Interactions among periodic waves and solitary waves of the (N + 1)-dimensional sine-Gordon field.

Exact solutions of the (n + 1)-dimensional sine-Gordon field equation are studied with help of those of the cubic nonlinear Klein-Gordon fields. The mapping relations among the sine-Gordon field equation and the cubic nonlinear Klein-Gordon fields are pure algebraic. By solving the cubic nonlinear Klein-Gordon equations, many new types of exact explicit solutions such as the periodic-periodic interaction waves, periodic-kink interaction waves, periodic perturbed "snake" shape solitary waves, etc., are displayed both analytically and graphically.

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