| Abstract: |
| We investigate blow-up phenomena in an integrable system with a singular integral, which is described by the equation $u_t = -2A u_x T u_x - V u_x + D u_{xx}$. Here, $ T $ is a singular integral operator with a $\coth$-type kernel, which incorporates nonlocal effects. $ A $, $V $, and $ D $ are constants. By utilizing exact solutions, we examine four aspects: (1) determining the conditions under which blow-up occurs; (2) identifying the locations of blow-up points; (3) analyzing the form of the blow-up solution; and (4) exploring the system`s behavior after blow-up. Furthermore, we apply these findings to a traffic flow model, where blow-up corresponds to complete congestion in high-density regions. |
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