Based on the Hirota bilinear method and Wronskian technique, two different classes of sufficient conditions consisting of linear partial differential equations system are presented, which guarantee that the Wronskian determinant is a solution to the corresponding Hirota bilinear equation of a (

Wronskian formulations are a common feature for soliton equations, and it is a powerful tool to construct exact solutions to the soliton equations [

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

Under a scale transformation

As we know, in the process of employing Wronskian technique, the main difficulty lies in looking for the linear differential conditions, which the functions in the Wronskian determinant should satisfy. Moreover, the differential conditions for the Wronskian determinant solutions of many soliton equations are not unique [

The

In this section, we present the first class of linear differential conditions for the Wronskian determinant solutions of (

Let a set of functions

The proof of Theorem

Set

Under the condition (

Under the properties of the Wronskian determinant and the conditions (

The condition (

For example, when

Using the linear differential conditions (

As an example, in the special case of

If we let the coefficient matrix

For example, when

In this section, we show another linear differential condition to the Wronskian determinant solutions of (

Let a group of functions

Under the properties of the Wronskian determinant and the condition (

The condition (

In particular, we can have the following Wronskian solutions of (

In summary, we have established two different kinds of linear differential conditions for the Wronskian determinant solutions of the (3+1)-dimensional generalized shallow water equation (

This work was supported in part by the National Science Foundation of China (under Grant nos. 11172233, 11102156, and 11002110) and Northwestern Polytechnical University Foundation for Fundamental Research (no. GBKY1034).