Bifurcations and Exact Solutions of the Coupled Nonlinear Generalized Zakharov Equations with Anti-Cubic Nonlinearity: Dynamical System Approach
Abstract
:1. Introduction
2. Bifurcations of Phase Portraits of the System (4)
- (i)
- If and then has two positive real zeros. Furthermore,
- (i1)
- If and then has three positive real zeros satisfying for and for .
- (i2)
- If and either or then has a double positive real zero.
- (i3)
- If and , then has two positive real zeros satisfying for and for .
- (ii)
- When if then has an positive real zero.
- (I)
- The case that there exist three equilibrium points (including double points) of the system (4) in the positive axis.
- (II)
- The case that there exist two equilibrium points (including double points) of the system (4) in the positive axis.
- (III)
- The case that there exists one equilibrium point of the system (4) in the positive axis.
3. Explicit Exact Parametric Representations of the Solutions of the System (4) with Three Equilibrium Points and a5 >0
3.1. The Parametric Representations of the Bounded Orbits Given by Figure 1a
- (i)
- Corresponding to the level curves defined by with , there exist two families of open orbits of the system (4), for which one family tends to the singular straight line , when (see Figure 5a). This open orbit family gives rise to a family of compacton solutions. Correspondingly, (13) can be written asHere
- (ii)
- Corresponding to the level curves defined by with there exists a family of periodic orbits and two families of open orbits of the system (4) (see Figure 5b). For the family of periodic orbits, (13) can be written as
- (iii)
- The level curves defined by contain a homoclinic orbit to the equilibrium point and an open orbit (see Figure 5c). For the homoclinic orbit enclosing the equilibrium point , (13) can be written asIt follows that
- (iv)
- The level curves defined by with are two open curve families, for which one curve family tends to the singular straight line as . It gives rise to a compacton solution family having the same parametric representation as that of (14).
- (v)
- The level curves defined by are two stable manifolds and two unstable manifolds of the saddle point , for which two manifolds tend to the singular straight line as (see Figure 5e). For the left stable manifold, (13) can be written as
3.2. The Parametric Representations of the Heteroclinic Orbits Given by Figure 1b
3.3. The Parametric Representations of the Homoclinic Orbit Given by Figure 1c
3.4. The Parametric Representations of the Stable Manifold of the Cusp Point Given by Figure 1d
4. Explicit Exact Parametric Representations of the Solutions of the System (4) with Three Equilibrium Points and a5 < 0
4.1. The Parametric Representations of the Bounded Orbits Given by Figure 2a
- (i)
- (ii)
- Corresponding to the level curves defined by with there exist two families of periodic orbits of the system (4), enclosing the equilibrium point and , respectively. For the left family of periodic orbits enclosing the center , (13) can be written asWe notice that when h approaches , the periodic orbit in the left family defined by tends to the left homoclinic loop which is close to the singular straight line . Therefore, the left homoclinic orbit gives rise to an envelope pseudo-peakon solution, and the periodic orbit family gives rise to a pseudo-periodic peakon family (see Figure 6a,b).
- (iii)
- Corresponding to the level curves defined by , there exist two homoclinic orbits of the system (4) to the saddle point , enclosing the equilibrium point and , respectively. For the right homoclinic orbit, (13) can be written as
- (iv)
- Corresponding to the level curves defined by with , there exists a global family of periodic orbits enclosing three equilibrium points of the system (4). In this case, this family has the same parametric representation as that of (27).Note that there exists a segment of every periodic orbit in this periodic family for which it is very close to the singular straight line . This global periodic orbit family gives rise to a family of pseudo-periodic peakon solution of the system (4) (see Figure 6c).Similarly, for the orbits shown in Figure 2b,c, one can calculate their parametric representations. We skip it here.
4.2. The Parametric Representations of the Homoclinic Orbit to the Cusp Point Given by Figure 2e
5. The Parametric Representations of the Homoclinic Orbit and Periodic Orbits Given by Figure 3b
- (i)
- Corresponding to the level curves defined by with there exist a family of periodic orbits enclosing the equilibrium point of the system (4) and an open curve family which tends to the singular straight line as For the periodic family, (13) can be written as
- (ii)
- Corresponding to the level curves defined by there exist a homoclinic orbit of the system (4) to the saddle point enclosing the equilibrium point . Equation (13) can be written as
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Song, J.; Li, F.; Zhang, M. Bifurcations and Exact Solutions of the Coupled Nonlinear Generalized Zakharov Equations with Anti-Cubic Nonlinearity: Dynamical System Approach. Mathematics 2025, 13, 217. https://doi.org/10.3390/math13020217
Song J, Li F, Zhang M. Bifurcations and Exact Solutions of the Coupled Nonlinear Generalized Zakharov Equations with Anti-Cubic Nonlinearity: Dynamical System Approach. Mathematics. 2025; 13(2):217. https://doi.org/10.3390/math13020217
Chicago/Turabian StyleSong, Jie, Feng Li, and Mingji Zhang. 2025. "Bifurcations and Exact Solutions of the Coupled Nonlinear Generalized Zakharov Equations with Anti-Cubic Nonlinearity: Dynamical System Approach" Mathematics 13, no. 2: 217. https://doi.org/10.3390/math13020217
APA StyleSong, J., Li, F., & Zhang, M. (2025). Bifurcations and Exact Solutions of the Coupled Nonlinear Generalized Zakharov Equations with Anti-Cubic Nonlinearity: Dynamical System Approach. Mathematics, 13(2), 217. https://doi.org/10.3390/math13020217