The (2+1)-dimensional modified dispersive water wave equation^[1–8]

$$ \begin{eqnarray}\left\{\begin{array}{c}{u}_{yt}+{u}_{xxy}-2{w}_{xx}-2{(u{u}_{y})}_{x}=0,\\ {w}_{t}-{w}_{xx}-2{(uw)}_{x}=0\end{array}\right.\end{eqnarray}$$ (1)

In this work, we employ the bifurcation theory of dynamical systems^[9–18] to study the following (2+1)-dimensional generalized modified dispersive water wave (GMDWW) equation:

$$ \begin{eqnarray}\left\{\begin{array}{c}{u}_{yt}+{u}_{xxy}-a{w}_{xx}-a{(u{u}_{y})}_{x}=0,\\ {w}_{t}-{w}_{xx}-a{(uw)}_{x}=0,\end{array}\right.\end{eqnarray}$$ (2)

The remainder of this work is organized as follows. In Section 2, we study the phase portraits and bifurcation analysis for Eq. (2). In Section 3, we obtain abundant exact traveling wave solutions of Eq. (2). The profiles of some exact traveling wave solutions are given in Section 4. A brief conclusion is given in Section 5.

By using the following transformation:

$$ \begin{eqnarray}u=\phi (\zeta),\,w=\psi (\zeta),\,\zeta =x+y-ct,\end{eqnarray}$$ (3)

$$ \begin{eqnarray}\left\{\begin{array}{c}-c{\phi }^{\prime \prime }+{\phi }^{\prime \prime \prime}-a{\psi }^{\prime \prime}-a{(\phi {\phi }^{\prime})}^{\prime}=0,\\ -c{\psi }^{\prime}-{\psi }^{\prime \prime}-a{(\phi \psi)}^{\prime}=0.\end{array}\right.\end{eqnarray}$$ (4)

$$ \begin{eqnarray}-c\phi +{\phi }^{^{\prime} }-a\psi -\displaystyle \frac{a}{2}{\phi }^{2}=0.\end{eqnarray}$$ (5)

$$ \begin{eqnarray}-c\psi -{\psi }^{^{\prime} }-a\phi \psi ={g}_{1},\end{eqnarray}$$ (6)

Using Eqs. (5) and (6), we obtain

$$ \begin{eqnarray}{\phi }^{\prime \prime}=\displaystyle \frac{{a}^{2}}{2}{\phi }^{3}+\displaystyle \frac{3ac}{2}{\phi }^{2}+{c}^{2}\phi +g.\end{eqnarray}$$ (7)

From Eq. (7), we establish a planar system

$$ \begin{eqnarray}\left\{\begin{array}{l}\displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}\zeta }=\varphi,\\ \displaystyle \frac{{\rm{d}}\varphi }{{\rm{d}}\zeta }=\displaystyle \frac{{a}^{2}}{2}{\phi }^{3}+\displaystyle \frac{3ac}{2}{\phi }^{2}+{c}^{2}\phi +g.\end{array}\right.\end{eqnarray}$$ (8)

$$ \begin{eqnarray}H(\phi,\varphi)=\displaystyle \frac{1}{2}{\varphi }^{2}-\displaystyle \frac{{a}^{2}}{8}{\phi }^{4}-\displaystyle \frac{ac}{2}{\phi }^{3}-\displaystyle \frac{{c}^{2}}{2}{\phi }^{2}-g\phi =h,\end{eqnarray}$$ (9)

Suppose that

$$ \begin{eqnarray}f(\phi)=\displaystyle \frac{{a}^{2}}{2}{\phi }^{3}+\displaystyle \frac{3ac}{2}{\phi }^{2}+{c}^{2}\phi,\end{eqnarray}$$ (10)

$$ \begin{eqnarray}{\phi }_{0}=0,\,{\phi }_{1}=-\displaystyle \frac{c}{a},\,{\phi }_{2}=-\displaystyle \frac{2c}{a},\end{eqnarray}$$ (11)

$$ \begin{eqnarray}{\phi }_{\pm }^{\ast }=\displaystyle \frac{-3c\pm \sqrt{3}c}{3a},\end{eqnarray}$$ (12)

$$ \begin{eqnarray}{g}_{0}=f({\phi }_{-}^{\ast })=\displaystyle \frac{{c}^{3}}{3\sqrt{3}a},\,-{g}_{0}=f({\phi }_{+}^{\ast })=-\displaystyle \frac{{c}^{3}}{3\sqrt{3}a}.\end{eqnarray}$$ (13)

By using the qualitative theory of dynamical systems,^[19,20] we draw the phase portraits of system (8) in Fig. 1.

If let

$$ \begin{eqnarray}{h}_{i}=H({\phi }_{i},0),\,i=0,1,9,10,20,21,\end{eqnarray}$$ (14)

and

 1

are the roots of f(ϕ) + g = 0, 0 < g < g₀, ϕ20=−3c+3c3a and ϕ21=−3c−3c3a are the roots of f(ϕ) + g₀ = 0, then we can derive the relations between the orbits of system (8), traveling wave solutions of Eq. (2), and the Hamiltonian h_i as the the following propositions 1–4.

When g = 0: (i)Suppose that h = h₀, system (8) has two heteroclonic orbits Γ₁ and Γ¯1 corresponding to two kink wave solutions of Eq. (2) and four special orbits Γ2,Γ¯2, Γ₃, and Γ¯3 corresponding to two singular wave solutions of Eq. (2).(ii)Suppose that h₁ < h < h₀, system (8) has three periodic orbits Γ∼4,Γ4, and Γ¯4 corresponding to three periodic wave solutions of Eq. (2).(iii)Suppose that h ≤ h₁, system (8) has two periodic orbits Γ₅ and Γ¯5 corresponding to four periodic wave solutions of Eq. (2).(iv)Suppose that h > 0, system (8) has two special orbits Γ₊ and Γ_–.

When 0 < g < g₀: (i)Suppose that h = h₉, system (8) has a homoclinic orbit Γ₆ corresponding to a solitary wave solution of Eq. (2) and three special orbits Γ7,Γ8, and Γ¯8 corresponding to two singular wave solutions of Eq. (2).(ii)Suppose that h₁₀ < h < h₉, system (8) has three periodic orbits !, and Γ¯9 corresponding to three periodic wave solutions of Eq. (2).(iii)Suppose that h ≤ h₁₀, system (8) has two periodic orbits Γ₁₀ and Γ¯10 corresponding to two periodic wave solutions of Eq. (2).(iv)Suppose that h > h₉, system (8) does not hanve any closed orbit.

When g = g₀: (i)Suppose that h = h₂₀, system (8) has three special orbits Γ11,Γ12, and Γ¯12 corresponding to three singular wave solutions of Eq. (2).(ii)Suppose that h < h₂₀, system (8) has two special orbits Γ₁₃ and Γ¯13.(iii)Suppose that h₂₀ < h < h₂₁, system (8) has two special orbits Γ₁₄ and Γ¯14.(iv)Suppose that h ≥ h₂₁, system (8) does not have any closed orbit.

When g > g₀ and h is an arbitrary constant, system (8) does not have any closed orbit.

For the convenience of exposition, we will omit the expressions of w with w(x,y,t)=ψ(ζ)=1aϕ′(ζ)−12ϕ2(ζ)−caϕ(ζ) in this work.

For the given positive constant c and transformation ζ = x + y – ct, we have the following results.

(i) When g = 0, equation (2) has two kink wave solutions

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=\displaystyle \frac{2c\mathop{\phi }\limits^{\sim }}{(2c+a\mathop{\phi }\limits^{\sim })\exp (c\zeta)-a\mathop{\phi }\limits^{\sim }},\end{array}\end{eqnarray}$$ (15)

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=\displaystyle \frac{2c\mathop{\phi }\limits^{\sim }\exp (c\zeta)}{2c+a\mathop{\phi }\limits^{\sim }(1-\exp (c\zeta))},\end{array}\end{eqnarray}$$ (16)

, two singular wave solutions

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=\displaystyle \frac{2c\exp (c\zeta)}{a(1-\exp (c\zeta))},\\ & & \end{array}\end{eqnarray}$$ (17)

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=-\displaystyle \frac{2c}{a(1-\exp (c\zeta))},\end{array}\end{eqnarray}$$ (18)

four periodic singular periodic wave solutions

$$ \begin{eqnarray}\begin{array}{lll} & & {u}_{\pm }(x,y,t)=\displaystyle \frac{c}{a}(-1\pm \sqrt{2}\sec \displaystyle \frac{\sqrt{2}c\zeta }{2}),\\ & & \end{array}\end{eqnarray}$$ (19)

$$ \begin{eqnarray}\begin{array}{lll} & & {u}_{\pm }(x,y,t)=\displaystyle \frac{c}{a}(-1\pm \sqrt{2}\csc \displaystyle \frac{\sqrt{2}c\zeta }{2}),\end{array}\end{eqnarray}$$ (20)

and three periodic wave solutions

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=\displaystyle \frac{{\phi }_{5}({\phi }_{8}-{\phi }_{6})+{\phi }_{6}({\phi }_{5}-{\phi }_{8}){{\rm{sn}}}^{2}(\displaystyle \frac{a}{2{g}_{1}}\zeta,{k}_{1})}{{\phi }_{8}-{\phi }_{6}+({\phi }_{5}-{\phi }_{8}){{\rm{sn}}}^{2}(\displaystyle \frac{a}{2{g}_{1}}\zeta,{k}_{1})},\\ & & \end{array}\end{eqnarray}$$ (21)

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=\displaystyle \frac{{\phi }_{6}({\phi }_{5}-{\phi }_{7})+{\phi }_{5}({\phi }_{7}-{\phi }_{6}){{\rm{sn}}}^{2}(\displaystyle \frac{a}{2{g}_{1}}\zeta,{k}_{1})}{{\phi }_{5}-{\phi }_{7}+({\phi }_{7}-{\phi }_{6}){{\rm{sn}}}^{2}(\displaystyle \frac{a}{2{g}_{1}}\zeta,{k}_{1})},\\ & & \end{array}\end{eqnarray}$$ (22)

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=\displaystyle \frac{{\phi }_{8}({\phi }_{7}-{\phi }_{5})+{\phi }_{7}({\phi }_{5}-{\phi }_{8}){{\rm{sn}}}^{2}(\displaystyle \frac{a}{2{g}_{1}}\zeta,{k}_{1})}{{\phi }_{7}-{\phi }_{5}+({\phi }_{5}-{\phi }_{8}){{\rm{sn}}}^{2}(\displaystyle \frac{a}{2{g}_{1}}\zeta,{k}_{1})},\end{array}\end{eqnarray}$$ (23)

where h∈(−c48a2,0) is the Hamiltonian,

 1

(ii) When 0 < g < g₀, equation (2) has one solitary wave solution

$$ \begin{eqnarray}u(x,y,t)={\phi }_{9}+\displaystyle \frac{2\sqrt{2}\delta \beta \exp (\displaystyle \frac{\sqrt{2\delta }\zeta }{2})}{{a}^{2}{\phi }_{9}(a{\phi }_{9}+2c)(1+{e}^{\sqrt{2\delta }\zeta })-2\sqrt{2}a\beta (a{\phi }_{9}+c)\exp (\displaystyle \frac{\sqrt{2\delta }\zeta }{2})},\end{eqnarray}$$ (24)

$$ \begin{eqnarray}u(x,y,t)={\phi }_{9}-\displaystyle \frac{2\delta (2c+2a{\phi }_{9}+\sqrt{2\delta })\exp (\displaystyle \frac{\sqrt{2\delta }\zeta }{2})}{{a}^{2}{\phi }_{9}(2c+a{\phi }_{9})+a(\gamma -a{\phi }_{9}(2c+a{\phi }_{9}))\exp (\displaystyle \frac{\sqrt{2\delta }\zeta }{2})-a\gamma \exp (\sqrt{2\delta }\zeta)},\end{eqnarray}$$ (25)

is the initial value, β=−aϕ9(2c+aϕ9), δ=2c2+6acϕ9+3a2ϕ92,γ=4c2+10acϕ9+5a2ϕ92+(2c+2aϕ9)2δ, and three periodic wave solutions

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=\displaystyle \frac{{\phi }_{16}({\phi }_{17}-{\phi }_{15})+{\phi }_{17}({\phi }_{15}-{\phi }_{16}){{\rm{sn}}}^{2}(\displaystyle \frac{a}{2{g}_{2}}\zeta,{k}_{2})}{{\phi }_{17}-{\phi }_{15}+({\phi }_{15}-{\phi }_{16}){{\rm{sn}}}^{2}(\displaystyle \frac{a}{2{g}_{2}}\zeta,{k}_{2})},\\ & & \end{array}\end{eqnarray}$$ (26)

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=\displaystyle \frac{{\phi }_{17}({\phi }_{16}-{\phi }_{14})+{\phi }_{16}({\phi }_{14}-{\phi }_{17}){{\rm{sn}}}^{2}(\displaystyle \frac{a}{2{g}_{2}}\zeta,{k}_{2})}{{\phi }_{16}-{\phi }_{14}+({\phi }_{14}-{\phi }_{17}){{\rm{sn}}}^{2}(\displaystyle \frac{a}{2{g}_{2}}\zeta,{k}_{2})},\\ & & \end{array}\end{eqnarray}$$ (27)

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=\displaystyle \frac{{\phi }_{14}({\phi }_{15}-{\phi }_{17})+{\phi }_{15}({\phi }_{17}-{\phi }_{14}){{\rm{sn}}}^{2}(\displaystyle \frac{a}{2{g}_{2}}\zeta,{k}_{2})}{{\phi }_{15}-{\phi }_{17}+({\phi }_{17}-{\phi }_{14}){{\rm{sn}}}^{2}(\displaystyle \frac{a}{2{g}_{2}}\zeta,{k}_{2})},\end{array}\end{eqnarray}$$ (28)

where

 1

(iii) When g = g₀, equation (2) has three singular wave solutions

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=\displaystyle \frac{9(1+\sqrt{3})c-(3-\sqrt{3}){c}^{3}{\zeta }^{2}}{3a({c}^{2}{\zeta }^{2}-3)},\\ & & \end{array}\end{eqnarray}$$ (29)

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=\displaystyle \frac{12\sqrt{3}-6(1-\sqrt{3})c\zeta -(3-\sqrt{3}){c}^{2}{\zeta }^{2}}{3a\zeta (c\zeta -2\sqrt{3})},\\ & & \end{array}\end{eqnarray}$$ (30)

$$ \begin{eqnarray}\begin{array}{lll} & & u(x,y,t)=\displaystyle \frac{12\sqrt{3}+6(1-\sqrt{3})c\zeta -(3-\sqrt{3}){c}^{2}{\zeta }^{2}}{3a\zeta (c\zeta +2\sqrt{3})}.\end{array}\end{eqnarray}$$ (31)

Proof (i) There are two heteroclinic orbits Γ₁ and Γ1*, four special orbits Γ₂, Γ¯2, Γ₃, and Γ¯3 in Fig. 1(d). The orbits are defined by H(ϕ, φ) = H(ϕ₀,0), which can be reduced to

$$ \begin{eqnarray}\varphi =\pm \displaystyle \frac{a}{2}\phi (\phi -{\phi }_{2}),\end{eqnarray}$$ (32)

Substituting Eq. (32) into d ϕ / d ζ = φ and integrating along the above orbits, we can obtain the following integrals:

$$ \begin{eqnarray}\begin{array}{lll} & & \pm \displaystyle {\int }_{\mathop{\phi }\limits^{\sim }}^{\phi }\displaystyle \frac{1}{r(r-{\phi }_{2})}{\rm{d}}r=\displaystyle \frac{a}{2}\displaystyle {\int }_{0}^{\zeta }{\rm{d}}r,\end{array}\end{eqnarray}$$ (33)

$$ \begin{eqnarray}\begin{array}{lll} & & \pm \displaystyle {\int }_{\phi }^{+\infty }\displaystyle \frac{1}{r(r-{\phi }_{2})}{\rm{d}}r=\displaystyle \frac{a}{2}\displaystyle {\int }_{0}^{\zeta }{\rm{d}}r,\end{array}\end{eqnarray}$$ (34)

Completing the integrals (33), (34) and utilizing the transformation (3), we obtain two kink solutions (15), (16) and two singular wave solutions (17), (18).

There are two special orbits Γ₅ and Γ¯5 in Fig. 1(d). The orbits are defined by H(ϕ, φ) = H(ϕ₁,0), which can be reduced to

$$ \begin{eqnarray}\varphi =\pm \displaystyle \frac{a}{2}(\phi +{\phi }_{1})\sqrt{(\phi -{\phi }_{3})(\phi -{\phi }_{4})},\end{eqnarray}$$ (35)

Substituting Eq. (35) into d ϕ / d ζ = φ and integrating along the above orbits, we can obtain the following integrals:

$$ \begin{eqnarray}\begin{array}{lll} & & \pm \displaystyle {\int }_{{\phi }_{3}}^{\phi }\displaystyle \frac{1}{(r-{\phi }_{1})\sqrt{(r-{\phi }_{3})(r-{\phi }_{4})}}{\rm{d}}r=\displaystyle \frac{a}{2}\displaystyle {\int }_{0}^{\zeta }{\rm{d}}r,\\ & & \end{array}\end{eqnarray}$$ (36)

$$ \begin{eqnarray}\begin{array}{lll} & & \pm \displaystyle {\int }_{\phi }^{\infty }\displaystyle \frac{1}{(r-{\phi }_{1})\sqrt{(r-{\phi }_{3})(r-{\phi }_{4})}}{\rm{d}}r=\displaystyle \frac{a}{2}\displaystyle {\int }_{0}^{\zeta }{\rm{d}}r.\end{array}\end{eqnarray}$$ (37)

There are one periodic orbit and two special orbits Γ∼4, Γ₄, and Γ¯4 in Fig. 1(d). The orbits are given by H(ϕ, φ) = h, h∈(h₁,h₀), which can be converted to

$$ \begin{eqnarray}z=\pm \displaystyle \frac{a}{2}\sqrt{(\phi -{\phi }_{5})(\phi -{\phi }_{6})(\phi -{\phi }_{7})(\phi -{\phi }_{8})},\end{eqnarray}$$ (38)

 1

(ii) In Fig. 1(e), there are one homoclinic orbit Γ₇ and three special orbits Γ₆, Γ₈, and Γ¯8. The orbits are defined by H(ϕ, φ) = H(ϕ₉, 0), which can be converted to

$$ \begin{eqnarray}\varphi =\pm \displaystyle \frac{a}{2}(\phi -{\phi }_{9})\sqrt{(\phi -{\phi }_{12})(\phi -{\phi }_{13})},\end{eqnarray}$$ (42)

is the initial value,

 1

Substituting Eq. (42) into d ϕ / d ζ = φ, and integrating along the above orbits, we can obtain the following integrals:

$$ \begin{eqnarray}\begin{array}{lll} & & \pm \displaystyle {\int }_{{\phi }_{12}}^{\phi }\displaystyle \frac{1}{(r-{\phi }_{9})\sqrt{(r-{\phi }_{12})(r-{\phi }_{13})}}{\rm{d}}r=\displaystyle \frac{a}{2}\displaystyle {\int }_{0}^{\zeta }{\rm{d}}r,\\ & & \end{array}\end{eqnarray}$$ (43)

$$ \begin{eqnarray}\begin{array}{lll} & & \pm \displaystyle {\int }_{\phi }^{+\infty }\displaystyle \frac{1}{(r-{\phi }_{9})\sqrt{(r-{\phi }_{12})(r-{\phi }_{13})}}{\rm{d}}r=\displaystyle \frac{a}{2}\displaystyle {\int }_{0}^{\zeta }{\rm{d}}r.\end{array}\end{eqnarray}$$ (44)

Completing the integrals (43), (44) and using the transformation (3), we obtain the solitary wave solution (24) and the singular wave solution (25).

Similarly, there are a periodic orbit Γ₉, two special orbits Γ∼9 and Γ¯9 in Fig. 1(e). The orbits are defined by H(ϕ, φ) = h, h∈(h₁₀,h₉), which can be converted to

$$ \begin{eqnarray}\varphi =\pm \displaystyle \frac{a}{2}\sqrt{(\phi -{\phi }_{14})(\phi -{\phi }_{15})(\phi -{\phi }_{16})(\phi -{\phi }_{17})},\end{eqnarray}$$ (45)

 1

$$ \begin{eqnarray}\begin{array}{lll} & & \pm \displaystyle {\int }_{{\phi }_{14}}^{\phi }\displaystyle \frac{1}{\sqrt{(r-{\phi }_{14})(r-{\phi }_{15})(r-{\phi }_{16})(r-{\phi }_{17})}}{\rm{d}}r=\displaystyle \frac{a}{2}\displaystyle {\int }_{0}^{\zeta }{\rm{d}}r,\\ & & \end{array}\end{eqnarray}$$ (46)

$$ \begin{eqnarray}\begin{array}{lll} & & \pm \displaystyle {\int }_{{\phi }_{16}}^{\phi }\displaystyle \frac{1}{\sqrt{({\phi }_{14}-r)({\phi }_{15}-r)(r-{\phi }_{16})(r-{\phi }_{17})}}{\rm{d}}s=\displaystyle \frac{a}{2}\displaystyle {\int }_{0}^{\zeta }{\rm{d}}r,\end{array}\end{eqnarray}$$ (47)

$$ \begin{eqnarray}\begin{array}{lll} & & \pm \displaystyle {\int }_{{\phi }_{17}}^{\phi }\displaystyle \frac{1}{\sqrt{({\phi }_{14}-r)({\phi }_{15}-r)({\phi }_{16}-r)({\phi }_{17}-r)}}{\rm{d}}r=\displaystyle \frac{a}{2}\displaystyle {\int }_{0}^{\zeta }{\rm{d}}r.\end{array}\end{eqnarray}$$ (48)

Completing the integrals (46)–(48) and utilizing the transformation (3), we obtain the periodic wave solutions (26)–(28).

(iii) In Fig. 1(e), three are three special orbits Γ₁₁, Γ₁₂, and Γ¯12. The orbits are defined by H(ϕ, φ) = H(ϕ₂₀,0), which can be converted to

$$ \begin{eqnarray}\varphi =\pm \displaystyle \frac{a}{2}(\phi -{\phi }_{20})\sqrt{(\phi -{\phi }_{20})(\phi -{\phi }_{22})},\end{eqnarray}$$ (49)

Substituting Eq. (49) into d ϕ / d ζ = φ and integrating along the above orbits, we can obtain the following integrals:

$$ \begin{eqnarray}\begin{array}{lll} & & \pm \displaystyle {\int }_{{\phi }_{22}}^{\phi }\displaystyle \frac{1}{(r-{\phi }_{20})\sqrt{(r-{\phi }_{20})(r-{\phi }_{22})}}{\rm{d}}r=\displaystyle \frac{a}{2}\displaystyle {\int }_{0}^{\zeta }{\rm{d}}r,\\ & & \end{array}\end{eqnarray}$$ (50)

$$ \begin{eqnarray}\begin{array}{lll} & & \pm \displaystyle {\int }_{\phi }^{+\infty }\displaystyle \frac{1}{(r-{\phi }_{20})\sqrt{(r-{\phi }_{20})(r-{\phi }_{22})}}{\rm{d}}r=\displaystyle \frac{a}{2}\displaystyle {\int }_{0}^{\zeta }{\rm{d}}r.\end{array}\end{eqnarray}$$ (51)

In this section, we draw the profiles of some traveling wave solutions when some parameters take special values.

(i) When a = 2, c = 2, and ϕ∼=−1.8, we draw the profiles of the kink wave solution (15), singular wave solution (17), and periodic singular wave solution (19) in Fig. 2.

(ii) When a = 2, c = 2, ϕ₉ = –0.1, and h = –0.2, we draw the profiles of the periodic wave solution (22), solitary wave solution (24), and singular wave solution (29) in Fig. 3.

In this work, we utilize the bifurcation theory of planar dynamical systems to obtain the phase portraits and bifurcations analysis of the traveling wave system corresponding to Eq. (2). Furthermore, we have obtained abundant exact kink wave solutions, singular wave solutions, periodic wave solutions, periodic singular wave solutions, and solitary wave solutions. We are convinced that, in the future, the method can also be applied to other partial differential equations and may find more new results.