# Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA

@article{Hereman1990SolitaryWS, title={Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA}, author={Willy A. Hereman and Masanori Takaoka}, journal={Journal of Physics A}, year={1990}, volume={23}, pages={4805-4822} }

The direct algebraic method for constructing travelling wave solutions on nonlinear evolution and wave equations has been generalized and systematized. The class of solitary wave solutions is extended to analytic (rather than rational) functions of the real exponential solutions of the linearized equation. Expanding the solutions in an infinite series in these real exponentials, an exact solution of the nonlinear PDE is obtained, whenever the series can be summed. Methods for solving the…

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## References

SHOWING 1-10 OF 49 REFERENCES

### Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method

- Mathematics
- 1986

The authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear…

### The Construction of Implicit and Explicit Solitary Wave Solutions of Nonlinear Partial Differential Equations.

- Mathematics, Physics
- 1987

Abstract : By means of easy examples, such as the Korteweg-de Vries, the Harry Dym, the sine-Gordon equations, and the Hirota coupled system, it is shown how nonlinear partial differential equations…

### Exact and explicit solitary wave solutions for the generalised fisher equation

- Mathematics, Physics
- 1988

### A Straightforward Method for Finding Implicit Solitary Wave Solutions of Nonlinear Evolution and Wave Equations

- Mathematics
- 1990

The authors present a straightforward method for finding implicit solutions for nonlinear evolution and wave equations. The method is illustrated by finding implicit single solitary wave solutions…

### Weak Nonlinear Dispersive Waves: A Discussion Centered Around the Korteweg–De Vries Equation

- Mathematics
- 1972

This article is concerned with wave propagation processes where some balance occurs in the competition between a nonlinear effect and a higher order derivative effect which might be of a dispersive…

### On series expansions giving closed-form solutions of Korteweg-de Vries-like equations

- Physics
- 1990

Korteweg–deVries (KdV)-like equations with higher-degree nonlinearity are solved by a direct algebraic technique due to Hereman et al. [J. Phys. A, 19 (1986), pp. 607–628]. For two KdV-like…

### Nonlinear systems of partial differential equations in applied mathematics

- Mathematics
- 1986

These two volumes of 47 papers focus on the increased interplay of theoretical advances in nonlinear hyperbolic systems, completely integrable systems, and evolutionary systems of nonlinear partial…