Nonlinear Scalar Equations

Xiaoping Xu
Additional contact information
Xiaoping Xu: Academy of Mathematics and System Science, Institute of Mathematics

Chapter Chapter 5 in Algebraic Approaches to Partial Differential Equations, 2013, pp 141-178 from Springer

Abstract: Abstract First we perform a symmetry analysis on the KdV equation, and obtain the traveling-wave solutions in terms of the functions ℘(z),tan2 z,coth2 z, and $\operatorname{cn}^{2}(z\mid m)$ , respectively. The two-soliton solution is also determined via the Hirota bilinear presentation. The KP equation is an extension of the KdV equation. We use symmetry transformations to extend the solutions of the KdV equation to the more sophisticated solutions of the KP equation. Then we solve the KP equation for solutions that are polynomial in a spatial variable, and obtain another type of solution. We also find the Hirota bilinear presentation of the KP equation and obtain the “lump” solution. Using the stable range of the nonlinear term, symmetry transformations, and the generalized power series method, we find a family of singular solutions with seven arbitrary parameter functions of t and a family of analytic solutions with six arbitrary parameter functions of t for the equation of transonic gas flows. Similar solutions are also obtained for the short-wave equation and the Khokhlov–Zabolotskaya equation in nonlinear acoustics of bounded bundles. The symmetry transformations and two new families of exact solutions with multiple parameter functions for the geopotential equation are derived.

Keywords: Symmetry Transformation; Nonlinear Acoustics; Weak Shock Wave; Sophisticated Solution; Nonlinear Reflection (search for similar items in EconPapers)
Date: 2013
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-36874-5_5

Ordering information: This item can be ordered from
http://www.springer.com/9783642368745

DOI: 10.1007/978-3-642-36874-5_5

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().