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Introduction to the special issue "Nonlinear wave phenomena in continuum physics: Some recent findings"
March  2019, 8(1): 1-29. doi: 10.3934/eect.2019001

## Strongly nonlinear perturbation theory for solitary waves and bions

 Department of Climate and Space Sciences and Engineering, University of Michigan, Ann Arbor, MI 48109-2143, USA

Dedicated to the memory of Christo I. Christov.

Received  October 2017 Revised  February 2018 Published  March 2019 Early access  January 2019

Fund Project: The author is supported by NSF grant DMS1521158.

Strongly nonlinear perturbation theory would seem to be an oxymoron, that is, a contradiction of terms. Nonetheless, we here describe perturbation methods for wave categories that are intrinsically nonlinear including solitons (solitary waves), bound states of solitons (bions) and spatially periodic traveling waves (cnoidal waves). Examples include the Kortweg-deVries and Benjamin-Ono equations with general power law nonlinearity and the Fifth Order KdV equation. The perturbation strategies include (ⅰ) the Gorshkov-Ostrovsky-Papko near-equal-amplitude soliton interaction theory (ⅱ) perturbation series in the Newton-homotopy parameter and (ⅲ) approximations for large values of the nonlinearity exponent. A long section places strongly nonlinear perturbation theory for waves in a larger context as a subset of unconventional perturbation expansions including phase transition theory in $4 - \epsilon$ dimensions, the $\epsilon = 1/D$ expansion where $D$ is the dimension in quantum chemistry, the renormalized quantum anharmonic oscillator, the Yakhot-Orszag expansion in the exponent of the energy spectrum in hydrodynamic turbulence, and the Newton homotopy expansion.

Citation: 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
##### References:
 [1] D. M. Ambrose and J. Wilkening, Global paths of time-periodic solutions of the Benjamin-Ono equation connecting pairs of traveling waves, Commun. Appl. Math. Comput. Sci., 4 (2009), 177-215.  doi: 10.2140/camcos.2009.4.177.  Google Scholar [2] ——, Computation of time-periodic solutions of the Benjamin-Ono equation, J. Nonlinear Sci., 20 (2010), 375-378. Google Scholar [3] P. Amore and A. Aranda, Presenting a new method for the solution of nonlinear problems, Phys. Lett. A, 316 (2003), 218-225.  doi: 10.1016/j.physleta.2003.08.001.  Google Scholar [4] P. Amore and H. M. Lamas, High order analysis of nonlinear periodic differential equations, Phys. Lett. A, 327 (2004), 158-166.  doi: 10.1016/j.physleta.2004.05.016.  Google Scholar [5] D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, Numerically Solving Polynomial Systems with Bertini, SIAM, Philadelphia, 2013.  Google Scholar [6] C. M. Bender and H. F. Jones, Calculation of low-lying energy levels in quantum mechanics, J. Phys. A, 47 (2014), 395303, 16 pp. doi: 10.1088/1751-8113/47/39/395303.  Google Scholar [7] C. M. Bender, K. A. Milton, S. S. Pinsky and L. M. Jr. Simmons, A new perturbative approach to nonlinear problems, J. Math. Phys., 30 (1989), 1447-1455.  doi: 10.1063/1.528326.  Google Scholar [8] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. 594  Google Scholar [9] C. M. Bender, A. Pelster and F. Weissbach, Boundary-layer theory, strong-coupling series, and large-order behavior, J. Math. Phys., 43 (2002), 4202-4220.  doi: 10.1063/1.1490408.  Google Scholar [10] C. M. Bender and A. Tovbis, Continuum limit of lattice approximation schemes, J. Math. Phys., 38 (1997), 3700-3717.  doi: 10.1063/1.532063.  Google Scholar [11] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.   Google Scholar [12] H. Blasius, Grenzschichten in flüssigkeiten mit kleiner reibung, Zeitschrift fur Mathematik und Physik, 56 (1908), 1-37.   Google Scholar [13] ——, The Boundary Layers in Fluids with Llttle Friction, Technical Memorandum 1256, NASA, Washington, D. C., 1950. 57, English translation by J. Venier of Grenzschichten in Flüssigkeiten mit kleiner Reibung. Google Scholar [14] J. L. Bona and H. Kalisch, Singularity formation in the generalized Benjamin-Ono equation, Discrete Continuous Dyn. Sys., 11 (2004), 27-45.  doi: 10.3934/dcds.2004.11.27.  Google Scholar [15] J. P. Boyd, A Chebyshev polynomial method for computing analytic solutions to eigenvalue problems with application to the anharmonic oscillator, Journal of Mathematical Physics, 19 (1978), 1445-1456.  doi: 10.1063/1.523810.  Google Scholar [16] ——, Solitons from sine waves: analytical and numerical methods for non-integrable solitary and cnoidal waves, Physica D, 21 (1986), 227-246. Google Scholar [17] ——, Numerical computations of a nearly singular nonlinear equation: Weakly nonlocal bound states of solitons for the Fifth-Order Korteweg-deVries equation, J. Comput. Phys., 124 (1996), 55–70. doi: 10.1006/jcph.1996.0044.  Google Scholar [18] ——, Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics: Generalized Solitons and Hyperasymptotic Perturbation Theory, vol. 442 of Mathematics and Its Applications, Kluwer, Amsterdam, 1998. 608 doi: 10.1007/978-1-4615-5825-5.  Google Scholar [19] ——, The Blasius function in the complex plane, J. Experimental Math., 8 (1999), 381–394. doi: 10.1080/10586458.1999.10504626.  Google Scholar [20] ——, Chebyshev and Fourier Spectral Methods, Dover, New York, 2001.  Google Scholar [21] ——, Deleted residuals, the QR-factored Newton iteration, and other methods for formally overdetermined determinate discretizations of nonlinear eigenproblems for solitary, cnoidal, and shock waves, J. Comput. Phys., 179 (2002), 216-237. doi: 10.1006/jcph.2002.7052.  Google Scholar [22] ——, Why Newton's method is hard for traveling waves: Small denominators, KAM theory, Arnold's linear Fourier problem, non-uniqueness, constraints and erratic failure, Math. Comput. Simul., 74 (2007), 72–81. doi: 10.1016/j.matcom.2006.10.001.  Google Scholar [23] ——, The Blasius function: Computations before computers, the value of tricks, undergraduate projects, and open research problems, SIAM Rev., 50 (2008), 791-804. doi: 10.1137/070681594.  Google Scholar [24] ——, Solving Transcendental Equations: The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation Series and Oracles, SIAM, Philadelphia, 2014. doi: 10.1137/1.9781611973525.  Google Scholar [25] J. P. Boyd and Z. Xu, Comparison of three spectral methods for the Benjamin-Ono equation: Fourier pseudospectral, rational Christov functions and Gaussian radial basis functions, Wave Motion, 48 (2011), 702-706.  doi: 10.1016/j.wavemoti.2011.02.004.  Google Scholar [26] J. P. Boyd and Z. Xu, Numerical and perturbative computations of solitary waves of the Benjamin-Ono equation with higher order nonlinearity using Christov rational basis functions, J. Comput. Phys., 231 (2012), 1216-1229.  doi: 10.1016/j.jcp.2011.10.004.  Google Scholar [27] H. Chen and J. L. Bona, Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations, Adv. Diff. Eq., 3 (1998), 51-84.   Google Scholar [28] R. E. Davis and A. Acrivos, Solitary internal waves in deep water, J. Fluid Mech., 29 (1967), 593. Google Scholar [29] K. Dutta and M. K. Nandy, Perturbative Evaluation of Universal Numbers in Homogeneous Shear Turbulence, (19). Google Scholar [30] D. Z. Goodson and D. R. Herschbach, Summation methods for dimensional perturbation-theory, Phys. Rev. A, 46 (1992), 5428-5436.   Google Scholar [31] K. A. Gorshkov, L. A. Ostrovskii and V. V. Papko, Interactions and bound states of solitons as classical particles, Soviet Physics JETP, 44 (1976), 306-311.   Google Scholar [32] K. A. Gorshkov, L. A. Ostrovskii, V. V. Papko and A. S. Pikovsky, On the existence of stationary multisolitons, Phys. Lett. A, 74 (1979), 177-179.  doi: 10.1016/0375-9601(79)90763-1.  Google Scholar [33] K. A. Gorshkov and L. A. Ostrovsky, Interactions of solitons in noninterable systems: Direct perturbation method and applications, Physica D, 3 (1981), 428-438.   Google Scholar [34] K. A. Gorshkov and V. V. Papko, Dynamic and stochastic oscillations of soliton lattices, Soviet Physics JETP, 46 (1977), 92-97.   Google Scholar [35] R. H. J. Grimshaw and B. A. Malomed, A note on the interaction between solitary waves in a singularly-perturbed Korteweg-deVries equation, J. Phys. A, 26 (1993), 4087-4091.  doi: 10.1088/0305-4470/26/16/024.  Google Scholar [36] J. He, Homotopy perturbation method: A new nonlinear analytical technique, Appl. Math. Comput., 135 (2003), 73-79.  doi: 10.1016/S0096-3003(01)00312-5.  Google Scholar [37] ——, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26 (2005), 695–700. Google Scholar [38] ——, New interpretation of homotopy perturbation method, Inter. J. Modern Phys. B, 20 (2006), 2561-2568. doi: 10.1142/S0217979206034819.  Google Scholar [39] ——, Some asymptotic methods for strongly nonlinear equations, Inter. J. Modern Phys. B, 20 (2006), 1141-1199. doi: 10.1142/S0217979206033796.  Google Scholar [40] D. Herschbach, J. Avery and O. Goscinkski, eds., Dimensional Scaling in Chemical Physics, Kluwer, Dordrecht, The Netherlands, 1992. Google Scholar [41] D. R. Herschbach, Dimensional interpolation for two-electron atoms, J. Chem. Phys., 84 (1986), 838-851.   Google Scholar [42] ——, Dimensional scaling and renormalization, Int. J. Quantum Chem., 57 (1996), 295-308. Google Scholar [43] ——, Fifty years in physical chemistry: Homage to mentors, methods, and molecules, Ann. Rev. Phys. Chem., 51 (2000), 1-39. Google Scholar [44] F. T. Hioe and E. W. Montroll, Quantum theory of anharmonic oscillators. I. Energy levels of oscillators with positive quartic anharmonicity, J. Math. Phys., 16 (1975), 1945-1955.  doi: 10.1063/1.522747.  Google Scholar [45] R. Iacono and J. P. Boyd, Simple analytic approximations to the Blasius equation, Physica D, 310 (2015), 72-78.  doi: 10.1016/j.physd.2015.08.003.  Google Scholar [46] K. Kolossovski, A. R. Champneys, A. Buryak and R. A. Sammut, Multi-pulse embedded solitons as bound states of quasi-solitons, Physica D, 171 (2002), 153-177.  doi: 10.1016/S0167-2789(02)00563-8.  Google Scholar [47] R. H. Kraichnan, An interpretation of the Yakhot-Orszag turbulence theory, Phys. Fluids, 30 (1987), 2400-2405.   Google Scholar [48] H. Nagashima and M. Kuwahara, Computer-simulation of solutions to the nonlinear wave equation $u_{t}+u u_{x} - \gamma u_{5x} = 0$, J. Phys. Soc., 50 (1981), 3792-3800.  doi: 10.1143/JPSJ.50.3792.  Google Scholar [49] R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Springer-Verlag, Heidelberg, 2d ed., 1994.  Google Scholar [50] V. E. Shamanskii, A modification of Newton's method, Ukr. Mat. Zh., 19 (1967), 133-138.   Google Scholar [51] A. Sidi, Practical Extrapolation Methods: Theory and Applications, vol. 10 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546815.  Google Scholar [52] P. M. Stevenson, Optimized perturbation theory, Phys. Rev. D, 23 (1981), 2916-2944.   Google Scholar [53] D. Swade, The Cogwheel Brain: Charles Babbage and the Quest to Build the First Computer, Viking, New York, 2001.  Google Scholar [54] G. 't Hooft, QCD perturbation theory, Phys. B, 72 (1974), 461. Google Scholar [55] F. Vinette and J. Čižek, Upper and lower bounds of the ground state energy of anharmonic oscillators using renormalized inner projection, J. Math. Phys., 32 (1991), 3392-3404.  doi: 10.1063/1.529452.  Google Scholar [56] E. J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Computer Physics Reports, 10 (1989), 189-371.   Google Scholar [57] ——, Optimized perturbation theory, Annals Phys., 246 (1996), 133-165. Google Scholar [58] E. J. Weniger, J. Čižek and F. Vinette, The summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic, sextic, and octic anharmonic oscillators using nonlinear sequence transformation, J. Math. Phys., 34 (1993), 571-609.  doi: 10.1063/1.530262.  Google Scholar [59] G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974.  Google Scholar [60] K. G. Wilson, Renormalization group and critical phenomena. 1. Renormalization group and Kadanoff scaling picture, Phys. Rev. B, 4 (1971), 3174. Google Scholar [61] K. G. Wilson, Critical exponents in 3.99 dimensions, Physica, 73 (1974), 119-128.   Google Scholar [62] K. G. Wilson and M. E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett., 28 (1972), 240-243.   Google Scholar [63] E. Witten, Quarks, atoms, and the 1/N expansion, Phys. Today, 33 (1980), 38-43.   Google Scholar [64] V. Yakhot and S. A. Orszag, Renormalization group analysis of turbulence. I. Basic theory, J. Sci. Comput., 1 (1986), 3-51.  doi: 10.1007/BF01061452.  Google Scholar [65] V. Yakhot, S. A. Orszag, S. Thangam, T. B. Gatski and C. G. Speziale, Development of turbulence models for shear flows by a double expansion technique, Phys. Fluids A, 4 (1992), 1510-1520.  doi: 10.1063/1.858424.  Google Scholar [66] Y. Zhou, Renormalization group theory for fluid and plasma turbulence, Phys. Reports, 488 (2010), 1-49.  doi: 10.1016/j.physrep.2009.04.004.  Google Scholar

show all references

Dedicated to the memory of Christo I. Christov.

##### References:
 [1] D. M. Ambrose and J. Wilkening, Global paths of time-periodic solutions of the Benjamin-Ono equation connecting pairs of traveling waves, Commun. Appl. Math. Comput. Sci., 4 (2009), 177-215.  doi: 10.2140/camcos.2009.4.177.  Google Scholar [2] ——, Computation of time-periodic solutions of the Benjamin-Ono equation, J. Nonlinear Sci., 20 (2010), 375-378. Google Scholar [3] P. Amore and A. Aranda, Presenting a new method for the solution of nonlinear problems, Phys. Lett. A, 316 (2003), 218-225.  doi: 10.1016/j.physleta.2003.08.001.  Google Scholar [4] P. Amore and H. M. Lamas, High order analysis of nonlinear periodic differential equations, Phys. Lett. A, 327 (2004), 158-166.  doi: 10.1016/j.physleta.2004.05.016.  Google Scholar [5] D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, Numerically Solving Polynomial Systems with Bertini, SIAM, Philadelphia, 2013.  Google Scholar [6] C. M. Bender and H. F. Jones, Calculation of low-lying energy levels in quantum mechanics, J. Phys. A, 47 (2014), 395303, 16 pp. doi: 10.1088/1751-8113/47/39/395303.  Google Scholar [7] C. M. Bender, K. A. Milton, S. S. Pinsky and L. M. Jr. Simmons, A new perturbative approach to nonlinear problems, J. Math. Phys., 30 (1989), 1447-1455.  doi: 10.1063/1.528326.  Google Scholar [8] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. 594  Google Scholar [9] C. M. Bender, A. Pelster and F. Weissbach, Boundary-layer theory, strong-coupling series, and large-order behavior, J. Math. Phys., 43 (2002), 4202-4220.  doi: 10.1063/1.1490408.  Google Scholar [10] C. M. Bender and A. Tovbis, Continuum limit of lattice approximation schemes, J. Math. Phys., 38 (1997), 3700-3717.  doi: 10.1063/1.532063.  Google Scholar [11] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.   Google Scholar [12] H. Blasius, Grenzschichten in flüssigkeiten mit kleiner reibung, Zeitschrift fur Mathematik und Physik, 56 (1908), 1-37.   Google Scholar [13] ——, The Boundary Layers in Fluids with Llttle Friction, Technical Memorandum 1256, NASA, Washington, D. C., 1950. 57, English translation by J. Venier of Grenzschichten in Flüssigkeiten mit kleiner Reibung. Google Scholar [14] J. L. Bona and H. Kalisch, Singularity formation in the generalized Benjamin-Ono equation, Discrete Continuous Dyn. Sys., 11 (2004), 27-45.  doi: 10.3934/dcds.2004.11.27.  Google Scholar [15] J. P. Boyd, A Chebyshev polynomial method for computing analytic solutions to eigenvalue problems with application to the anharmonic oscillator, Journal of Mathematical Physics, 19 (1978), 1445-1456.  doi: 10.1063/1.523810.  Google Scholar [16] ——, Solitons from sine waves: analytical and numerical methods for non-integrable solitary and cnoidal waves, Physica D, 21 (1986), 227-246. Google Scholar [17] ——, Numerical computations of a nearly singular nonlinear equation: Weakly nonlocal bound states of solitons for the Fifth-Order Korteweg-deVries equation, J. Comput. Phys., 124 (1996), 55–70. doi: 10.1006/jcph.1996.0044.  Google Scholar [18] ——, Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics: Generalized Solitons and Hyperasymptotic Perturbation Theory, vol. 442 of Mathematics and Its Applications, Kluwer, Amsterdam, 1998. 608 doi: 10.1007/978-1-4615-5825-5.  Google Scholar [19] ——, The Blasius function in the complex plane, J. Experimental Math., 8 (1999), 381–394. doi: 10.1080/10586458.1999.10504626.  Google Scholar [20] ——, Chebyshev and Fourier Spectral Methods, Dover, New York, 2001.  Google Scholar [21] ——, Deleted residuals, the QR-factored Newton iteration, and other methods for formally overdetermined determinate discretizations of nonlinear eigenproblems for solitary, cnoidal, and shock waves, J. Comput. Phys., 179 (2002), 216-237. doi: 10.1006/jcph.2002.7052.  Google Scholar [22] ——, Why Newton's method is hard for traveling waves: Small denominators, KAM theory, Arnold's linear Fourier problem, non-uniqueness, constraints and erratic failure, Math. Comput. Simul., 74 (2007), 72–81. doi: 10.1016/j.matcom.2006.10.001.  Google Scholar [23] ——, The Blasius function: Computations before computers, the value of tricks, undergraduate projects, and open research problems, SIAM Rev., 50 (2008), 791-804. doi: 10.1137/070681594.  Google Scholar [24] ——, Solving Transcendental Equations: The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation Series and Oracles, SIAM, Philadelphia, 2014. doi: 10.1137/1.9781611973525.  Google Scholar [25] J. P. Boyd and Z. Xu, Comparison of three spectral methods for the Benjamin-Ono equation: Fourier pseudospectral, rational Christov functions and Gaussian radial basis functions, Wave Motion, 48 (2011), 702-706.  doi: 10.1016/j.wavemoti.2011.02.004.  Google Scholar [26] J. P. Boyd and Z. Xu, Numerical and perturbative computations of solitary waves of the Benjamin-Ono equation with higher order nonlinearity using Christov rational basis functions, J. Comput. Phys., 231 (2012), 1216-1229.  doi: 10.1016/j.jcp.2011.10.004.  Google Scholar [27] H. Chen and J. L. Bona, Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations, Adv. Diff. Eq., 3 (1998), 51-84.   Google Scholar [28] R. E. Davis and A. Acrivos, Solitary internal waves in deep water, J. Fluid Mech., 29 (1967), 593. Google Scholar [29] K. Dutta and M. K. Nandy, Perturbative Evaluation of Universal Numbers in Homogeneous Shear Turbulence, (19). Google Scholar [30] D. Z. Goodson and D. R. Herschbach, Summation methods for dimensional perturbation-theory, Phys. Rev. A, 46 (1992), 5428-5436.   Google Scholar [31] K. A. Gorshkov, L. A. Ostrovskii and V. V. Papko, Interactions and bound states of solitons as classical particles, Soviet Physics JETP, 44 (1976), 306-311.   Google Scholar [32] K. A. Gorshkov, L. A. Ostrovskii, V. V. Papko and A. S. Pikovsky, On the existence of stationary multisolitons, Phys. Lett. A, 74 (1979), 177-179.  doi: 10.1016/0375-9601(79)90763-1.  Google Scholar [33] K. A. Gorshkov and L. A. Ostrovsky, Interactions of solitons in noninterable systems: Direct perturbation method and applications, Physica D, 3 (1981), 428-438.   Google Scholar [34] K. A. Gorshkov and V. V. Papko, Dynamic and stochastic oscillations of soliton lattices, Soviet Physics JETP, 46 (1977), 92-97.   Google Scholar [35] R. H. J. Grimshaw and B. A. Malomed, A note on the interaction between solitary waves in a singularly-perturbed Korteweg-deVries equation, J. Phys. A, 26 (1993), 4087-4091.  doi: 10.1088/0305-4470/26/16/024.  Google Scholar [36] J. He, Homotopy perturbation method: A new nonlinear analytical technique, Appl. Math. Comput., 135 (2003), 73-79.  doi: 10.1016/S0096-3003(01)00312-5.  Google Scholar [37] ——, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26 (2005), 695–700. Google Scholar [38] ——, New interpretation of homotopy perturbation method, Inter. J. Modern Phys. B, 20 (2006), 2561-2568. doi: 10.1142/S0217979206034819.  Google Scholar [39] ——, Some asymptotic methods for strongly nonlinear equations, Inter. J. Modern Phys. B, 20 (2006), 1141-1199. doi: 10.1142/S0217979206033796.  Google Scholar [40] D. Herschbach, J. Avery and O. Goscinkski, eds., Dimensional Scaling in Chemical Physics, Kluwer, Dordrecht, The Netherlands, 1992. Google Scholar [41] D. R. Herschbach, Dimensional interpolation for two-electron atoms, J. Chem. Phys., 84 (1986), 838-851.   Google Scholar [42] ——, Dimensional scaling and renormalization, Int. J. Quantum Chem., 57 (1996), 295-308. Google Scholar [43] ——, Fifty years in physical chemistry: Homage to mentors, methods, and molecules, Ann. Rev. Phys. Chem., 51 (2000), 1-39. Google Scholar [44] F. T. Hioe and E. W. Montroll, Quantum theory of anharmonic oscillators. I. Energy levels of oscillators with positive quartic anharmonicity, J. Math. Phys., 16 (1975), 1945-1955.  doi: 10.1063/1.522747.  Google Scholar [45] R. Iacono and J. P. Boyd, Simple analytic approximations to the Blasius equation, Physica D, 310 (2015), 72-78.  doi: 10.1016/j.physd.2015.08.003.  Google Scholar [46] K. Kolossovski, A. R. Champneys, A. Buryak and R. A. Sammut, Multi-pulse embedded solitons as bound states of quasi-solitons, Physica D, 171 (2002), 153-177.  doi: 10.1016/S0167-2789(02)00563-8.  Google Scholar [47] R. H. Kraichnan, An interpretation of the Yakhot-Orszag turbulence theory, Phys. Fluids, 30 (1987), 2400-2405.   Google Scholar [48] H. Nagashima and M. Kuwahara, Computer-simulation of solutions to the nonlinear wave equation $u_{t}+u u_{x} - \gamma u_{5x} = 0$, J. Phys. Soc., 50 (1981), 3792-3800.  doi: 10.1143/JPSJ.50.3792.  Google Scholar [49] R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Springer-Verlag, Heidelberg, 2d ed., 1994.  Google Scholar [50] V. E. Shamanskii, A modification of Newton's method, Ukr. Mat. Zh., 19 (1967), 133-138.   Google Scholar [51] A. Sidi, Practical Extrapolation Methods: Theory and Applications, vol. 10 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546815.  Google Scholar [52] P. M. Stevenson, Optimized perturbation theory, Phys. Rev. D, 23 (1981), 2916-2944.   Google Scholar [53] D. Swade, The Cogwheel Brain: Charles Babbage and the Quest to Build the First Computer, Viking, New York, 2001.  Google Scholar [54] G. 't Hooft, QCD perturbation theory, Phys. B, 72 (1974), 461. Google Scholar [55] F. Vinette and J. Čižek, Upper and lower bounds of the ground state energy of anharmonic oscillators using renormalized inner projection, J. Math. Phys., 32 (1991), 3392-3404.  doi: 10.1063/1.529452.  Google Scholar [56] E. J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Computer Physics Reports, 10 (1989), 189-371.   Google Scholar [57] ——, Optimized perturbation theory, Annals Phys., 246 (1996), 133-165. Google Scholar [58] E. J. Weniger, J. Čižek and F. Vinette, The summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic, sextic, and octic anharmonic oscillators using nonlinear sequence transformation, J. Math. Phys., 34 (1993), 571-609.  doi: 10.1063/1.530262.  Google Scholar [59] G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974.  Google Scholar [60] K. G. Wilson, Renormalization group and critical phenomena. 1. Renormalization group and Kadanoff scaling picture, Phys. Rev. B, 4 (1971), 3174. Google Scholar [61] K. G. Wilson, Critical exponents in 3.99 dimensions, Physica, 73 (1974), 119-128.   Google Scholar [62] K. G. Wilson and M. E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett., 28 (1972), 240-243.   Google Scholar [63] E. Witten, Quarks, atoms, and the 1/N expansion, Phys. Today, 33 (1980), 38-43.   Google Scholar [64] V. Yakhot and S. A. Orszag, Renormalization group analysis of turbulence. I. Basic theory, J. Sci. Comput., 1 (1986), 3-51.  doi: 10.1007/BF01061452.  Google Scholar [65] V. Yakhot, S. A. Orszag, S. Thangam, T. B. Gatski and C. G. Speziale, Development of turbulence models for shear flows by a double expansion technique, Phys. Fluids A, 4 (1992), 1510-1520.  doi: 10.1063/1.858424.  Google Scholar [66] Y. Zhou, Renormalization group theory for fluid and plasma turbulence, Phys. Reports, 488 (2010), 1-49.  doi: 10.1016/j.physrep.2009.04.004.  Google Scholar
Both graphs show the FKdV soliton for unit phase speed. The solitary wave decays proportionally to $\exp(- |X| / \sqrt{2}) \cos(|X|/\sqrt{2} - \mbox{constant})$. The oscillations are invisible for large $X$ on a plot with a linear scale (top), so the absolute value of the solitary wave is plotted on a logarithmic scale in the bottom graph
The left two plots show the potential energy $V(s)$ for the FKdV soliton of unit phase speed. Although one minimum is visible in the upper left plot, that at $s = 6.38$, the logarithmic scale plot (lower left) shows that the oscillations continue to all $X$. (Each downward spike on the log plot indicates a zero of $V(s)$.) Right: same but for the KdV soliton
Left: the solid curve is the FKdV bion for unit phase speed; the dashed curve is the superposition of two solitary waves with their centers at $X = \pm 3.19$ predicted by the Gorshkov-Ostrovsky-Papko perturbation theory. Right: the difference between the two curves on the left. The maximum error of the approximation of the bion by two solitons separated by s = 6.38 is only 5.7 % of the maximum of the bion
$L_{\infty}$ error norms [$\max_{X \in [-\infty, \infty]}$] in perturbative continuation of various orders from the Benjamin-Ono soliton to the cubic Benjamin-Ono soliton
Padé approximations formed from the perturbation series in $\delta$ for the quadratically-to-cubically-nonlinear Benjamin-Ono homotopy
Solid curve: Unit phase speed in the cubic BO soliton, solving $( u^2 - 1) u_{X} + \mathcal{H}(u_{XX}) = 0$. The dashed curve shows the soliton for the ordinary, quadratically-nonlinear Benjamin-Ono equation. Only positive $X$ is shown because both curves are symmetric about the origin
Higher order Korteweg-deVries equation with $m = 10$. Black: solitary wave of unit phase speed. Red dashed: Green's function multiplied by a constant; this curve is also the asymptotic (large $|X|$) approximation to the solitary wave. Blue dotted: the nonlinear term $u^{11}$, scaled by a constant so as to fit on the same graph as the solitary wave.
Solitons of unit phase speed. Generalized Benjamin-Ono.
Octic BO soliton [black solid curve] and the Green's function approximation [red dashed curve] and the inverse-square [$0.4611/X^{2}$] approximation [gold dot-dash curve]. In the outer region, the soliton and Green's function approximation are graphically indistinguishable
Zoom of the previous plot, showing the inner region. The inverse-square asymptotic (large $|X|$) approximation [gold dashed curve in the previous figure] is not shown because it lies outside the chosen axes. The blue dotted curve is the nonlinear term, $u^{8}/8$, scaled by a constant to fit on the same graph as the soliton [black solid curve] and the Green's function approximation [red dashed curve]. The vertical green dotted line is the boundary between the inner and outer regions. Its positioning is somewhat arbitrary because theory constrains the width of the inner region only in magnitude, which is $O(1/m^{2})$ where here $1/m^{2} = 1/49$
Unorthodox Perturbation Parameters
 phase speed difference $\epsilon \equiv \exp( - \mbox{constant} [c_{1}-c_{2}] )$ Gorshkov et al [31] [Time-dependent FKdV] bion overlap parameter $\varepsilon \equiv U_{1}(X) U_{2}(X-s)$ [31]; FKdV bion delta expansion $\delta$ of nonlinear $u^{1+\delta}$ Bender, Milton, Pinsky & Simmons [7,52] inverse nonlinear exponent $\mu=1/m$ in $u^{m}$ Boyd & Xu [26] dimensional $\epsilon=4 - D$, $D$ is dimension Wilson & Fisher [62] phase transitions [61,63] $1/D$ [inverse dimension] $\epsilon \equiv 1/\mbox{dimension}$ [63,41] $1/N_{q}$ [1/(quark number)] $\varepsilon \equiv 1/\mbox{number of quark species}$ t'Hooft [54,63] homotopy parameter $\delta$ in $(1-\delta)\mathfrak{Q}(u) + \delta \mathfrak{N}(u)=0$ [39,38,26] $\mathfrak{Q}(u)=0$ has known solution $\mathfrak{N}(u)=0$ is target problem strong coupling $\lambda^{-2/3}$ Symanzik [44,15] quartic oscillator renormalized modified $\lambda$ Civek & Vinette [55,58] quartic oscillator ground state series $E_{0}$, the smallest eigenvalue Bender & Jones [6] lattice limit $h$, the grid spacing Bender & Tovbis [10] energy spectrum exponent turbulent $E(k) \propto k^{1-(2/3) \epsilon}$ Yakhot & Orszag [64]
 phase speed difference $\epsilon \equiv \exp( - \mbox{constant} [c_{1}-c_{2}] )$ Gorshkov et al [31] [Time-dependent FKdV] bion overlap parameter $\varepsilon \equiv U_{1}(X) U_{2}(X-s)$ [31]; FKdV bion delta expansion $\delta$ of nonlinear $u^{1+\delta}$ Bender, Milton, Pinsky & Simmons [7,52] inverse nonlinear exponent $\mu=1/m$ in $u^{m}$ Boyd & Xu [26] dimensional $\epsilon=4 - D$, $D$ is dimension Wilson & Fisher [62] phase transitions [61,63] $1/D$ [inverse dimension] $\epsilon \equiv 1/\mbox{dimension}$ [63,41] $1/N_{q}$ [1/(quark number)] $\varepsilon \equiv 1/\mbox{number of quark species}$ t'Hooft [54,63] homotopy parameter $\delta$ in $(1-\delta)\mathfrak{Q}(u) + \delta \mathfrak{N}(u)=0$ [39,38,26] $\mathfrak{Q}(u)=0$ has known solution $\mathfrak{N}(u)=0$ is target problem strong coupling $\lambda^{-2/3}$ Symanzik [44,15] quartic oscillator renormalized modified $\lambda$ Civek & Vinette [55,58] quartic oscillator ground state series $E_{0}$, the smallest eigenvalue Bender & Jones [6] lattice limit $h$, the grid spacing Bender & Tovbis [10] energy spectrum exponent turbulent $E(k) \propto k^{1-(2/3) \epsilon}$ Yakhot & Orszag [64]
Coefficients $c_{n}$ of the renormalized Vinette-Čižek series for the quartic quantum oscillator
 $n$ $c_{n}$ 0 1 1 -1/4 2 -5/240 3 5/320 4 -0.02860966435185185185185185 5 0.0657642505787037037037037037037037 6 -0.1836971078880529835390916502 7 0.6040323830435796039094650206 8 -2.285197581882939035618855738 9 9.777776663767784547740376371
 $n$ $c_{n}$ 0 1 1 -1/4 2 -5/240 3 5/320 4 -0.02860966435185185185185185 5 0.0657642505787037037037037037037037 6 -0.1836971078880529835390916502 7 0.6040323830435796039094650206 8 -2.285197581882939035618855738 9 9.777776663767784547740376371
Errors in Padé approximants to the ground state eigenvalue $E_{quartic}$ in $u_{yy}+(E_{quartic } - y^{4}) u = 0$ derived from the renormalized expansion in the limit $\kappa = 1$ where $\kappa(\epsilon)$ is the coupling constant of the renormalized series
 Padé degree Error Approximation $[1/1]$ 0.011 1.0489 $[1/2]$ -0.0117 1.0721 $[2/2]$ 0.00111 1.0592 $[2/3]$ -0.00141 1.06177 $[3/3]$ 0.000203 1.060159 $[3/4]$ -0.000239 1.0606011 $[4/4]$ 0.0000521 1.0603099 $[4/5]$ -0.0000513 1.060413 Exact 0 1.06036209048418289983
 Padé degree Error Approximation $[1/1]$ 0.011 1.0489 $[1/2]$ -0.0117 1.0721 $[2/2]$ 0.00111 1.0592 $[2/3]$ -0.00141 1.06177 $[3/3]$ 0.000203 1.060159 $[3/4]$ -0.000239 1.0606011 $[4/4]$ 0.0000521 1.0603099 $[4/5]$ -0.0000513 1.060413 Exact 0 1.06036209048418289983
FKdV soliton in Matlab
 function U = FKdVOP32soliton(X); a=[-0.5453314648796651, 0.5068203053504623, ...-0.4402630181529713, 0.349096468941388, ...-0.2460937274312324, 0.1505301481748724, ...-0.07877103488351303, 0.03576658123700287, ...-0.01508939706498649, 0.006484221100529809, ...-0.002787043402654402, 0.001076066017735743, ...-0.0003885102078872311, 0.0001579017580647854, ...-6.120851442708405e-05, 1.828270102148695e-05, ...-7.60499954586267e-06, 3.475951657348701e-06, ...-4.851950450442076e-07, 3.264844822058454e-07, ...-2.778993470149972e-07, -6.662198834658049e-08, ...-2.495375367409503e-08, 3.678002950984579e-08, ... 2.716273301091234e-08, 9.834431689322261e-09, ...-4.488746695840646e-09, -7.448008666537661e-09, ...-4.578973978364871e-09, -7.149230751276156e-10, ... 1.377693304434431e-09, 1.596376433909487e-09, ... 8.662998026533877e-10, 8.76005697976315e-11, ...-3.238108570125493e-10, -3.638592525678838e-10, ...-2.109894058947881e-10, -3.82707672046e-11, ... 6.47290253283e-11, 8.76009399732e-11, ... 6.10548488901e-11, -2.14300062965e-11, ...-7.9943985917e-12, -2.00450976361e-11, -1.8306686011e-11, ...-1.01476636914e-11, -1.8476681323e-12, 3.3274735468e-12, ... 4.8889141218e-12, 3.9414102155e-12, 1.9920356858e-12, ... 2.017704407e-13, -8.678528362e-13, -1.1691087695e-12, ...-9.423374250e-13, -4.983807316e-13, -8.04581170e-14, ... 1.859262284e-13, 2.805047562e-13, 2.471215398e-13, ... 1.508907701e-13, 4.81962648e-14, -2.69479308e-14, ... -6.34741232e-14, -6.65839414e-14, -4.91445812e-14, ...-2.46054163e-14, -2.8465775e-15, 1.11350982e-14, ... 1.65791376e-14, 1.53985988e-14, 1.05396256e-14, ... 4.7517599e-15, -7.10186e-17, -3.0443617e-15, ...-4.1160885e-15, -3.7440569e-15, -2.5741786e-15, ...-1.2016066e-15, -4.15608e-17, 7.028038e-16, ... 1.0071960e-15, 9.624651e-16, 7.082454e-16, ... 3.810491e-16, 8.34033e-17, -1.27058e-16, ...-2.339024e-16, -2.502104e-16, -2.046779e-16];L=12, % map parametert=acot(X/L); U=0;for j=1:90, U=U + a(j)* (cos(2*j*t)-1); endend
 function U = FKdVOP32soliton(X); a=[-0.5453314648796651, 0.5068203053504623, ...-0.4402630181529713, 0.349096468941388, ...-0.2460937274312324, 0.1505301481748724, ...-0.07877103488351303, 0.03576658123700287, ...-0.01508939706498649, 0.006484221100529809, ...-0.002787043402654402, 0.001076066017735743, ...-0.0003885102078872311, 0.0001579017580647854, ...-6.120851442708405e-05, 1.828270102148695e-05, ...-7.60499954586267e-06, 3.475951657348701e-06, ...-4.851950450442076e-07, 3.264844822058454e-07, ...-2.778993470149972e-07, -6.662198834658049e-08, ...-2.495375367409503e-08, 3.678002950984579e-08, ... 2.716273301091234e-08, 9.834431689322261e-09, ...-4.488746695840646e-09, -7.448008666537661e-09, ...-4.578973978364871e-09, -7.149230751276156e-10, ... 1.377693304434431e-09, 1.596376433909487e-09, ... 8.662998026533877e-10, 8.76005697976315e-11, ...-3.238108570125493e-10, -3.638592525678838e-10, ...-2.109894058947881e-10, -3.82707672046e-11, ... 6.47290253283e-11, 8.76009399732e-11, ... 6.10548488901e-11, -2.14300062965e-11, ...-7.9943985917e-12, -2.00450976361e-11, -1.8306686011e-11, ...-1.01476636914e-11, -1.8476681323e-12, 3.3274735468e-12, ... 4.8889141218e-12, 3.9414102155e-12, 1.9920356858e-12, ... 2.017704407e-13, -8.678528362e-13, -1.1691087695e-12, ...-9.423374250e-13, -4.983807316e-13, -8.04581170e-14, ... 1.859262284e-13, 2.805047562e-13, 2.471215398e-13, ... 1.508907701e-13, 4.81962648e-14, -2.69479308e-14, ... -6.34741232e-14, -6.65839414e-14, -4.91445812e-14, ...-2.46054163e-14, -2.8465775e-15, 1.11350982e-14, ... 1.65791376e-14, 1.53985988e-14, 1.05396256e-14, ... 4.7517599e-15, -7.10186e-17, -3.0443617e-15, ...-4.1160885e-15, -3.7440569e-15, -2.5741786e-15, ...-1.2016066e-15, -4.15608e-17, 7.028038e-16, ... 1.0071960e-15, 9.624651e-16, 7.082454e-16, ... 3.810491e-16, 8.34033e-17, -1.27058e-16, ...-2.339024e-16, -2.502104e-16, -2.046779e-16];L=12, % map parametert=acot(X/L); U=0;for j=1:90, U=U + a(j)* (cos(2*j*t)-1); endend
FKdV bion in Matlab
 function U = FKdVOP32bion(X); a=[-0.6249529522855067, -0.06248613190603774, ...0.617299971308332, -0.678152415917394, ...0.3136709320967295, 0.09735853435905277, ...-0.2482896407689298, 0.1463681479501488, ...0.001660678333513156, -0.05396361762649843, ...0.02849417718501095, 0.001302906101052367, ...-0.007702923501187634, 0.003346664847975391, ...-9.327765530010755e-05, -0.000782021001205553, ...0.0005243572937128794, -1.961663775942396e-05, ...-0.0001312662112695258, 3.890073621307305e-05, ...4.45019977705866e-06, -4.90156843400462e-06, ...6.788989042114139e-06, -3.19484355347982e-07, ...-2.06495521142274e-06, -3.009462285770579e-07, ...-1.574599685925207e-07, 8.614181320294967e-08, ...2.69200464278439e-07, 1.269674376990847e-07, ...-2.783175287572208e-10, -4.719125664354351e-08, ...-5.042442417958271e-08, -2.535915798256805e-08, ...1.135883504776753e-10, 1.159080056928546e-08, ...1.165851313528745e-08, 6.375735163162946e-09, ...8.309234298651179e-10, -2.303376033217859e-09, ...-2.879341193338522e-09, -1.952336751484576e-09, ...-6.589060519867656e-10, 2.855915705868245e-10, ...6.672122868160802e-10, 6.062853737640122e-10, ...3.385894573618819e-10, 6.517373474838492e-11, ...-1.077636147721495e-10, -1.626684976080264e-10, ...-1.33972796931821e-10, -7.0305728960486e-11, ...-1.01147892835e-11, 2.717980489038125e-1, 3.90182292235e-11, 3.26957968995e-11, 1.83118278535e-11, 4.1017935603e-12, ...-5.4228334397e-12, -9.2492259114e-12, -8.6123719612e-12, ... -5.6020544748e-12, 2.1349588146e-12, 5.560290722e-13, ... 1.9954191691e-12, 2.2782816883e-12, 1.7966058889e-12, ... 9.986262724e-13, 2.365948023e-13, -2.923085132e-13, ...-5.360409404e-13, -5.416314510e-13, -4.014029598e-13, ...-2.095235315e-13, -3.622205926e-14, 8.094449569e-14, ...1.336638605e-1, 1.332772705e-13, 1.000697622e-13, ... 5.46382681e-14, 1.265945082e-14, -1.70294246e-14, ...-3.19117103e-14, -3.39481150e-14, -2.73870326e-14, ... -1.68901894e-14, -6.2912288e-15, 1.9728521e-15, ...6.8905583e-15, 8.5696771e-15, 7.7958865e-15, ...5.6076546e-15, 2.9756802e-15, 6.173997e-16, ...-1.06448571e-15, -1.9532122e-15, -2.1391070e-15, ... -1.82619267e-15, -1.2492804e-15, -6.148978e-16];L=12; t=acot(X/L); U=0;for j=1:length(a), U=U + a(j)* (cos(2*j*t)-1); end; end
 function U = FKdVOP32bion(X); a=[-0.6249529522855067, -0.06248613190603774, ...0.617299971308332, -0.678152415917394, ...0.3136709320967295, 0.09735853435905277, ...-0.2482896407689298, 0.1463681479501488, ...0.001660678333513156, -0.05396361762649843, ...0.02849417718501095, 0.001302906101052367, ...-0.007702923501187634, 0.003346664847975391, ...-9.327765530010755e-05, -0.000782021001205553, ...0.0005243572937128794, -1.961663775942396e-05, ...-0.0001312662112695258, 3.890073621307305e-05, ...4.45019977705866e-06, -4.90156843400462e-06, ...6.788989042114139e-06, -3.19484355347982e-07, ...-2.06495521142274e-06, -3.009462285770579e-07, ...-1.574599685925207e-07, 8.614181320294967e-08, ...2.69200464278439e-07, 1.269674376990847e-07, ...-2.783175287572208e-10, -4.719125664354351e-08, ...-5.042442417958271e-08, -2.535915798256805e-08, ...1.135883504776753e-10, 1.159080056928546e-08, ...1.165851313528745e-08, 6.375735163162946e-09, ...8.309234298651179e-10, -2.303376033217859e-09, ...-2.879341193338522e-09, -1.952336751484576e-09, ...-6.589060519867656e-10, 2.855915705868245e-10, ...6.672122868160802e-10, 6.062853737640122e-10, ...3.385894573618819e-10, 6.517373474838492e-11, ...-1.077636147721495e-10, -1.626684976080264e-10, ...-1.33972796931821e-10, -7.0305728960486e-11, ...-1.01147892835e-11, 2.717980489038125e-1, 3.90182292235e-11, 3.26957968995e-11, 1.83118278535e-11, 4.1017935603e-12, ...-5.4228334397e-12, -9.2492259114e-12, -8.6123719612e-12, ... -5.6020544748e-12, 2.1349588146e-12, 5.560290722e-13, ... 1.9954191691e-12, 2.2782816883e-12, 1.7966058889e-12, ... 9.986262724e-13, 2.365948023e-13, -2.923085132e-13, ...-5.360409404e-13, -5.416314510e-13, -4.014029598e-13, ...-2.095235315e-13, -3.622205926e-14, 8.094449569e-14, ...1.336638605e-1, 1.332772705e-13, 1.000697622e-13, ... 5.46382681e-14, 1.265945082e-14, -1.70294246e-14, ...-3.19117103e-14, -3.39481150e-14, -2.73870326e-14, ... -1.68901894e-14, -6.2912288e-15, 1.9728521e-15, ...6.8905583e-15, 8.5696771e-15, 7.7958865e-15, ...5.6076546e-15, 2.9756802e-15, 6.173997e-16, ...-1.06448571e-15, -1.9532122e-15, -2.1391070e-15, ... -1.82619267e-15, -1.2492804e-15, -6.148978e-16];L=12; t=acot(X/L); U=0;for j=1:length(a), U=U + a(j)* (cos(2*j*t)-1); end; end
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