Residual symmetry, interaction solutions, and conservation laws of the (2+1)-dimensional dispersive long-wave system

2017 ◽  
Vol 26 (3) ◽  
pp. 030202 ◽  
Author(s):  
Ya-rong Xia ◽  
Xiang-peng Xin ◽  
Shun-Li Zhang
2018 ◽  
Vol 75 (8) ◽  
pp. 2625-2628 ◽  
Author(s):  
Hui Wang ◽  
Yun-Hu Wang ◽  
Huan-He Dong

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 878
Author(s):  
Alexei Cheviakov ◽  
Denys Dutykh ◽  
Aidar Assylbekuly

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.


Author(s):  
Peter J. Olver

AbstractThe BBM or Regularized Long Wave Equation is shown to possess only three non-trivial independent conservation laws. In order to prove this result, a new theory of Euler-type operators in the formal calculus of variations will be developed in detail.


2020 ◽  
Vol 34 (12) ◽  
pp. 2050117 ◽  
Author(s):  
Xianglong Tang ◽  
Yong Chen

Utilizing the Hirota bilinear method, the lump solutions, the interaction solutions with the lump and the stripe solitons, the breathers and the rogue waves for a (3[Formula: see text]+[Formula: see text]1)-dimensional Kudryashov–Sinelshchikov equation are constructed. Two types of interaction solutions between the lumps and the stripe solitons are exhibited. Some different breathers are given by choosing special parameters in the expressions of the solitons. Through a long wave limit of breathers, the lumps and rogue waves are derived.


2009 ◽  
Vol 64 (9-10) ◽  
pp. 597-603 ◽  
Author(s):  
Zhong Zhou Dong ◽  
Yong Chen

By means of the generalized direct method, we investigate the (2+1)-dimensional dispersive long wave equations. A relationship is constructed between the new solutions and the old ones and we obtain the full symmetry group of the (2+1)-dimensional dispersive long wave equations, which includes the Lie point symmetry group S and the discrete groups D. Some new forms of solutions are obtained by selecting the form of the arbitrary functions, based on their relationship. We also find an infinite number of conservation laws of the (2+1)-dimensional dispersive long wave equations.


Author(s):  
Gerassimos Papadopoulos

According to Imamura (1937: 123), the term tunami or tsunami is a combination of the Japanese word tu (meaning a port) and nami (a long wave), hence long wave in a harbour. He goes on to say that the meaning might also be defined as a seismic sea-wave since most tsunamis are produced by a sudden dip-slip motion along faults during major earthquakes. Other submarine or coastal phenomena, however, such as volcanic eruptions, landslides, and gas escapes, are also known to cause tsunamis. According to Van Dorn (1968), ‘tsunami’ is the Japanese name for the gravity wave system formed in the sea following any large-scale, short-duration disturbance of the free surface. Tsunamis fall under the general classification of long waves. The length of the waves is of the order of several tens or hundreds of kilometres and tsunamis usually consist of a series of waves that approach the coast with periods ranging from 5 to 90 minutes (Murty 1977). Some commonly used terms that describe tsunami wave propagation and inundation are illustrated in Figure 17.2. Because of the active lithospheric plate convergence, the Mediterranean area is geodynamically characterized by significant volcanism and high seismicity as discussed in Chapters 15 and 16 respectively. Furthermore, coastal and submarine landslides are quite frequent and this is partly in response to the steep terrain of much of the basin (Papadopoulos et al. 2007a). Tsunamis are among the most remarkable phenomena associated with earthquakes, volcanic eruptions, and landslides in the Mediterranean basin. Until recently, however, it was widely believed that tsunamis either did not occur in the Mediterranean Sea, or they were so rare that they did not pose a threat to coastal communities. Catastrophic tsunamis are more frequent on Pacific Ocean coasts where both local and transoceanic tsunamis have been documented (Soloviev 1970). In contrast, large tsunami recurrence in the Mediterranean is of the order of several decades and the memory of tsunamis is short-lived. Most people are only aware of the extreme Late Bronge Age tsunami that has been linked to the powerful eruption of Thera volcano in the south Aegean Sea (Marinatos 1939; Chapter 15).


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