Multi-soliton interactions and the inverse problem of wave crests

A thesis accepted by Tallinn Technical University for the Degree of Doctor of Philosophy in Natural Sciences

Pearu Peterson, MSc.

Tallinn Technical University, Estonia

Jüri Engelbrecht, DSc., Prof.

Institute of Cybernetics at Tallinn Technical University, Estonia

Embrecht van Groesen, DSc.Ir., Prof.

University of Twente, The Netherlands

Roger Grimshaw, DSc., Prof.

Loughborough University, United Kingdom

Tarmo Soomere, DSc.

Estonian Marine Institute, Estonia


The aim of this thesis is to find a detailed description of multi-soliton interactions. For that multi-soliton solutions of KdV type equations are defined, constructed, and analyzed in phase variables. As a result of the analysis, the concept of an interaction soliton is introduced and a novel decomposition of multi-soliton solutions is proposed. According to the decomposition, a multi-soliton solution is a linear superposition of solitons and interaction solitons. The concept of the interaction soliton is exploited to interpret multi-soliton interactions. A geometric representation of interaction patterns is introduced and an algorithmic way to construct the interaction patterns for an arbitrary number of solitons is proposed and implemented. All new concepts are illustrated for two-soliton interactions, examples are given also for three- and five-soliton interactions. For exemplifying models of soliton interactions, the KdV, KdV-Sawada-Kotera, and KP equations are used. As a practical application of these findings, an inverse problem of wave crests is introduced. The uniqueness of a solution to the inverse problem is proved for the KP two-soliton interactions. Sensitivity of this solution is analyzed against possible measurement errors.


The presentation of the thesis took place on November 7th, 2001, at Institute of Cybernetics at TTU, Tallinn, Estonia.



Overview: pdf (541 KB), ps.gz (766 KB)

Foreword. Introduction. Solitons and their interactions. Soliton interaction patterns. Application: The inverse problem of wave crests. Discussion. Summary. References. Acknowledgments.

Publication I: pdf (2114 KB), ps.gz (2058 KB)

P. Peterson and E. van Groesen. A direct and inverse problem for wave crests modelled by interactions of two solitons. Physica D, 141:316--332, 2000.

Publication II: pdf (526 KB), ps.gz (587 KB)

P. Peterson and E. van Groesen. Sensitivity of the inverse wave crest problem. Wave Motion, 34(4):391--399, 2001.

Publication III: pdf (155 KB), ps.gz (151 KB)

P. Peterson. Construction and decomposition of multi-soliton solutions of KdV type equations.

Publication IV: pdf (632 KB), ps.gz (1769 KB)

P. Peterson. Reconstruction of multi-soliton interactions using crest data for (2+1)-dimensional KdV type equations. Physica D, 171(4):221-235, 2002.


ISSN 1406-4723, ISBN 9985-59-233-6


Pearu Peterson, 2001

Releated talks:

P. Peterson and E. van Groesen. Wave interaction patterns and prediction of wave parameters. In Symposium on Mathematical Support for Hydrodynamic Laboratories (LabMath), Institut Teknologi Bandung, Indonesia, September 9--11, 2001.

P. Peterson and E. van Groesen. The direct and inverse problem of wave crests. In 20th International Congress of Theoretical and Applied Mechanics - ICTAM 2000, Chicago, USA, August 27--September 2, 2000