^{1}

^{1}

A Shallow Water Wave-like nonlinear differential equation is considered by using the generalized bilinear equation with the generalized bilinear derivatives
D
_{3,x} and
D
_{3,t}, which possesses the same bilinear form as the standard shallow water wave bilinear equation. By symbolic computation, four presented classes of rational solutions contain all rational solutions to the resulting Shallow Water Wave-like equation, which generated from a search for polynomial solutions to the corresponding generalized bilinear equation.

In recent years, numerous scientists committed to the research of water waves, the shallow water wave can not only describe the freedom of the shallow surface under the gravitational influence of one-way transmission, but also produce on the bottom of the deep sea. The most important of the shallow water wave in the ocean is tsunami, which generated by the huge initial disturbance, such as earthquake, leading to the large wavelength and small height ocean wave. Therefore, the wave solutions, especially the rational solutions to nonlinear differential equations have attracted more and more attentions in the worldwide. What’s more, rogue wave solutions play an important role in rational solutions, which describe significant nonlinear wave phenomena in oceanography [

Rational solutions to the non-integrable (3 + 1)-dimensional KP I [

In this article, we introduce a Shallow Water Wave (SWW)-like nonlinear differential equation in terms of a generalized bilinear differential equation of Shallow Water Wave type using three generalized bilinear differential operators D 3 , x and D 3 , t . We will search for polynomial solutions to the corresponding generalized bilinear equation by Maple symbolic computation and generate four classes of rational solutions to the resulting Shallow Water Wave-like equation. Four particular rational solutions will be plotted to exhibit different distributions of singularities.

Let us consider a generalized bilinear differential equation of SWW type:

( D 3 , x D 3 , t − D 3 , t D 3 , x 3 + D 3 , x 2 ) f ⋅ f = 2 f x t f − 2 f x f t − 6 f x x f x t + 6 f x x t f x + 2 f x x f − 2 f x 2 = 0. (2.1)

This is the same type bilinear equation as the SWW equation [

( D x D t − D t D x 3 + D x 2 ) f ⋅ f = 0 , (2.2)

with the corresponding nonlinear differential equation

u t − u x x t − 3 u u t + 3 u x ∫ x ∞ u t d x + u x = 0. (2.3)

The above differential operators are some kind of generalized bilinear differential operators introduced in [

D p , x m D p , t n f ⋅ f = ( ∂ ∂ x + α p ∂ ∂ x ′ ) m ( ∂ ∂ t + α p ∂ ∂ t ′ ) n f ( x , t ) f ( x ′ , t ′ ) | x ′ = x , t ′ = t = ∑ i = 0 m ∑ j = 0 n ( m i ) ( n j ) α p i α p j ∂ m − i ∂ x m − i ∂ i ∂ x ′ ( i ) ∂ n − j ∂ t n − j ∂ j ∂ t ′ ( j ) f ( x , t ) f ( x ′ , t ′ ) | x ′ = x , t ′ = t = ∑ i = 0 m ∑ j = 0 n ( m i ) ( n j ) α p i α p j ∂ m + n − i − j f ( x , t ) ∂ x m − i ∂ t n − j ∂ i + j f ( x , t ) ∂ x i ∂ t j , m , n ≥ 0. (2.4)

where α p s is computed as follows:

α p s = ( − 1 ) r p ( s ) , s = r p ( s ) mod p , (2.5)

it is necessary to point out that

α p i α p j ≠ α p i + j , i , j ≥ 0.

Setting p = 3 , we have

α 3 = − 1 , α 3 2 = 1 , α 3 3 = 1 , α 3 4 = − 1 , α 3 5 = 1 , α 3 6 = 1.

and thus

D 3 , x D 3 , t f ⋅ f = 2 f x t f − 2 f x f t , D 3 , t D 3 , x 3 f ⋅ f = − 6 f x x f x t + 6 f x x t f x , D 3 , x 2 f ⋅ f = 2 f x x f − 2 f x 2 .

In the case of p = 2 , which is the Hirota case, the following equations are true:

D 2 , x D 2 , t f ⋅ f = 2 f x t f − 2 f x f t , D 2 , x 2 f ⋅ f = 2 f x x f − 2 f x 2 , D 2 , t D 2 , x 3 f ⋅ f = 2 f x x x t f + 6 f x x f x t − 2 f x x x f t ,

which generates the standard bilinear SWW equation [

Motivated by the introduction on a general Bell polynomial theory [

u = 2 ( ln f ) x , (2.6)

and then can directly show that the generalized bilinear Equation (2.1) is linked to a SWW-like scalar nonlinear differential equation

u t + u x + 3 2 u u x t − 3 2 u x u t + 1 2 u 2 u t = 0 , (2.7)

from the generalized bilinear Equation (2.1). Through the transformation (2.6), the following equality can be deduced:

( D 3 , x D 3 , t − D 3 , t D 3 , x 3 ) f ⋅ f f 2 = u t + u x + 3 2 u u x t − 3 2 u x u + t 1 2 u 2 u t , (2.8)

and thus, f solves (2.1) if and only if u = 2 ( ln f ) x presents a solution to the SWW-like Equation (2.7).

In [

By symbolic computation with Maple, we look for polynomial solutions, with degree of x and t being less than 3:

f = ∑ i = 0 3 ∑ j = 0 3 c i , j x i x j , (3.1)

where the c i , j ’s are constants, and present 4 classes of polynomial solutions to the generalized bilinear Equation (2.1), based on the powers of x , the solutions could be divided into three categories: cubic polynomials, linear polynomials and the trivial solutions. Besides the trivial solutions, those solutions, in turn, lead to four classes of rational solutions to the SWW-like (2.5) through the transformation (2.4). We list those classes of rational solutions as follows. The first class of rational solutions to (2.5) is

u 1 = 2 p q , (3.2)

where

p = c 10 c 12 + c 11 c 12 t + c 12 2 t 2 , q = c 10 c 12 x + c 11 c 12 x t + c 12 2 x t 2 + c 10 ( c 11 + c 02 ) + ( c 11 c 02 − c 10 c 12 + c 11 2 ) t + c 02 c 12 t 2 − c 12 2 t 3 .

The second class of rational solutions to (2.5) is

u 2 = 2 ( c 11 c 10 + c 11 2 t ) c 11 c 10 x + c 11 2 x t + c 10 ( c 01 + c 10 ) + c 01 c 11 t − c 11 2 t 2 , (3.3)

The third class of rational solutions to (2.5) is

u 3 = 6 p q , (3.4)

where

p = c 20 2 x 2 + 2 c 10 c 20 x − 2 c 20 2 x t + c 10 2 − 2 c 20 c 10 t + c 20 2 t 2 , p = c 20 2 x 3 + 3 c 10 c 20 x 2 − 3 c 20 2 x 2 t + 3 c 10 2 x − 6 c 20 c 10 x t + 3 c 20 2 x t 2 − 3 ( c 10 2 − 6 c 20 2 ) t + 3 c 20 c 10 t 2 − c 20 2 t 3 + 3 c 00 c 10 .

The fourth class of rational solutions to (2.5) is

u 4 = 36 p q , (3.5)

where

p = 3 c 01 x 2 − 6 c 01 x t + 6 c 10 x t 2 , q = c 01 x 3 − 3 c 01 x 2 t + 3 c 01 x 2 t 2 + 18 c 01 t − c 01 t 3 + 18 c 00 .

We took a SWW-like nonlinear differential equation into consideration by the generalized bilinear equation of SWW type. Furthermore, we constructed two classes of rational solutions to the resulting SWW-like equation. A kind of generalized bilinear differential operators, which introduced in [

It is parallel to the discussion, a kind of generalized tri-linear differential equ-

ations and their resonant solutions was considered in [

This work is supported by Guangxi College Enhancing Youths Capacity Project (KY2016LX315) and Guangxi University of Finance and Economics Youth Progresss Project (2016QNB22).

Wei, M. and Cai, J.N. (2017) The Exact Rational Solutions to a Shallow Water Wave-Like Equation by Generalized Bilinear Method. Journal of Applied Mathematics and Physics, 5, 715-721. https://doi.org/10.4236/jamp.2017.53060