Exact Solutions > Nonlinear Partial Differential Equations > Second-Order Hyperbolic Partial Differential Equations

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2. Nonlinear Hyperbolic Equations

2.1. Nonlinear Wave Equations of the Form w_tt = aw_xx + f(w)

1. w_tt = aw_xx + aw + bwⁿ.    Klein--Gordon equation with a power-law nonlinearity.
2. w_tt = w_xx + awⁿ + bw²ⁿ⁻¹.    Klein--Gordon equation with a power-law nonlinearity.
3. w_tt = a²w_xx + be^βw.    Modified Liouville equation.
4. w_tt = w_xx + ae^βw + be^2βw.    Klein--Gordon equation with a exponential nonlinearity.
5. w_tt = aw_xx + b sinh(λw).    Sinh--Gordon equation.
6. w_tt = aw_xx + b sin(λw).    Sine--Gordon equation.
7. w_tt = w_xx + f(w).    Nonlinear Klein--Gordon equation.

2.2. Other Nonlinear Hyperbolic Equations

1. w_tt = a(ww_x)_x.
2. w_tt = a(wⁿw_x)_x + bw^k.
3. w_tt = a(e^λww_x)_x.
4. w_tt = ax⁻ⁿ(xⁿw_x)_x + f(w).
5. w_tt = [a(x + b)ⁿw_x]_x + f(w).
6. w_tt = a(e^λxw_x)_x + f(w).
7. w_tt = [f(w)w_x]_x.