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Published online by Cambridge University Press:  30 June 2017

Thomas J. Bridges
University of Surrey
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[1] M.B., Abbott. Computational Hydraulics, Pitman Publishers: London (1979).
[2] M.J., Ablowitz. Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons, Cambridge University Press: Cambridge (2011).
[3] M.J., Ablowitz & D.J., Benney. The evolution of multi-phase modes for nonlinear dispersive waves, Stud. Appl. Math. 49, 225–238 (1970).Google Scholar
[4] M.J., Ablowitz & P., Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering, Lecture Notes in Mathematics 149, Cambridge University Press: Cambridge (1991).
[5] M.J., Ablowitz & H., Segur. On the evolution of packets of water waves, J. Fluid Mech. 92 691–715 (1979).Google Scholar
[6] T.R., Akylas. Three-dimensional long water-wave phenomena, Ann. Rev. Fluid Mech. 26 191–210 (1994).Google Scholar
[7] I.M., Anderson. Introduction to the variational bicomplex, in Mathematical Aspects of Classical Field Theory, Contemp. Math. 132, 51–73 (1992).Google Scholar
[8] V.I., Arnol'D, V.V., Kozlov & A.I., Neishtadt. Mathematical Aspects of Classical and Celestial Mechanics, Encycl. Math. Sci. 3, Springer: Berlin (1993).
[9] C., Baesens & R.S., Mackay. Uniformly travelling water waves from a dynamical systems viewpoint: some insights into bifurcation from Stokes' family, J. Fluid Mech. 241 333–347 (1992).Google Scholar
[10] T.B., Benjamin. Impulse, flow force, and variational principles, IMA J. Appl. Math. 32 3–68 (1984).Google Scholar
[11] T.B., Benjamin & M.J., Lighthill. On cnoidal waves and bores, Proc. Roy. Soc. Lond. A 224 448–460 (1954).Google Scholar
[12] T.B., Benjamin & P.J., Olver. Hamiltonian structure, symmetries and conservation laws for water waves, J. Fluid Mech. 125 137–185 (1982).Google Scholar
[13] S., Benzoni-GAVAGE, C., Mietka & L.M., Rodrigues. Co-periodic stability of periodic waves in some Hamiltonian PDEs, Nonlinearity 29 3241–3308 (2016).Google Scholar
[14] S., Benzoni-GAVAGE, P., Noble & L.M., Rodrigues. Slow modulations of periodic waves in Hamiltonian PDEs, with application to capillary fluids, J. Nonl. Sci. 24 711–768 (2014).Google Scholar
[15] A.J., Bernoff. Slowly varying fully nonlinear wavetrains in the GinzburgLandau equation, Physica D 30 363–381 (1988).Google Scholar
[16] F., BÉTHUEL, P., Gravejat, J.-C., Saut & D., Smets. On the Korteweg de Vries long-wave approximation of the Gross–Pitaevskii equation I, Int. Math. Res. Not. 14 2700–2748 (2009).Google Scholar
[17] F., BÉTHUEL, P., Gravejat, J.-C., Saut & D., Smets. On the Korteweg de Vries long-wave approximation of the Gross–Pitaevskii equation II, Comm. PDEs 35 113–164 (2010).Google Scholar
[18] E., Binz, J., Sniatycki & H.R., Fischer. Geometry of Classical Fields, North-Holland, Elsevier Science Publishers: New York (1988).
[19] G., Biondini, G.A., El, M.A., Hoefer & P.D., Miller. Dispersive Hydrodynamics: Preface, Physica D 333 1–5 (2016).Google Scholar
[20] G., Biondini & D.E., Pelinovsky. Kadomtsev-Petviashvili equation, Scholarpedia 3 6539 (2008).Google Scholar
[21] F., Bisshopp. A modified stationary principle for nonlinear waves, J. Diff. Eqns. 5 592–605 (1969).Google Scholar
[22] J.L., Bona, M., Chen & J.C., Saut. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: the nonlinear theory, Nonlinearity 17 925–952 (2004).Google Scholar
[23] J.L., Bona, M., Chen & J.C., Saut. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: derivation and linear theory, J. Nonlinear Sci. 12 283–318 (2002).Google Scholar
[24] J.V., Boussinesq. Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contene dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pure Appl. 17 55–108 (1872).Google Scholar
[25] H.R., Brand. Phase dynamics – the concept and some recent developments, pp. 25–34 in Patterns, Defects, and Material Instabilities, eds. D., Walgraef & N.M., Ghonlem, Kluwer Academic Publishing Netherlands (1990).
[26] F.P., Bretherton & C.J.R., Garrett. Wavetrains in inhomogeneous moving media, Proc. Roy. Soc. Lond. A 302 529–554 (1968).Google Scholar
[27] T.J., Bridges. Geometric aspects of degenerate modulation equations, Stud. Appl. Math. 91 125–151 (1994).Google Scholar
[28] T.J., Bridges. Periodic patterns, linear instability, symplectic structure and mean-flow dynamics for three-dimensional surface waves, Phil. Trans. Roy. Soc. Lond. A 354 533–574 (1996).Google Scholar
[29] T.J., Bridges. Multi-symplectic structures and wave propagation, Math. Proc. Camb. Phil. Soc. 121 147–190 (1997).Google Scholar
[30] T.J., Bridges. A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities, Proc. Roy. Soc. Lond. A 453 1365–1395 (1997).Google Scholar
[31] T.J., Bridges. Transverse instability of solitary-wave states of the water-wave problem, J. Fluid Mech. 439 255–278 (2001).Google Scholar
[32] T.J., Bridges. Superharmonic instability, homoclinic torus bifurcation and water-wave breaking, J. Fluid Mech. 505 153–162 (2004).Google Scholar
[33] T.J., Bridges. Canonical multi-symplectic structure on the total exterior algebra bundle, Proc. Roy. Soc. Lond. A 462 1531–1551 (2006).Google Scholar
[34] T.J., Bridges. Geometric lift of paths of Hamiltonian equilibria and homoclinic bifurcation, Int. J. Bifurcation Chaos 22 1250304 (2012).Google Scholar
[35] T.J., Bridges. Emergence of unsteady dark solitary waves from coalescing spatially-periodic patterns, Proc. Roy. Soc. Lond. A 468 3784–3803 (2012).Google Scholar
[36] T.J., Bridges. A universal form for the emergence of the Korteweg-de Vries equation, Proc. Roy. Soc. Lond. A 469 20120707 (2013).Google Scholar
[37] T.J., Bridges. Bifurcation from rolls to multi-pulse planforms via reduction to a parabolic Boussinesq model, Physica D 275 8–18 (2014).Google Scholar
[38] T.J., Bridges. Emergence of dispersion in shallow water hydrodynamics via modulation of uniform flow, J. Fluid Mech. 761 R1-R9 (2014).Google Scholar
[39] T.J., Bridges. Breakdown of the Whitham modulation theory and the emergence of dispersion, Stud. Appl. Math. 135 277–294 (2015).Google Scholar
[40] T.J., Bridges & G., Derks. Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry, Proc. Roy. Soc. Lond. A 455 2427– 2469 (2001).Google Scholar
[41] T.J., Bridges & N.M., Donaldson. Degenerate periodic orbits and homoclinic torus bifurcation, Phys. Rev. Lett. 95 104301 (2005).Google Scholar
[42] T.J., Bridges & N.M., Donaldson. Secondary criticality of water waves. Part 1: Definition, bifurcation and solitary waves, J. Fluid Mech. 565 381–417 (2006).Google Scholar
[43] T.J., Bridges & N.M., Donaldson. Criticality manifolds and their role in the generation of solitary waves for two-layer flow with a free surface, Euro. J. Mech. B/Fluids 28 117–126 (2009).Google Scholar
[44] T.J., Bridges & N.M., Donaldson. Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves, Phys. Fluids 19 072111 (2007).Google Scholar
[45] T.J., Bridges & N.M., Donaldson. Variational principles for water waves from the viewpoint of a time-dependent moving mesh,Mathematika 57, 147–173 (2011).Google Scholar
[46] T.J., Bridges, P.E., Hydon, & J.K., Lawson. Multisymplectic structures and the variational bicomplex, Math. Proc. Camb. Phil. Soc. 148, 159–178 (2010).Google Scholar
[47] T.J., Bridges & F.E., Laine-PEARSON. Multisymplectic relative equilibria, multiphase wavetrains, and coupled NLS equations, Stud. Appl. Math. 107, 137–155 (2001).Google Scholar
[48] T.J., Bridges, J., Pennant & S., Zelik. Degenerate hyperbolic conservation laws with dissipation: reduction to and validity of a class of Burgers-type equations, Arch. Rat.Mech. Anal. 214 671–716 (2014).Google Scholar
[49] T.J., Bridges & D.J., Ratliff. Double criticality and the two-way Boussinesq equation in stratified shallow water hydrodynamics, Phys. Fluids 28 0162103 (2016).Google Scholar
[50] H.W., Broer, S.-N., Chow, Y.-I., Kim & G., Vegter. The Hamiltonian double-zero eigenvalue, Fields Inst. Comm. 4, 1–19 (1995).Google Scholar
[51] J.C., Bronski, V.M., Hur & M.A., Johnson. Modulational instability in equations of KdV type, in New Approaches to Nonlinear Waves, ed. E. TOBISCH, Lect. Notes Phys. 908 83–133 (2016).Google Scholar
[52] B., Buffoni, M.D., Groves, S.M., Sun & E. WAHLÉN. Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves, J. Diff. Eqns. 254 1006–1096 (2013).Google Scholar
[53] F., Cantrijn, A., Ibort & M. DE, LeÓN. On the geometry of multisymplectic manifolds, J. Austral. Math. Soc. (Ser. A) 66 303–330 (1999).Google Scholar
[54] H., ChatÉ & P., Manneville. Phase turbulence, pp. 67–74 in Turbulence: A Tentative Dictionary, ed. P., Tabeling & O., Cardoso, NATO ASI Series B 341, Plenum Press: New York (1994).
[55] M., Chirlius-BRUCKNER, W.-P., DÜLL & G., Schneider. Validity of the KdV equation for the modulation of periodic traveling waves in the NLS equation, J. Math. Anal. Appl. 414 166–175 (2014).Google Scholar
[56] D., Chiron & F., Rousset. The KdV/KP-I limit of the nonlinear Schrödinger equation, SIAM J. Math. Anal. 42 64–96 (2010).Google Scholar
[57] V.H., Chu & C.C., Mei. On slowly-varying Stokes waves, J. Fluid Mech. 41 873–887 (1970).Google Scholar
[58] R.R., Cordeiro & R. VIEIRA, Martins. Effect of Krein signatures on the stability of relative equilibria, Cel. Mech. Dyn. Astron. 61 217–238 (1995).Google Scholar
[59] M.C., Cross & H., Greenside. Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press: Cambridge (2009).
[60] M.C., Cross & A.C., Newell. Convection patterns in large aspect ratio systems, Physica D 10 299–328 (1984).Google Scholar
[61] R., Cushman & J.A., Sanders. Invariant theory and normal form of Hamiltonian vectorfields with nilpotent linear part, Can. Math. Soc. Conf. Proc. 8 353–371 (1987).Google Scholar
[62] R.W.R., Darling. Differential Forms and Connections, Cambridge University Press: Cambridge (1994).
[63] B., Deconinck. The initial-value problem for multiphase solutions of the Kadomtsev-Petviashvili equation, PhD Thesis, AppliedMathematics, University of Colorado (1998).
[64] G., Derks & G., Gottwald. A robust numerical method to study oscillatory instability of gap solitary waves, SIAM J. Appl. Dyn. Syst. 4 140–158 (2005).Google Scholar
[65] M.W., Dingemans. Water wave propagation over uneven bottoms. Part 2 – Non-linear wave propagation, World Scientific Publisher: Singapore (1997).
[66] V.D., Djordjevic & L.G., Redekopp. On two-dimensional packets of capillary-gravity waves, J. Fluid Mech. 79 703–714 (1977).Google Scholar
[67] A., Doelman, B., Sandstede, A., Scheel & G., Schneider. The dynamics of modulated wave trains, AMS Memoirs 934, AmericanMathematical Society: Providence (2009).
[68] P.G., Drazin & R.S., Johnson. Solitons: An Introduction, Cambridge University Press: Cambridge (1989).
[69] B.A., Dubrovin, R., Flickinger & H., Segur. Three-phase solutions of the Kadomtsev-Petviashvili equation, Stud. Appl. Math. 99 137–203 (1997).Google Scholar
[70] W.-P., DÜLL & G., Schneider. Validity of Whitham's equations for the modulation of periodic traveling waves in the NLS equation, J. Nonl. Sci. 19 453–466 (2009).Google Scholar
[71] W.-P., DÜLL, G., Schneider & C.E., Wayne. Justification of the nonlinear Schrödinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth, Arch. Rat.Mech. Anal. 220 543–602 (2016).Google Scholar
[72] K.B., Dysthe. A note on the application of Whitham's method to nonlinear waves in dispersive media, J. Plasma Phys. 11 63–76 (1974).Google Scholar
[73] W., Eckhaus & G., Iooss. Strong selection or rejection of spatially periodic patterns in degenerate bifurcations, Physica D 39 124–146 (1989).Google Scholar
[74] G.A., El. Resolution of a shock in hyperbolic systems modified by weak dispersion, Chaos 15 037103 (2005).Google Scholar
[75] G.A., El, A.L., Krylov & S., Venakides. Unified approach to KdV modulations, Comm. Pure Appl. Math. 54 1243–1270 (2001).Google Scholar
[76] N.M., Ercolani, R., Indik, A.C., Newell & T., Passot. The geometry of the phase-diffusion equation, J. Nonl. Sci. 10 223–274 (2000).Google Scholar
[77] A.D., Gilbert. An examination of Whitham's exact averaged variational principle, Proc. Camb. Phil. Soc. 76 327–344 (1974).Google Scholar
[78] J.E., Gilbert & M., Murray. Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studes in Advanced Mathematics, 26, Cambridge University Press: Cambridge (1991).Google Scholar
[79] YU.G., Gladush, G.A., El, A., Gammal & A.M., Kamchatnov. Radiation of linear waves in the stationary flow of a Bose-Einstein condensate past an obstacle, Phys. Rev. A 75 022619 (2007).Google Scholar
[80] M.J., Gotay, J., Isenberg, J.E., Marsden & R., Montgomery. Momentum maps and classical fields. Part I: Covariant field theory, arXiv preprint physics/9801019 (1998).
[81] R.H.J., Grimshaw. Weakly non-linear, slowly varying waves and their instabilities, Proc. Camb. Phil. Soc. 72 95–104 (1972).Google Scholar
[82] R.H.J., Grimshaw. Models for long-wave instability due to a resonance between two waves, in Trends in Appl. of Math. to Mech., eds. G., Iooss, O., Gues & A., Nouri, Chapman & Hall/CRC Monographs 106 183–192 (2000).
[83] R.H.J., Grimshaw & P., Christodoulides. Short-wave instability in a three-layer stratified shear flow, Quart. J. Mech. Appl. Math. 54 375–388 (2001).Google Scholar
[84] R.H.J., Grimshaw & Y., Skyrnnikov. Long-wave instability in a three-layer stratified shear flow, Stud. Appl. Math. 108 77–88 (2002).Google Scholar
[85] R.H.J., Grimshaw. Korteweg-de Vries equation, in Nonlinear Waves in Fluids: Recent Advances and Modern Applications, ed. R.H.J., Grimshaw, CISM: Springer-Verlag, pp. 1–28 (2005.
[86] E. VAN, Groesen. On variational formulation of periodic and quasi-periodic Hamiltonian motions as relative equilibria, in Geometry and Analysis in Nonlinear Dynamics, eds. H., Broer & F., Takens, Longman/Pitman, 22–33 (1992).
[87] A.V., Gurevich & L.P., Pitaevskii. Nonstationary structure of a collisionless shock wave, Sov. Phys. JETP 38 291–297 (1974).Google Scholar
[88] K., Habermann & L., Habermann. Introduction to Symplectic Dirac Operators, Lect. Notes Math. 1887, Springer-Verlag: Berlin (2006).Google Scholar
[89] J., Hammack, N., Scheffner & H., Segur. Two-dimensional periodic waves in shallow water, J. Fluid Mech. 209 567–589 (1989).Google Scholar
[90] J., Hammack, D., Mccallister, N., Scheffner & H., Segur. Twodimensional periodic waves in shallow water. Part 2. Asymmetric waves, J. Fluid Mech. 285 95–122 (1995).Google Scholar
[91] W.D., Hayes. Conservation of action and modal wave action, Proc. Roy. Soc. Lond. A 320 187–208 (1970).Google Scholar
[92] W.D., Hayes. Group velocity and nonlinear dispersive wave propagation, Proc. Roy. Soc. Lond. A 332 199–221 (1973).Google Scholar
[93] F.M., Henderson. Open Channel Flow, MacMillan Publishers: London (1966).
[94] R., Hirota. Exact N-soliton solutions of the wave equation of long waves in shallow water and in nonlinear lattices, J. Math. Phys. 14 810–814 (1973).Google Scholar
[95] D.D., Holm. Geometric Mechanics, Part I: Dynamics and Symmetry, Imperial College Press: London (2008).
[96] H.W., Hoogstraten & R. VAN DER, Heide. A perturbation method for nonlinear dispersive waves with an application to water waves, J. Eng. Math. 6 341–353 (1972).Google Scholar
[97] M.S., Howe. Nonlinear theory of open-channel steady flow past a solid surface of finite-wave-group shape, J. Fluid Mech. 30 497–512 (1967).Google Scholar
[98] M.S., Howe. Phase jumps, J. Fluid Mech. 32 779–789 (1968).Google Scholar
[99] R.H., Hoyle. Pattern Formation: An Introduction to Methods, Cambridge University Press: Cambridge (2006).
[100] P.E., Hydon. Multisymplectic conservation laws for differential and differential-difference equations, Proc. Roy. Soc. Lond. A 461 1627–1637 (2005).Google Scholar
[101] A., Ibort & C. MARTINEZ, Ontalba. Periodic orbits of Hamiltonian systems and symplectic reduction, J. Phys. A: Math. Gen. 29 675–687 (1996).Google Scholar
[102] E., Infeld. On the three-dimensional generalization of the Boussinesq and Korteweg-de Vries equations, Quart. Appl. Math. 38 377–387 (1980).Google Scholar
[103] E., Infeld. Three-dimensional stability of Korteweg-de Vries waves and solitons, III. Lagrangian methods, KdV with positive dispersion, Acta Phys. Polon. A60 623–643 (1981).Google Scholar
[104] E., Infeld & G., Rowlands. Three-dimensional stability of Korteweg-de Vries waves and solitons, II, Acta Phys. Polon. A56 329–332 (1979).Google Scholar
[105] E., Infeld & G., Rowlands. Nonlinear Waves, Solitons, and Chaos, Cambridge University Press: Cambridge, 2nd Edition (2000).
[106] E., Infeld, G., Rowlands & M., Hen. Three-dimensional stability of Korteweg-de Vries waves and solitons, Acta Phys. Polon. A54 131–139 (1978).Google Scholar
[107] E., Infeld, G., Rowlands & A., Senatorski. Instabilities and oscillations of one- and two-dimensional Kadomtsev–Petviashvili waves and solitons, Proc. Roy. Soc. Lond. A 455 4363–4381 (1992).Google Scholar
[108] G., Iooss. Global characterization of the normal form for a vector field near a closed orbit, J. Diff. Eqns. 76 47–76 (1988).Google Scholar
[109] E.M. DE, Jager. 2011 On the origin of the Korteweg-de Vries equation, Forum der Berliner Mathematischen Gesellschaft 19, 171–195 (2011). ( Scholar
[110] P.A.E.M., Janssen. Stability of steep gravity waves and the average Lagrangian method, KNMI Preprint (1989).
[111] R.S., Johnson. Water waves and Korteweg-de Vries equations, J. Fluid Mech. 97 701–719 (1980).Google Scholar
[112] R.S., Johnson. A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press: Cambridge (1997).
[113] B., Kabil & L.M., Rodrigues. Spectral validation of the Whitham equations for periodic waves of lattice dynamical systems, J. Diff. Eqns. 260 2994–3028 (2016).Google Scholar
[114] B.B., Kadomtsev & V.I., Petviashvili. On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 15 539–541 (1970).Google Scholar
[115] A.M., Kamchatnov. Nonlinear PeriodicWaves and TheirModulations,World Scientific: Singapore (2000).
[116] I., Kanatchikov. Canonical structure of classical field theory in the polymomentum phase space, Rep. Math. Phys. 41 49–90 (1998).Google Scholar
[117] T., Kapitula. The Krein signature, Krein eigenvalues, and the Krein oscillation theorem, Indiana U. Math. J. 59 1245–1276 (2010).Google Scholar
[118] N., Karjanto & E. VAN, Groesen. Note on wavefront dislocation in surface water waves, Phys. Lett. A 371 173–179 (2007).Google Scholar
[119] T., Kataoka. On the superharmonic instability of surface gravity waves on fluid of finite depth, J. Fluid Mech. 547 175–184 (2006).Google Scholar
[120] B., Kim & T.R., Akylas. On gravity-capillary lumps, J. Fluid Mech. 540 337– 351 (2005).Google Scholar
[121] P., Kirrmann, G., Schneider & A., Mielke. The validity of modulation equations for extended systems with cubic nonlinearities, Proc. R. Soc. Edin. 122A 85–91 (1992).Google Scholar
[122] Y.S., Kivshar. Dark-soliton dynamics and shock waves induced by the stimulated Raman effect in optical fibers, Phys. Rev A 42 1757–1761 (1990).Google Scholar
[123] N., Kopell & L.N., Howard. Slowly varying waves and shock structures in reaction-diffusion equations, Stud. Appl. Math. 56 95–145 (1977).Google Scholar
[124] D.J., Korteweg & G. DE, Vries. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. 39 422–443 (1895).Google Scholar
[125] Y., Kuramoto. Chemical Oscillations, Waves and Turbulence, Springer- Verlag: New York (1984).
[126] Y., Kuramoto & T., Tsuzuki. Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys. 55 356– 369 (1976).Google Scholar
[127] E.A., Kuznetsov & S.K., Turitsyn. Two- and three-dimensional solitons in weakly dispersive media, Sov. Phys. JETP 55 844–847 (1982).Google Scholar
[128] J.K., Lawson. A frame-bundle generalization of multisymplectic geometry, Rep. Math. Phys. 45 183–205 (2000).Google Scholar
[129] M. DE, LeÓN, M., Mclean, L.K., Norris, A., Rey-ROCA & M., Salgado. Geometric structures in field theory, Preprint math-ph/0208036 (2002).
[130] M.J., Lighthill. A discussion on nonlinear theory of wave propagation in dispersive systems, Proc. Roy. Soc. Lond. A 299 1–145 (1957).Google Scholar
[131] M.J., Lighthill. Contribution to the theory of waves in non-linear dispersive systems, J. Inst.Math. Appl. 1 269–306 (1965).Google Scholar
[132] P., Lounesto. Clifford Algebras and Spinors, LMS Lecture Notes 239, Cambridge University Press: Cambridge (1997).
[133] J.C., Luke. A perturbation method for nonlinear dispersive wave problems, Proc. Roy. Soc. Lond. A 292 403–412 (1966).Google Scholar
[134] J.C., Luke. A variational principle for a fluid with a free surface, J. Fluid Mech. 27 395–397 (1967).Google Scholar
[135] W.-X., Ma. Comment on the 3+1 dimensional Kadomtsev-Petviashvili equation, Comm. Nonl. Sci. Numer. Sim. 16 2663–2666 (2011).Google Scholar
[136] J.H., Maddocks. Stability and folds, Arch. Rat. Mech. Anal. 99 301–328 (1987).Google Scholar
[137] J.H., Maddocks & R.L., Sachs. On the stability of KdV multi-solitons, Comm. Pure Appl. Math. 46 867–901 (1993).Google Scholar
[138] J.R., Magnus & H., Neudecker. Matrix Differential Calculus, Wiley- Blackwell: London (1988).
[139] J.E., Marsden. Lectures on Mechanics, London Mathematical Society Lecture Notes 174, Cambridge University Press.
[140] J.E., Marsden, R., Montgomery & T., Ratiu. Reduction, symmetry, and phases in mechanics, Mem. American Mathematical Society 88, No. 436 Amer. Math. Soc: Providence (1990).Google Scholar
[141] J.E., Marsden & T., Ratiu. Introduction to Mechanics and Symmetry, Springer: Berlin (1999).
[142] P.C., Matthews & S.M., Cox. Pattern formation with a conservation law, Nonlinearity 13 1293–1320 (2000).Google Scholar
[143] C.C., Mei. The Applied Dynamics of Ocean Surface Waves, World Scientific: Singapore (1989).
[144] K.R., Meyer & G.R., Hall. Introduction to Hamiltonian dynamical systems and the n-body problem, Appl. Math. Sci. 90, Springer-Verlag: New York (1992).
[145] A., Mielke. A spatial center manifold approach to steady bifurcations from spatially periodic patterns, Pitman Res. Notes in Math. 352 209–277, Addison Wesley Longman Ltd: Essex (1996).
[146] A.A., Minzoni & N.F., Smyth. Modulation theory, dispersive shock waves and Gerald Beresford Whitham, Physica D 333 6–10 (2016).Google Scholar
[147] J., Montaldi. Relative equilibria and conserved quantities in symmetric Hamiltonian systems, in Peyresq Lectures on Nonlinear Phenomena, eds. R., Kaiser & J., Montaldi, World Scientific: River Edge NJ 239–280 (2000).
[148] S., Morita Geometry of Differential Forms, American Mathematical Society: Providence (2001).
[149] A.C., Newell & Y., Pomeau. Phase diffusion and phase propagation: interesting connections, Physica D 87 216–232 (1995.Google Scholar
[150] P.K., Newton & J.B., Keller. Stability of periodic plane waves, SIAM J. Appl. Math. 47 959–964 (1987).Google Scholar
[151] L.K., Norris. Generalized symplectic geometry on the frame bundle of a manifold, Proc. Symp. Pure Math. 54, Amer. Math. Soc. Publ. 435–465 (1993).
[152] A.R., Osborne. Nonlinear Ocean Waves and the Inverse Scattering Transform, Academic Press, Elsevier: London (2010).
[153] C., Paufler & H., RÖMER. Geometry of Hamiltonian n-vector fields in multisymplectic field theory, J. Geom. Phys. 44 52–69 (2002).Google Scholar
[154] D.E., Pelinovsky, Y.A., Stepanyants & Y.S., Kivshar. Self-focusing of plane dark solitons in nonlinear defocusing media, Phys. Rev. E 51 5016–5026 (1995).Google Scholar
[155] D.H., Peregrine. Wave jumps and caustics in the propagation of finiteamplitude water waves, J. Fluid Mech. 136 435–452 (1983).Google Scholar
[156] Y., Pomeau & P., Manneville. Stability and fluctuations of a spatially periodic convection flow, J. Phys. (Paris) Lett. 40 609–612 (1979).Google Scholar
[157] D.J., Ratliff. Phase dynamics of periodic wavetrains leading to the fifth-order KdV equation, Preprint, University of Surrey (2017).
[158] D.J., Ratliff. Multiphase modulation, universality, and nonlinear waves, PhD Thesis, University of Surrey (2017).
[159] D.J., Ratliff & T.J., Bridges. Phase dynamics of periodic waves leading to the Kadomtsev-Petviashvili equation in 3+1 dimensions, Proc. Roy. Soc. Lond. A 471 20150137 (2015).Google Scholar
[160] D.J., Ratliff & T.J., Bridges. Whitham modulation equations, coalescing characteristics, and dispersive Boussinesq dynamics, Physica D 333 107–116 (2016).Google Scholar
[161] M., Roberts, C., Wulff & J., Lamb. Hamiltonian systems near relative equilibria, J. Diff. Eqns. 179 562–604 (2002).Google Scholar
[162] P.G., Saffman. The superharmonic instablility of finite-amplitude water waves, J. Fluid Mech. 159 169–174 (1985).Google Scholar
[163] B., Sanstede & A., Scheel. Defects in oscillatory media: towards a classification, SIAM J. Appl. Dyn. Sys. 3 1–68 (2004).Google Scholar
[164] D.J., Saunders. The Geometry of Jet Bundles, Cambridge University Press: Cambridge (1989).
[165] G., Schneider & C.E., Wayne. The long wave limit for the water wave problem. I. The case of zero surface tension, Comm. Pure Appl.Math. 53 1475–1535 (2000).Google Scholar
[166] YU. V., Sedletsky. Addition of dispersive terms to the method of averaged Lagrangian, Phys. Fluids 24 062105 (2012).Google Scholar
[167] YU. V., Sedletsky. Dispersive terms in the averaged Lagrangian method, Int. J. Nonl. Mech. 57 140–145 (2013).Google Scholar
[168] YU. V., Sedletsky. Variational approach to the derivation of the Davey- Stewartson equation, Fluid. Dyn. Res. 48 015506 (2016).Google Scholar
[169] A., Senatorski & E., Infeld. Breakup of two-dimensional into threedimensional Kadomtsev-Petviashvili solitons, Phys. Rev. E 57 6050–6055 (1998).Google Scholar
[170] A.E., Taylor & D.C., Lay. Introduction to Functional Analysis, Krieger Publishers (1986).
[171] W.M., Tulczyjew. The Euler-Lagrange Resolution, Lecture Notes in Mathematics 836, Springer-Verlag, New York (1980) pp. 22–48.
[172] J.-M., Vandenbroeck. Some new gravity waves in water of finite depth, Phys. Fluids 26 2385–2387 (1983).
[173] A.M., Vinogradov. A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints, Sov. Math. Dokl. 19 144–148 (1978).Google Scholar
[174] A.M., Vinogradov. The C-spectral sequence, Lagrangian formalism and conservation laws I, II, J. Math. Anal. Appl. 100 1–129 (1984).Google Scholar
[175] T.I., Vogel. On constrained extrema, Pacific J. Math. 176 557–561 (1996).Google Scholar
[176] A., Weinstein. Bifurcation and Hamilton's principle, Math. Z. 159 235–248 (1978).Google Scholar
[177] G.B., Whitham. A general approach to linear and nonlinear dispersive waves using a Lagrangian, J. Fluid Mech. 22 273–283 (1965).Google Scholar
[178] G. B., Whitham. Non-linear dispersion of water waves, J. Fluid Mech. 27 399–412 (1967).Google Scholar
[179] G.B., Whitham. Two-timing, variational principles and waves, J. Fluid Mech. 44 373–395 (1970).Google Scholar
[180] G.B., Whitham. Linear and Nonlinear Waves, Wiley-Interscience: New York (1974).
[181] B., Willink. The collaboration between Korteweg and de Vries – an enquiry into personalities, arXiv:0710.5227 (2007). Published as Chapter 3 of Physics as a Calling Science for Society, eds. A., Maas & H., Schatz, Leiden University Press (2013.
[182] H.C., Yuen & B.M., Lake. Nonlinear deep water waves: theory and experiment, Phys. Fluids 18 956–960 (1975).Google Scholar
[183] V.E., Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 2 190–194 (1968).Google Scholar
[184] V.E., Zakharov & L.A., Ostrovsky. Modulation instability: the beginning, Physica D 238 540–548 (2009).Google Scholar
[185] J.A., Zufiria. Weakly nonlinear non-symmetric gravity waves on water of finite depth, J. Fluid Mech. 180 371–385 (1987).Google Scholar

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  • Thomas J. Bridges, University of Surrey
  • Book: Symmetry, Phase Modulation and Nonlinear Waves
  • Online publication: 30 June 2017
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  • References
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  • Book: Symmetry, Phase Modulation and Nonlinear Waves
  • Online publication: 30 June 2017
  • Chapter DOI:
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