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The relations between

The studies of exact solutions of nonlinear partial differential equations (NPDEs) have received considerable attention in connection with the important problems that arise in scientific applications. Many powerful methods have been proposed to obtain exact solutions of (NPDEs); a series of methods have been proposed, such as Painléve test [

In order to seek the periodic solutions of nonlinear evolution equations, Porubov and Parker proposed Weierstrass elliptic function expansion method [

In fact, solving nonlinear equations (especially nonlinear partial differential equations) is very difficult, and there is no unified method. The present methods can only be applied to a certain equation or some equations. So the work of continuing to find some effective method of solving nonlinear equations is important and meaningful. Recently, Ma put forward generalized bilinear differential operators named

The paper is structured as follows. In Section

It is known to us that Hirota bilinear

Based on the Hirota

Obviously, the case of

In particular, when

Now, under

In fact, if we seek the bilinear form with

As we all know, Bell proposed three kinds of exponent form polynomials. Later, Wang and Chen generalized the third type of Bell polynomials in [

For example,

Then we have

From (

In this section, we will construct the bilinear forms for Kdv equation, (2+1)-dimensional Kdv equation, and (2+1)-dimensional Sawada-Kotera equation with the

Consider

Setting

Based on (

Consider

Setting

Consider

From the above computation process for seeking the bilinear forms of three nonlinear equation, we can find that the bilinear forms with

In this section, firstly, we will give the bilinear form of a (3+1)-dimensional generalized shallow water equation with the help of

The following is (3+1)-dimensional generalized shallow water equation:

In what follows we construct the one-periodic wave solutions of (

Riemann theta function (

A one-periodic wave (

To this end, the soliton solution of (

And we can write

In this paper, we investigate a (3+1)-dimensional generalized shallow water wave equation (

There are many other interesting questions on bilinear differential equations; for example, can the approach be generalized to solve trilinear equations with trilinear differential operators? How to apply the

The authors declare that there is no conflict of interests regarding the publication of this paper.