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

Abstract

The non-linear fifth order differential equations:
u_t+(105⁄16)α²·uu_x−u_xxxxx=0,
and
v_t+(105⁄16)γ²·vv_x+(13⁄4)δ·v_xxx−v_xxxxx=0,
have solutions:
(Remark: Graphics omitted.),
and
(Remark: Graphics omitted.),
respectively. Here cn (z, k) is the Jacobi cn-function of modulus k, with ξ=x−(21αβt⁄8), η=x−(9δ²t⁄4), positive constants α, β, γ, and δ; and any constants ξ₀ and η₀.