The Interactions of N-Soliton Solutions for the Generalized 2 + 1-Dimensional Variable-Coefficient Fifth-Order KdV Equation

A generalized (2 + 1)-dimensional variable-coefficient KdV equation is introduced, which can describe the interaction between a water wave and gravity-capillary waves better than the (1 + 1)-dimensional KdV equation. The N-soliton solutions of the (2 + 1)dimensional variable-coefficient fifth-order KdV equation are obtained via the Bell-polynomial method. Then the soliton fusion, fission, and the pursuing collision are analyzed depending on the influence of the coefficient eij ; when eij = 0, the soliton fusion and fission will happen; when eij ̸ = 0, the pursuing collision will occur. Moreover, the Bäcklund transformation of the equation is gotten according to the binary Bell-polynomial and the period wave solutions are given by applying the Riemann theta function method.

Soliton interaction can be split into elastic and inelastic.As for the elastic, the amplitudes, velocities, and shapes of the soliton can be brought into correspondence with the initial soliton, but for the inelastic collision, after the interaction, one soliton can be divided into two or more solitons, a phenomenon called soliton fission, or contrarily, two or more solitons can be merged into one soliton which is called soliton fusion.What is more, the variable coefficient of the equation can lead to the soliton fission and fusion.In [14], the variablecoefficient KdV equation can be applied to describe the largeamplitudes internal waves of the atmosphere and the ocean.In recent years, the general KdV equation had been expanded to the fifth-order KdV equation and the generalized (2 + 1)-dimensional Korteweg-de Vries equation whose bilinear Bäcklund transformation and Darboux covariant Lax pair have been obtained.
(3) When () = 3, () = , () = 2, () = 1, () = 2/15, and () = () = ℎ() = () = 0, (1) takes the form of which occurs to a weakly nonlinear long-wave approximation to the general gravity-capillary water-wave problem and  is a real scaled parameter.The focus of the paper is to get the -soliton solutions of the generalized variable-coefficient fifth-order (2 + 1)dimensional equation and analyze the interaction of the water wave and the gravity-capillary wave [26][27][28][29][30].The details of the paper are as follows: Section 2 introduces a variablecoefficient fifth-order (2 + 1)-dimensional KdV equation.The approach and the properties of the Bell-polynomial are presented in Section 3 and then give rise to the -soliton solutions of the equation based on the Hirota approach.In the final part, we explain the soliton fission and fusion and the soliton pursuing collision of the variable-coefficient fifth-order (2 + 1)-dimensional KdV equation according to the different coefficients    .Furthermore, the Bäcklund transformation is given.

The Introduction of the Bell-Polynomial
To start with, we briefly introduce the basic concepts and the properties of the Bell-polynomial.
Moreover, we can calculate the initial explicit expressions by the definition as follows: (2) Take  as a  ∞ multivariables function; then the definition of the multivariables Bell-polynomial is as follows: As for a special function  with the variables , , we give rise to the following several initial values under the definition of the multivariables Bell-polynomial: In view of the multivariables Bell-polynomial, the multivariables binary Bell-polynomial can be defined as follows: where  and  both are the  ∞ function with the variables  1 ,  2 , . . .,   ; likewise, we can take some of the expressions depending on (9); for example, Next, we study the proposition of the Bell-polynomial.
Proposition 1. Bell-polynomial ( 9) can be written as the Hirota -operator through a transformational identity: Advances in Mathematical Physics 3 where the Hirota operator is defined by Especially when  = , (11) can be read as In ( 13), the Bell-polynomial is significant if and only if when Σ  =1    V, one redefines a -polynomial: when  1 +  2 + . . .+   is even.The initial few -polynomials are As for the NLEEs, ( 9) and ( 15) are important to get the soliton solutions; one can get the bilinear equation provided that the NLEEs can express the linear combination of the polynomials.

The 𝑁-Soliton Solutions of the Variable-Coefficient Fifth-Order KdV Equation
As for (1), if we take then (1) can be rewritten as By virtue of the transformation  =   , ( 17) can be changed into What is more, we can obtain the following formula by making use of the integral to (18) with respect to the variable : Expression ( 19) is changed as follows with the aid of polynomial (15): which cannot be written as the linear combination of polynomials, so we construct an auxiliary variable  which satisfies From ( 21), we can get a pair of -polynomials as follows: Once the bilinear representation of ( 19) is given, we can present the -soliton solutions of (23) with the help of Hirota's bilinear approach and the symbolic computation.
After that, we begin to solve (17) on account of the Hirota method; set Substitute ( 24) into (23) and compare the powers of ; then the -soliton solutions of ( 19) are gotten by making  = 1 as follows: with where   and   are both the constants and ∑   ,  =0,1 indicate summation over all the different possible cases   ,   = 0, 1 (,  = 1, 2, 3, . ..).
For  = 1, we can read the one-soliton solution as For  = 2, the two-soliton solutions can be written as For  = 3, we can obtain the three-soliton solutions as

The Bäcklund Transformation of the Variable-Coefficient Fifth-Order KdV Equation
For the NLEEs, the BT method provides a new idea to construct the solutions by the Bell-polynomial.In this section, we will obtain the BT of the known (2 + 1)-dimensional variable-coefficient fifth-order KdV equation.Suppose ,  are the two different solutions of (19), and consider the following form: In order to obtain the BT of ( 30 where Finally, we derive the BT by introducing a spectrum parameter equation as and have then the Bell-polynomial-typed BT of the (2+1)-dimensional variable-coefficient KdV equation is the following: where () is a function about the variable .
The BT of (36) can be read with the help of the expression of ( 11), (37)

Advances in Mathematical Physics
Therefore, we can obtain the period wave solution of (17) in Figure 1.

The Interaction of the Soliton Waves
In this part, we will discuss the interaction of the soliton waves.From Section 3, we get the -soliton solutions, which will show soliton fusion or fission when    = 0; on the other hand, when    ̸ = 0, there will occur the soliton pursuing collision; after the collision, the waves are still spread along the previous direction but cannot keep the previous amplitude.Then we can describe the interaction of the soliton waves by Figures 2-5.
If    = 0, two-soliton solutions (27) become the following resonance solution: We can analyze the soliton fusion and fission under the approximation form.Next, we discuss the influence of the variable coefficient; due to the different coefficient, the different waves shapes will occur; the specific progress is as shown in Figure 3.
In the end, we provide the interaction of the soliton solutions based on the coefficient    ̸ = 0.As time goes on, there will happen the soliton pursuing collision because of the different soliton speed; the wave with the faster speed will catch up with the slower speed wave; then collision of the twosoliton solution can happen; after the collision, some of the solitons can spread along the previous direction, but the other soliton will be far from the previous direction and spread with a new direction; the faster speed wave will be in front of the slower waves; Figures 4 and 5 give a visual description of the collision.
As for two-soliton solutions (27), we get the soliton pursuing collision in Figure 4.
As for three-soliton solutions (28), we describe the threesoliton pursuing collision in Figure 5.

Conclusion
In this paper, we first introduce a generalized (2 + 1)dimensional variable-coefficient KdV equation, which can describe the interaction between a water wave and gravitycapillary waves better than the (1 + 1)-dimensional KdV equation.Secondly, we get the -soliton solutions of the (2+1)-dimensional variable-coefficient KdV equation via the Bell-polynomial approach and explain the interactions of the -soliton solutions.The main conclusions of this paper can be summarized as follows: (1) The Bell-polynomial of the (2 + 1)-dimensional variable-coefficient fifth-order KdV equation has ), we introduce the mixing variables as  = ln ( ⋅ ) ,  = ln (   ) ,  =  +  = 2 ln ,  =  −  = 2 ln ;

Figure 2 (
Figure2(a) indicates that there appear two waves on the time 45; in Figure2(b), it causes the collision between the two waves; during the collision, it creates new waves because of the acting force; in Figure2(c), the two waves merge into one wave after the collision and spread placidly.Next, we discuss the influence of the variable coefficient; due to the different coefficient, the different waves shapes will occur; the specific progress is as shown in Figure3.In the end, we provide the interaction of the soliton solutions based on the coefficient    ̸ = 0.As time goes on, there will happen the soliton pursuing collision because of the different soliton speed; the wave with the faster speed will catch up with the slower speed wave; then collision of the twosoliton solution can happen; after the collision, some of the solitons can spread along the previous direction, but the other soliton will be far from the previous direction and spread with a new direction; the faster speed wave will be in front of the slower waves; Figures4 and 5give a visual description of the collision.