Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation
The generalized forced Boussinesq equation, utt−uxx+[f(u)]xx+uxxxx=h0, and its periodic traveling wave solutions are considered. Using the transform z=x−ωt, the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the o...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1999-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171299226439 |
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| Summary: | The generalized forced Boussinesq equation, utt−uxx+[f(u)]xx+uxxxx=h0, and its periodic traveling wave solutions are considered. Using the transform
z=x−ωt, the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the ordinary differential equation and a Hammerstein type integral equation is then established by using the Green's function method. This integral equation generates compact operators in a Banach space of real-valued continuous
periodic functions with a given period 2T. The Schauder's fixed point theorem is then used to prove the existence of solutions to the integral equation. Therefore, the existence of nonconstant
periodic traveling wave solutions to the generalized forced Boussinesq equation is established. |
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| ISSN: | 0161-1712 1687-0425 |