Kawahara equation

The Kawahara equation is a partial differential equation that arises in various fields of mathematical physics, particularly in the study of wave phenomena. Named after the Japanese mathematician T. Kawahara, who first introduced it in the context of fluid dynamics and nonlinear wave propagation, this equation extends the well-known Korteweg-de Vries (KdV) equation by incorporating higher-order derivatives. This inclusion allows for the modeling of more complex wave behaviors, capturing phenomena such as wave shape distortion and the emergence of solitary waves in dispersive systems.^[1]^[2]

The Kawahara equation is a fifth-order KdV equation and typically expressed as:^[3]

$\partial _{t}\phi +\alpha \phi \partial _{x}\phi +\beta \partial _{x}^{3}\phi -\gamma \,\partial _{x}^{5}\phi =0,$

where waves described by function $\phi (x,t)$ and $\alpha ,\,\beta$ , and $\gamma >0$ are constant.

References

^ Biswas, Anjan (2009). "Solitary wave solution for the generalized Kawahara equation". Applied Mathematics Letters. 22 (2): 208–210 – via Elsevier Science Direct.
^ Haghighatdoost Gh., M Bazghandi and F. Pashahie (2025). "A finite generating set of differential invariants for Lie symmetry group of the fifth-order KdV types". Computational Methods for Differential Equations. 11 (4): 803–810.
^ Kawahara, T. (1972). "Oscillatory solitary waves in dispersive media". Journal of the Physical Society of Japan. 33 (1): 260–264.

[1] Biswas, Anjan (2009). "Solitary wave solution for the generalized Kawahara equation". Applied Mathematics Letters. 22 (2): 208–210 – via Elsevier Science Direct.

[2] Haghighatdoost Gh., M Bazghandi and F. Pashahie (2025). "A finite generating set of differential invariants for Lie symmetry group of the fifth-order KdV types". Computational Methods for Differential Equations. 11 (4): 803–810.

[3] Kawahara, T. (1972). "Oscillatory solitary waves in dispersive media". Journal of the Physical Society of Japan. 33 (1): 260–264.

[1]

[2]

[3]

See also

References