2. Processing Techniques of the Riccati Equation
Due to its very simple structure, the Riccati equation is an ideal auxiliary equation for solving NLEEs. Therefore, it is often used to solve the constant coefficient, variable coefficient, and high-dimensional nonlinear evolution equations. In this paper, we first use the Ricatti equation in the following form [
27,
28,
29,
30]:
where
p1 and
q1 are the coefficients of the equation. Equation (2) has well-known solutions
tanh(
ξ) and
coth(
ξ) when
p1 = −1 and
q1 = 1. Directly applying this equation or its deformation as an auxiliary equation to solve NLEEs is widely used, and relevant research can be found in Refs [
17,
27,
28,
29,
30]. The solution of Equation (2) has also been extended to a series of trigonometric and hyperbolic functions. In fact, as a nonlinear equation, the solution to Equation (2) is much better than that. In order to solve Equation (2) and explore its new exact solutions, we introduce another auxiliary equation that satisfies the following form
where
p2 and
q2 are the coefficients of the equation. Equation (3) has the following hyperbolic function solutions:
Among these solutions, Equations (6) and (7) have rarely been reported in other studies. Equation (3) cannot be directly used as an auxiliary equation to solve Equation (1), as the order of
g(
ξ) is the same as that of
g′(
ξ), which would result in Equation (1) having no solution. To apply Equation (3) to solve Equation (1), it is necessary to make certain transformations. In this article, we first apply the known solutions of Equation (3) to solve Equation (2), and then use Equation (2) as an auxiliary equation to solve Equation (1). We first assume that
and
satisfy the following form of relationship
where
r,
k, and
δ are pending constants. If
r = 0,
k = 0, and
δ = 0, Equation (8) can be transformed into the
/
expansion method described in Ref. [
23]. By substituting Equation (8) into Equation (2) and using Equation (3) and its derivation, we can obtain a series of algebraic equations about
,
k,
δ,
r,
p1, and
q1. Then, we collect all terms with the same power of
and set each coefficient to zero. We can obtain the following solutions
Therefore, we have the exact solutions for Equation (2) as follows:
In these solutions, we discarded the meaningless solutions of Equation (8) in the imaginary space. Next, we use another form of the relationship between
and
to solve Equation (2) as follows:
where
r,
k, and
δ are pending constants. If
k = 0, Equation (22) degenerates into a part of Equation (8). Therefore, we consider the case where
k 0. By applying the same method as described above and solving Equations (2), (3) and (22), we have
where
r is an arbitrary constant. However, to make them meaningful, there must be
> 0. Therefore, the following exact solutions of Equation (2) are obtained:
where
r is an arbitrary constant. If
r = 0, Equation (24) degenerates into Equation (16).
where
r < −1 or
r > 1.
where
r < −
or
r >
.
among them, if
= −1,
r is an arbitrary constant, and if
= 1,
r < −
or
r > .
According to the symmetry of the Riccati Equation (2),
is also its solution when
, and the following equations are also the solutions of Equation (2)
where
r is an arbitrary constant. If
r = 0, Equation (35) degenerates into Equation (29).
where
r < −1 or
r > 1.
where r < −
or r >
.
among them, if
= −1,
r is an arbitrary constant, and if
= 1,
r < −
or
r > .
In these solutions –– are the new exact solutions of Equation (2), rarely reported in the other literature. Due to the very simple structure of Equation (2), using it as an auxiliary equation to solve Equation (1) can greatly simplify the solution process.
3. Application to Reaction–Diffusion Equation
We assume that Equation (1) has the following traveling wave solution
where
μ and
c are wave parameters to be determined. Substituting Equation (39) into Equation (1) yields
Making
, the above equation is transformed into
Balancing
with
gives
so that
n = 1.
Therefore, we assume that Equation (45) has the formal solution as follows
Substituting Equation (43) into Equation (41) and setting the coefficients of to zero, then solving the resulting equations about , , , , and , we can obtain
- (1)
The first set:
- (2)
The second set:
- (3)
The third set:
According to the first set, we can obtain the following solutions for the reaction–diffusion system
where
. This is a set of traveling wave solutions for the reaction–diffusion equation. It is singular and blows up at
, indicating the explosive instability of non-trivial diffusion coefficient problems.
Figure 1a provides a three-dimensional diagram for Equation (47), which represents the traveling wave of Equation (41) with singularities when
.
Figure 1b shows the two-dimensional image of the traveling wave, representing the evolution of the traveling wave from having singularities to having no singularities.
where
. This is a set of traveling wave solutions for the reaction–diffusion equation. It is singular and blows up at
, indicating the explosive instability of non-trivial diffusion coefficient problems.
Figure 2a provides the three-dimensional diagram for Equation (48), representing the traveling wave of Equation (41) with singularities when
.
Figure 2b shows a two-dimensional image of the traveling wave, representing the evolution of the traveling wave from having no singularities to having singularities.
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 3a provides a three-dimensional diagram for Equation (49), representing the traveling wave solution of Equation (41) with singularities when
.
Figure 3b shows a two-dimensional image of the traveling wave solution, representing the evolution of the traveling wave from having singularities to having no singularities.
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 4a provides the three-dimensional diagram for Equation (50), representing the traveling wave of Equation (41).
Figure 4b shows a two-dimensional image of the traveling wave, which is the same as that in
Figure 3.
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 5a provides a three-dimensional diagram for Equation (51), representing the traveling wave of Equation (41).
Figure 5b shows a two-dimensional image of the traveling wave, which is the same as that in
Figure 1.
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 6a provides a three-dimensional diagram for Equation (52), representing the traveling wave of Equation (41).
Figure 6b shows a two-dimensional image of the traveling wave, which is the same as that in
Figure 3.
where
and
= 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 7a,b show three-dimensional and two-dimensional diagrams for Equation (53), representing the traveling wave of Equation (41), which is the same as that in
Figure 3.
where
and
= 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 8a,b show three-dimensional and two-dimensional diagrams for Equation (54), representing the traveling wave of Equation (41). It can be seen that the traveling wave shown in
Figure 8 is the same as that shown in
Figure 3.
where
and
= 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 9a,b show a three-dimensional diagram and two-dimensional image of Equation (55), representing the traveling wave of Equation (41), which is the same as that in
Figure 3.
where
r is an arbitrary constant,
and
= 1.This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 10a,b show three-dimensional and two-dimensional diagrams for Equation (56), representing the traveling wave of Equation (41). In this set of solutions, different values of
r can cause changes in the shape of the wave.
where
r < −1 or
r > 1,
, and
= 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 11a,b show three-dimensional and two-dimensional diagrams for Equation (57), representing the traveling wave of Equation (41). In this set of solutions, different values of
r can cause changes in the shape of the wave.
where
r < −
or
r > ,
, and
= 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 12a,b show three-dimensional and two-dimensional diagrams for Equation (58), representing the traveling wave of Equation (41). In this set of solutions, different values of
r can cause changes in the shape of the wave.
where
,
=1
1; if
= −1,
r is an arbitrary constant, and if
= 1,
r < −
or
r > .This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 13a,b show three-dimensional and two-dimensional diagrams for Equation (59), representing the traveling wave of Equation (41).
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 14a,b show three-dimensional and two-dimensional diagrams for Equation (60), representing the traveling wave of Equation (41), which is the same as that in
Figure 1.
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 15a,b show three-dimensional and two-dimensional diagrams for Equation (61), representing the traveling wave of Equation (41), which is the same as that in
Figure 1.
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 16a,b show three-dimensional and two-dimensional diagrams for Equation (62), representing the traveling wave of Equation (41), which is the same as that in
Figure 3.
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 17a,b show three-dimensional and two-dimensional diagrams for Equation (63), representing the traveling wave of Equation (41), which is the same as that in
Figure 1.
where
1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 18 a,b show three-dimensional and two-dimensional diagrams for Equation (64), representing the traveling wave of Equation (41), which is the same as that in
Figure 3.
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 19a,b show three-dimensional and two-dimensional diagrams for Equation (65), representing the traveling wave of Equation (41), which is the same as that in
Figure 1.
where
1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 20a,b show three-dimensional and two-dimensional diagrams for Equation (66), representing the traveling wave of Equation (41), which is the same as that in
Figure 1.
where
r is an arbitrary constant,
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 21a,b show three-dimensional and two-dimensional diagrams for Equation (67), representing the traveling wave of Equation (41).
where
r < −1 or
r > 1,
1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 22a,b show three-dimensional and two-dimensional diagrams for Equation (68), representing the anti-kink wave of Equation (41).
where
r < −
or
r > . This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 23a,b show three-dimensional and two-dimensional diagrams for Equation (69), representing the traveling wave of Equation (41).
where
r < −
or
r > 1; if
= −1,
r is an arbitrary constant, and if
= 1,
r < −
or
r > . This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 24a,b show three-dimensional and two-dimensional diagrams for Equation (70), representing the traveling wave of Equation (41).
The solutions of the reaction–diffusion equation in the second set are the same as those in the first set, since
is already included in the solutions. According to the third set, we can obtain the following solutions of the reaction–diffusion equation
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 25a,b show three-dimensional and two-dimensional diagrams for Equation (71), representing the traveling wave of Equation (41).
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 26a,b show three-dimensional and two-dimensional diagrams for Equation (72), representing the traveling wave of Equation (41).
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 27a,b show three-dimensional and two-dimensional diagrams for Equation (72), which represent the traveling wave of Equation (41).
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 28a,b show three-dimensional and two-dimensional diagrams for Equation (74), representing the traveling wave of Equation (41).
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 29a,b show three-dimensional and two-dimensional diagrams for Equation (75), representing the kink solitary wave of Equation (41).
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 30a,b show three-dimensional and two-dimensional diagrams for Equation (76), representing the traveling wave of Equation (41).
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 31a,b represent three-dimensional and two-dimensional diagrams for Equation (77), representing the traveling wave of Equation (41).
where
. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 32a,b show three-dimensional and two-dimensional diagrams for Equation (78), representing the traveling wave of Equation (41).
where
, and
r is an arbitrary constant. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 33a,b represents three-dimensional and two-dimensional diagrams for Equation (79), which represents the traveling wave of Equation (41).
where
, and
r < −1 or
r > 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 34a,b presents three-dimensional and two-dimensional diagrams for Equation (80), which represents the traveling wave of Equation (41).
where
, and
r < −
or
r > . This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 35a,b represent three-dimensional and two-dimensional diagrams for Equation (81), representing the traveling wave of Equation (41).
where
, and
; if
= −1,
r is an arbitrary constant and if
= 1,
r < −
or
r > . This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at
.
Figure 36a,b show three-dimensional and two-dimensional diagrams for Equation (82), representing the traveling wave of Equation (41) with singularities in certain spatio-temporal positions.