Consider a line of pendula, hanging on a straight line, in constant gravity. Connect the bobs of the pendula together by a string in constant tension. Let the angle of the pendulum at location x be φ φ , then schematically, the dynamics of the line of pendulum follows Newton's second law: m φ φ t t ⏟ ⏟ mass times acceleration = T φ φ x x ⏟ ⏟ tension − − m g sin ⁡ ⁡ φ φ ⏟ ⏟ gravity and this is the sine-Gordon equation, after scaling time and distance appropriately.