On a Solution on Non-Linear Time-Evolution Equation of Fifth Order

The non-linear fifth order differential equations: u t + (105/16)α 2 · u u x - u x x x x x =0, and v t +(105/16)γ 2 · v v x +(13/4)δ· v x x x - v x x x x x =0, have solutions: \(u= \frac{\beta }{\alpha } \cdot \text{cn}^{4} \left( \frac{1}{2\sqrt{2}} {\alpha \beta }^{1/4} \cdot (\xi -\xi _{0}), \sqrt{\frac{1}{2}} \right),\) and \(v= \left( \frac{\delta }{\gamma } \right)^{2} \cdot \text{sech}^{4} \left( \frac{1}{4} \delta ^{1/2} \cdot (\eta -\eta _{0}) \right),\) respectively. Here cn ( z , k ) is the Jacobi cn-function of modulus k , with ξ= x -(21αβ t /8), η= x -(9δ 2 t /4), positive constants α, β, γ, and δ; and any constants ξ 0 and η 0 .