摘要
On studying traveling waves on a nonlinearly suspended bridge,the following partial differential equation has been considered:\$\$u\-\{tt\}+u\-\{xxxx\}+f(u)=0,\$\$where f(u)=u\++-1 .Here the bridge is seen as a vibrating beam supported by cables,which are treated as a spring with a one\|sided restoring force.The existence of a traveling wave solution to the above piece\|wise linear equation has been proved by solving the equation explicitly (McKenna & Walter in 1990).Recently the result has been extended to a group of equations with more general nonlinearities such as f(u)=u\++-1+g(u) (Chen & McKenna,1997).However,the restrictions on g(u) do not allow the resulting restoring force function to increase faster than the linear function u-1 for u >1.Since an interesting “multiton” behavior,that is ,two traveling waves appear to emerge intact after interacting nonlinearly with each other,has been observed in numerical experiments for a fast\|increasing nonlinearity f(u)=e u-1 -1 ,it hints that the conclusion of the existence of a traveling wave solution with fast\|increasing nonlinearities shall be valid as well.\;In this paper,the restoring force function of the form f(u)=u·h(u)-1 is considered.It is shown that a traveling wave solution exists when h(u)≥1 for u≥1 (with other assumptions which will be detailed in the paper),and hence allows f to grow faster than u-1 .It is shown that a solution can be obtained as a saddle point in a variational formulation.It is also easy to construct such fast\|increasing f(u) 's for more numerical tests.
On studying traveling waves on a nonlinearly suspended bridge,the following partial differential equation has been considered:\$\$u\-\{tt\}+u\-\{xxxx\}+f(u)=0,\$\$where f(u)=u\++-1 .Here the bridge is seen as a vibrating beam supported by cables,which are treated as a spring with a one\|sided restoring force.The existence of a traveling wave solution to the above piece\|wise linear equation has been proved by solving the equation explicitly (McKenna & Walter in 1990).Recently the result has been extended to a group of equations with more general nonlinearities such as f(u)=u\++-1+g(u) (Chen & McKenna,1997).However,the restrictions on g(u) do not allow the resulting restoring force function to increase faster than the linear function u-1 for u >1.Since an interesting “multiton” behavior,that is ,two traveling waves appear to emerge intact after interacting nonlinearly with each other,has been observed in numerical experiments for a fast\|increasing nonlinearity f(u)=e u-1 -1 ,it hints that the conclusion of the existence of a traveling wave solution with fast\|increasing nonlinearities shall be valid as well.\;In this paper,the restoring force function of the form f(u)=u·h(u)-1 is considered.It is shown that a traveling wave solution exists when h(u)≥1 for u≥1 (with other assumptions which will be detailed in the paper),and hence allows f to grow faster than u-1 .It is shown that a solution can be obtained as a saddle point in a variational formulation.It is also easy to construct such fast\|increasing f(u) 's for more numerical tests.
基金
Project supported by National Natural Science Foundation of China! (19701029)
by Outstanding Young Teacher Foundation of Chi
