Bifurcation, Bi-instability and Area Principle for the Solitary Waves of the Nonlinear Wave Equation with Quartic Polynomial Potential

For the nonlinear wave equation with quartic polynomial potential, bifurcation, bi-instability and solitary waves are investigated. An area principle based on the bifurcation diagram is found for the existence of bright and dark solitary waves and shock waves. The simple forms of solitary wave solutions are given by an approximate analytic method.