# American Institute of Mathematical Sciences

July  2012, 11(4): 1497-1522. doi: 10.3934/cpaa.2012.11.1497

## A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques

 1 School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom

Received  June 2011 Revised  September 2011 Published  January 2012

The methods of analysis based on asymptotic expansions, with a small parameter, are briefly outlined. These techniques are then applied to three examples in the theory of water waves, the aim being to demonstrate the effectiveness of this approach. Throughout, we relate this procedure to more rigorous methods.
The classical problem of water waves is formulated, and then various parameter choices are incorporated. The first problem examines the properties of small-amplitude, periodic waves over constant vorticity and, invoking the ideas of breakdown, scaling and matching, the possibility of stagnation is investigated. In the second case, a Camassa-Holm equation is derived; this is found to be relevant only for the horizontal velocity component in the flow at a specific depth. However, this special, integrable equation requires the retention of a number of terms of different asymptotic order--which is the least satisfactory way to use these methods. The final example, which shows how some new aspects of a familiar problem can be obtained by these methods, develops an asymptotic description of edge waves. This leads to a single equation that captures all aspects of the run-up pattern at a shoreline.
Citation: R. S. Johnson. A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1497-1522. doi: 10.3934/cpaa.2012.11.1497
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##### References:
 [1] John Boyd. Strongly nonlinear perturbation theory for solitary waves and bions. Evolution Equations & Control Theory, 2019, 8 (1) : 1-29. doi: 10.3934/eect.2019001 [2] Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607 [3] Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239 [4] Shu-Ming Sun. Existence theory of capillary-gravity waves on water of finite depth. Mathematical Control & Related Fields, 2014, 4 (3) : 315-363. doi: 10.3934/mcrf.2014.4.315 [5] Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103 [6] John M. Hong, Cheng-Hsiung Hsu, Bo-Chih Huang, Tzi-Sheng Yang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1501-1526. doi: 10.3934/cpaa.2013.12.1501 [7] Walter A. Strauss. Vorticity jumps in steady water waves. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1101-1112. doi: 10.3934/dcdsb.2012.17.1101 [8] Vera Mikyoung Hur. On the formation of singularities for surface water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1465-1474. doi: 10.3934/cpaa.2012.11.1465 [9] Jerry L. Bona, Henrik Kalisch. Models for internal waves in deep water. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 1-20. doi: 10.3934/dcds.2000.6.1 [10] Martina Chirilus-Bruckner, Guido Schneider. Interaction of oscillatory packets of water waves. Conference Publications, 2015, 2015 (special) : 267-275. doi: 10.3934/proc.2015.0267 [11] Xiaoyu Zeng. Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1749-1762. doi: 10.3934/dcds.2017073 [12] Nghiem V. Nguyen, Zhi-Qiang Wang. Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1005-1021. doi: 10.3934/dcds.2016.36.1005 [13] Calin Iulian Martin. A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 387-404. doi: 10.3934/dcds.2017016 [14] Anca-Voichita Matioc. On particle trajectories in linear deep-water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1537-1547. doi: 10.3934/cpaa.2012.11.1537 [15] Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 599-628. doi: 10.3934/dcds.2013.33.599 [16] Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano. Structural stability for the splash singularities of the water waves problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 4997-5043. doi: 10.3934/dcds.2014.34.4997 [17] Mats Ehrnström, Gabriele Villari. Recent progress on particle trajectories in steady water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 539-559. doi: 10.3934/dcdsb.2009.12.539 [18] David M. Ambrose, Jerry L. Bona, David P. Nicholls. Well-posedness of a model for water waves with viscosity. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1113-1137. doi: 10.3934/dcdsb.2012.17.1113 [19] David Henry, Bogdan--Vasile Matioc. On the regularity of steady periodic stratified water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1453-1464. doi: 10.3934/cpaa.2012.11.1453 [20] Gerhard Tulzer. On the symmetry of steady periodic water waves with stagnation points. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1577-1586. doi: 10.3934/cpaa.2012.11.1577

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