  A highly nonlinear, fourth-order wave equation that models the continuum
theory of long wavelength pulses in weakly compressed, discrete, homogeneous
chains with a general power-law contact interaction is studied. For this wave
equation, all solitary wave solutions and all nonlinear periodic wave
solutions, along with all conservation laws, are derived. The solutions are
explicitly parameterized in terms of the asymptotic value of the wave amplitude
in the case of solitary waves and the peak of the wave amplitude in the case of
nonlinear periodic waves. All cases in which the solution expressions can be
stated in an explicit analytic form using elementary functions are worked out.
In these cases, explicit expressions for the total energy and total momentum
for all solutions are obtained as well. The derivation of the solutions uses
the conservation laws combined with an energy analysis argument to reduce the
wave equation directly to a separable first-order differential equation which
determines the wave amplitude in terms of the travelling wave variable. This
method can be applied more generally to other highly nonlinear wave equations.
