scholarly journals Rational solution to a shallow water wave-like equation

2016 ◽  
Vol 20 (3) ◽  
pp. 875-880
Author(s):  
Hong-Cai Ma ◽  
Ke Ni ◽  
Guo-Ding Ruan ◽  
Ai-Ping Deng
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Shallow Water ◽  
Prime Number ◽  
Water Wave ◽  
Rational Solution ◽  
Rational Solutions ◽  
Shallow Water Wave ◽  
Linear Differential ◽  
New Equation

Two classes of rational solutions to a shallow water wave-like non-linear differential equation are constructed. The basic object is a generalized bilinear differential equation based on a prime number, p = 3. Through this new transformation and with the help of symbolic computation with MAPLE, both the new equation and its rational solutions are obtained.

scholarly journals Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Elliptic Integral ◽  
Momentum Conservation ◽  
Two Dimensional ◽  
Order Ordinary Differential Equation ◽  
Closed Form Solutions ◽  
Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

The lump, lumpoff and rouge wave solutions of a (3+1)-dimensional generalized shallow water wave equation

2019 ◽  
Vol 33 (17) ◽  
pp. 1950190
Author(s):  
Jin-Jie Yang ◽  
Shou-Fu Tian ◽  
Wei-Qi Peng ◽  
Zhi-Qiang Li ◽  
Tian-Tian Zhang
Keyword(s):  
Shallow Water ◽  
Bilinear Form ◽  
Water Wave ◽  
Rational Solution ◽  
Bell Polynomials ◽  
Ocean Engineering ◽  
Shallow Water Wave ◽  
Propagation Behavior ◽  
Wave Solutions ◽  
Lump Wave

We consider the (3[Formula: see text]+[Formula: see text]1)-dimensional generalized shallow water wave (GSWW) equation. By virtue of the binary Bell polynomials theory, we obtain the bilinear form of the equation. Then its lump wave solutions, a kind of rational solution localized in all directions of the space, are derived by employing its bilinear form at the special situation for [Formula: see text]. Furthermore, it is worth noting that the lump wave solutions can interact with single-stripe soliton waves and double-stripe solution waves to generate lumpoff waves and a kind of predictable rouge waves, respectively. Especially, it is interesting that we can predict when and where the peculiar rouge waves will occur. Moreover, in order to understand the dynamics and propagation of the lump waves and the interaction solution, some graphic analyses are exhibited by selecting special parameters. The results of this work can be used to understand the propagation behavior of these solutions of the GSWW equation, which is of great significance for ocean engineering.

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