The Gardner equation $u_t+au{u}_x+b{u}^2{u}_x+\mu u_{xxx}=0$ is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation when the effects of higher-order nonlinearity become significant. Using the so-called traveling wave ansatz $u(x,t)=\phi \left(\xi \right)$, $\xi =x-vt$ (where $v$ is the velocity of the wave) we convert the (1+1)-dimensional partial differential equation to a second-order ordinary differential equation in $\varphi$ with an arbitrary constant and treat the latter equation by the methods of the dynamical systems theory. With some special attention on the equilibrium points of the equation, we derive an analytical constraint for admissible values of the parameters $a$, $b$, and $\mu$. From the Hamiltonian form of the system we confirm that, in addition to the usual bright soliton solution, the equation can be used to generate three different varieties of internal waves of which one is a dark soliton recently observed in water [A. Chabchoub et al., Phys. Rev. Lett. 110, 124101 (2013)].

• Received 22 September 2013
• Revised 8 January 2014

DOI:https://doi.org/10.1103/PhysRevE.89.023204