

pp. 137-144 ◽

Cited By ~ 2

Author(s):

Sheng Zhang ◽

Mingying Liu ◽

Bo Xu

Keyword(s):

Shallow Water ◽

Water Waves ◽

Water Wave ◽

Wave Equations ◽

Soliton Solutions ◽

Bilinear Forms ◽

Shallow Water Waves ◽

Bilinear Method ◽

Shallow Water Wave ◽

Shallow Water Wave Equations

In this paper, new and more general Whitham-Broer-Kaup equations which can describe the propagation of shallow-water waves are exactly solved in the framework of Hirota?s bilinear method and new multi-soliton solutions are obtained. To be specific, the Whitham-Broer-Kaup equations are first reduced into Ablowitz- Kaup-Newell-Segur equations. With the help of this equations, bilinear forms of the Whitham-Broer-Kaup equations are then derived. Based on the derived bilinear forms, new one-soliton solutions, two-soliton solutions, three-soliton solutions, and the uniform formulae of n-soliton solutions are finally obtained. It is shown that adopting the bilinear forms without loss of generality play a key role in obtaining these new multi-soliton solutions.

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X-type Soliton, Resonance Y-type Soliton and Hybrid Solutions of a (2+1)-dimensional Hirota-Satsuma-Ito System for the Shallow Water Waves

10.21203/rs.3.rs-751414/v1 ◽

2021 ◽

Author(s):

Yuan Shen ◽

Bo Tian ◽

Tian-Yu Zhou ◽

Xiao-Tian Gao

Keyword(s):

Shallow Water ◽

Symbolic Computation ◽

Water Waves ◽

Water Wave ◽

Soliton Solutions ◽

Hirota Method ◽

Shallow Water Waves ◽

Soliton Resonance ◽

Hybrid Solutions

Abstract Water waves are observed in the rivers, lakes, oceans, etc. Under investigation in this paper is a (2+1)-dimensional Hirota-Satsuma-Ito system arising in the shallow water waves. Via the Hirota method and symbolic computation, we derive some X-type and resonance Y-type soliton solutions. We also work out some hybrid solutions consisting of the resonance Y-type solitons, solitons, breathers and lumps. Graphics we present reveal that the hybrid solutions consisting of the resonance Y-type solitons and solitons/breathers/lumps describe the interactions between the resonance Y-type solitons and solitons/breathers/lumps, respectively. The obtained results rely on the water-wave coefficient in that system.

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Water Waves and Light: Two Unlikely Partners

Nonlinear Optics - From Solitons to Similaritons ◽

10.5772/intechopen.95431 ◽

2021 ◽

Author(s):

Georgios N. Koutsokostas ◽

Theodoros P. Horikis ◽

Dimitrios J. Frantzeskakis ◽

Nalan Antar ◽

İlkay Bakırtaş

Keyword(s):

Shallow Water ◽

Water Waves ◽

Water Wave ◽

Soliton Solutions ◽

Type Equation ◽

Beam Propagation ◽

Generic Model ◽

Two Dimensional ◽

Shallow Water Waves ◽

Nonlocal Nonlinear

We study a generic model governing optical beam propagation in media featuring a nonlocal nonlinear response, namely a two-dimensional defocusing nonlocal nonlinear Schrödinger (NLS) model. Using a framework of multiscale expansions, the NLS model is reduced first to a bidirectional model, namely a Boussinesq or a Benney-Luke-type equation, and then to the unidirectional Kadomtsev-Petviashvili (KP) equation – both in Cartesian and cylindrical geometry. All the above models arise in the description of shallow water waves, and their solutions are used for the construction of relevant soliton solutions of the nonlocal NLS. Thus, the connection between water wave and nonlinear optics models suggests that patterns of water may indeed exist in light. We show that the NLS model supports intricate patterns that emerge from interactions between soliton stripes, as well as lump and ring solitons, similarly to the situation occurring in shallow water.

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SHOALING AND REFLECTION OF NONLINEAR SHALLOW WATER WAVES

Coastal Engineering Proceedings ◽

10.9753/icce.v20.60 ◽

1986 ◽

Vol 1 (20) ◽

pp. 60 ◽

Cited By ~ 3

Author(s):

Padmaraj Vengayil ◽

James T. Kirby

Keyword(s):

Solitary Wave ◽

Shallow Water ◽

Water Waves ◽

Water Wave ◽

Periodic Waves ◽

Shallow Water Waves ◽

Reflected Waves ◽

Wave Shoaling ◽

Backward Propagation ◽

Resonant Reflection

The formulation for shallow water wave shoaling and refraction diffraction given by Liu et al (1985) is extended to include reflected waves. The model is given in the form of coupled K-P equations for forward and backward propagation. Shoaling on a plane beach is studied using the forward-propagating model alone. Non-resonant reflection of a solitary wave from a slope and resonant reflection of periodic waves by sinusoidal bars are then studied.

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On the Multipeakon Dissipative Behavior of the Modified Coupled Camassa-Holm Model for Shallow Water System

Mathematical Problems in Engineering ◽

10.1155/2013/107450 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 4

Author(s):

Zhixi Shen ◽

Yujuan Wang ◽

Hamid Reza Karimi ◽

Yongduan Song

Keyword(s):

Differential Equations ◽

Energy Loss ◽

Shallow Water ◽

Water Waves ◽

Wave Breaking ◽

Water Wave ◽

Water System ◽

Shallow Water Waves ◽

Two Component ◽

Shallow Water System

This paper investigates the multipeakon dissipative behavior of the modified coupled two-component Camassa-Holm system arisen from shallow water waves moving. To tackle this problem, we convert the original partial differential equations into a set of new differential equations by using skillfully defined characteristic and variables. Such treatment allows for the construction of the multipeakon solutions for the system. The peakon-antipeakon collisions as well as the dissipative behavior (energy loss) after wave breaking are closely examined. The results obtained herein are deemed valuable for understanding the inherent dynamic behavior of shallow water wave breaking.

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SCOUR OF SAND BEACHES IN FRONT OF SEAWALLS

Coastal Engineering Proceedings ◽

10.9753/icce.v11.40 ◽

1968 ◽

Vol 1 (11) ◽

pp. 40 ◽

Cited By ~ 5

Author(s):

John B. Herbich ◽

Stephen C. Ko

Keyword(s):

Mathematical Model ◽

Shallow Water ◽

Water Waves ◽

Water Wave ◽

Wave Theory ◽

Beach Erosion ◽

Theoretical Development ◽

Scour Depth ◽

Shallow Water Waves ◽

Boundary Layer Equations

Many previous studies were confined to problem of beach erosion due to waves breaking on the structure. The investigation reported here involved regular non-breaking, shallow water waves progressing toward a seawall. An analytical solution was developed and compared with laboratory- scale experiments. The shallow-water wave theory and boundary layer equations were used in theoretical development, which resulted in a mathematical model for the ultimate scour depth in front of a seawall.

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