The Korteweg–de Vries–Burgers equation is a nonlinear partial differential equation:
u t + α u x x x + u u x − β u x x = 0. {\displaystyle u_{t}+\alpha u_{xxx}+uu_{x}-\beta u_{xx}=0.}
The equation gives a description for nonlinear waves in dispersive-dissipative media by combining the nonlinear and dispersive elements from the KdV equation with the dissipative element from Burgers' equation. The modified KdV–Burgers equation can be written as:
u t + a u x x x + u 2 u x − b u x x = 0. {\displaystyle u_{t}+au_{xxx}+u^{2}u_{x}-bu_{xx}=0.}
See also Burgers' equation Korteweg–de Vries equation modified KdV equation
Notes
References Polyanin, Andrei D.; Zaitsev, Valentin F. (2003). "9.1.7. Burgers–Korteweg–de Vries Equation and Other Equation". Handbook of Nonlinear Partial Differential Equations. Boca Raton, Fla: Chapman and Hall/CRC. ISBN 978-1-58488-355-5. Wang, Mingliang (1996). "Exact solutions for a compound KdV–Burgers equation". Physics Letters A. 213 (5–6): 279–287. Bibcode:1996PhLA..213..279W. doi:10.1016/0375-9601(96)00103-X.