Dynamics of Nonlinear Systems: Integrable and Chaotic Solutions
175
Fig. 1 Soliton solution for KP equation
Fig. 2 Soliton solution for mKP equation
Using the same integration techniques as in the previous case, we reach at the solution
f (ψ) = −(c − 12)sech(
(c − 12)(ψ − ψ 0 ))
(39)
This gives the solitary wave solution for mKP equations and its graphical representation is given in Fig. 2.
This section discussed the propagation of wave through a two-dimensional lattice.
From the equation of motion, we derived the modelling equation for wave propagation for three cases: quadratic nonlinearity, cubic nonlinearity and their combination.
The first two cases we found soliton solutions and for the third case we could not find
a soliton. Thus, we conclude that the lattice dynamic system with quadratic nonlinearity is completely integrable while the system with cubic nonlinearity is partially
integrable. When the quadratic and cubic nonlinearities act together, the system is not
integrable and may become chaotic. As expected, nonlinearity in evolution equations
may cause different behaviours, from regular motion to chaotic motion.
175
Fig. 1 Soliton solution for KP equation
Fig. 2 Soliton solution for mKP equation
Using the same integration techniques as in the previous case, we reach at the solution
f (ψ) = −(c − 12)sech(
(c − 12)(ψ − ψ 0 ))
(39)
This gives the solitary wave solution for mKP equations and its graphical representation is given in Fig. 2.
This section discussed the propagation of wave through a two-dimensional lattice.
From the equation of motion, we derived the modelling equation for wave propagation for three cases: quadratic nonlinearity, cubic nonlinearity and their combination.
The first two cases we found soliton solutions and for the third case we could not find
a soliton. Thus, we conclude that the lattice dynamic system with quadratic nonlinearity is completely integrable while the system with cubic nonlinearity is partially
integrable. When the quadratic and cubic nonlinearities act together, the system is not
integrable and may become chaotic. As expected, nonlinearity in evolution equations
may cause different behaviours, from regular motion to chaotic motion.