# Numerical Study of Breakup in Generalized Korteweg–de Vries and Kawahara Equations

## Abstract

*Comm. Math. Phys.*, 267 (2006), pp. 117–139] on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasi-linear transport equation. The regularizations are characterized by two arbitrary functions of one variable, where the condition of integrability implies that one of these functions must not vanish. It is shown numerically for a large class of equations that the local behavior of their solution near the point of gradient catastrophe for the transport equation is described by a special solution of a Painlevé-type equation. This local description holds also for solutions to equations where blowup can occur in finite time. Furthermore, it is shown that a solution of the dispersive equations away from the point of gradient catastrophe is approximated by a solution of the transport equation with the same initial data, modulo terms of order $\epsilon^2$, where $\epsilon^2$ is the small dispersion parameter. Corrections up to order $\epsilon^4$ are obtained and tested numerically.

### MSC codes

### Keywords

## Get full access to this article

View all available purchase options and get full access to this article.

## References

*Non-dissipative and low-dissipative shocks with regular and stochastic structures in non-linear media with dispersion*, in Proceedings of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, A. V. Borisov, V. V. Kozlov, I. S. Mamaev, and M. A. Sokolovskiy, eds., IUTAM Bookseries 6, Springer, New York, 2008.

*Radiation and modulational instability described by the fifth-order Korteweg–DeVries equation*, in Mathematical Problems in the Theory of Water Waves, F. Dias, J.-M. Ghidaglia, and J.-C. Saut, eds., Contemp. Math. 200, AMS, Providence, RI, 1996.

*On the nonlinear evolution of perturbation in plasma and other dispersive media*, Zh. Elek. Tekh. Fiz., 51 (1966), 1557–1568 (translated as Soviet Phys. JETP, 24 (1967)).

*Ann. Inst. Fourier (Grenoble)*, 55 (2005), pp. 1943–2000.

*Phys. Lett. B*, 242 (1990), pp. 35–38.

*Comm. Math. Phys.*, 286 (2009), pp. 979–1009.

*Nonlinearity*, 20 (2007), pp. 1163–1184.

*J. Comput. Phys.*, 176 (2002), pp. 430–455.

*Spectral sequences and the inverse problem of the calculus of variations*, in Differential Geometrical Methods in Mathematical Physics, Geometry, Topology, and Mathematical Physics, Lecture Notes in Math. 836, Springer, New York, 1980, pp. 498–503.

*Comm. Math. Phys.*, 267 (2006), pp. 117–139.

*On universality of critical behaviour in Hamiltonian PDEs*, in Amer. Math. Soc. Transl. 224, AMS, Providence, RI, 2008, pp. 59–109.

*Hamiltonian perturbations of hyperbolic PDEs: From classification results to the properties of solutions*, in New Trends in Mathematical Physics. Selected Contributions of the XVth International Congress on Mathematical Physics, V. Sidoravicius, ed., Springer, Dordrecht, The Netherlands, 2009, pp. 231–276.

*Russian Math. Surv.*, 44 (1989), pp. 29–98.

*Resolution of a shock in hyperbolic systems modified by weak dispersion*, Chaos, 15 (2005), paper 037103.

*Internat. Math. Res. Notices*, 14 (2003), pp. 755–820.

*Arch. Ration. Mech. Anal.*, 121 (1992), pp. 235–265.

*Comm. Pure Appl. Math.*, 60 (2007), pp. 1623–1664.

*Numerical study of a multiscale expansion of KdV and Camassa-Holm equation*, in Integrable Systems and Random Matrices, J. Baik, T. Kriecherbauer, L.-C. Li, K. D. T.-R. McLaughlin, and C. Tomei, eds., Contemp. Math. 458, AMS, Providence, RI, 2008, pp. 81–99.

*Sov. Phys. JETP Lett.*, 38 (1974), pp. 291–297.

*Math. Tables Aids Comput.*, 9 (1955), pp. 92–96.

*Comm. Pure Appl. Math.*, 44 (1991), pp. 1–40.

*Phys. D*, 32 (1988), pp. 253–268.

*Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 187 (1991); English translation in J. Math. Sci.*, 73 (1995), pp. 468–481.

*J. Phys. Soc. Japan*, 33 (1972), pp. 260–264.

*Electron. Trans. Numer. Anal.*, 29 (2008), pp. 116–135.

*Obstacles to asymptotic integrability*, in Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl. 26, Birkhäuser, Boston, MA, 1997, pp. 173–204.

*Phys. Lett. A*, 221 (1996), pp. 204–208.

*Nonlinear Anal.*, 36 (1999), pp. 213–230.

*Ann. of Math. (2)*, 155 (2002), pp. 235–280.

*J. Differential Equations*, 102 (1993), pp. 360–381.

*J. Exp. Theor. Phys.*, 78 (1994), pp. 583–587.

*J. Comput. Phys.*, 50 (1983), pp. 138–161.

*Phys. Rev. Lett.*, 15 (1965), pp. 240–243.

## Information & Authors

### Information

#### Published In

#### Copyright

#### History

**Submitted**: 4 January 2011

**Accepted**: 28 March 2011

**Published online**: 7 July 2011

#### MSC codes

#### Keywords

### Authors

## Metrics & Citations

### Metrics

### Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

#### Cited By

- Spectral approach to Korteweg-de Vries equations on the compactified real lineApplied Numerical Mathematics, Vol. 177 | 1 Jul 2022
- Numerical Study of the Generalized Korteweg–de Vries Equations with Oscillating Nonlinearities and Boundary ConditionsWater Waves, Vol. 4, No. 1 | 11 April 2022
- Numerical study of break-up in solutions to the dispersionless Kadomtsev–Petviashvili equationLetters in Mathematical Physics, Vol. 111, No. 5 | 30 August 2021
- Asymptotic behaviours given by elliptic functions in ${\rm P_I}$ –${\rm P_V}$Nonlinearity, Vol. 31, No. 8 | 4 July 2018
- Numerical study of the Kadomtsev–Petviashvili equation and dispersive shock wavesProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 474, No. 2210 | 14 February 2018
- On Whitham and Related EquationsStudies in Applied Mathematics, Vol. 140, No. 2 | 7 December 2017
- Shock Waves in Dispersive Hydrodynamics with Nonconvex DispersionSIAM Journal on Applied Mathematics, Vol. 77, No. 1 | 3 January 2017
- Damping to prevent the blow-up of the korteweg-de vries equationCommunications on Pure & Applied Analysis, Vol. 16, No. 4 | 1 Jan 2017
- On critical behaviour in generalized Kadomtsev–Petviashvili equationsPhysica D: Nonlinear Phenomena, Vol. 333 | 1 Oct 2016
- On the generation of dispersive shock wavesPhysica D: Nonlinear Phenomena, Vol. 333 | 1 Oct 2016
- Numerical study of the long wavelength limit of the Toda latticeNonlinearity, Vol. 28, No. 8 | 17 July 2015
- On Critical Behaviour in Systems of Hamiltonian Partial Differential EquationsJournal of Nonlinear Science, Vol. 25, No. 3 | 11 February 2015
- Numerical study of blow-up and dispersive shocks in solutions to generalized Korteweg–de Vries equationsPhysica D: Nonlinear Phenomena, Vol. 304-305 | 1 Jun 2015
- Gradient catastrophes and sawtooth solutions for a generalized Burgers equation on an intervalJournal of Geometry and Physics, Vol. 85 | 1 Nov 2014
- Fourier Splitting Method for Kawahara Type EquationsJournal of Computational Methods in Physics, Vol. 2014 | 26 Aug 2014
- Asymptotic behavior of a generalized Burgers' equation solutions on a finite intervalJournal of Physics: Conference Series, Vol. 482 | 5 March 2014
- Group classification of variable coefficient generalized Kawahara equationsJournal of Physics A: Mathematical and Theoretical, Vol. 47, No. 4 | 3 January 2014
- Numerical study of shock formation in the dispersionless Kadomtsev–Petviashvili equation and dispersive regularizationsPhysica D: Nonlinear Phenomena, Vol. 265 | 1 Dec 2013
- A numerical study of the small dispersion limit of the Korteweg–de Vries equation and asymptotic solutionsPhysica D: Nonlinear Phenomena, Vol. 241, No. 23-24 | 1 Dec 2012
- On the critical behavior in nonlinear evolutionary PDEs with small viscosityRussian Journal of Mathematical Physics, Vol. 19, No. 4 | 8 December 2012