Strongly nonlinear perturbation theory for solitary waves and bions

John Boyd

doi:10.3934/eect.2019001

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March 2019, 8(1): 1-29. doi: 10.3934/eect.2019001

Strongly nonlinear perturbation theory for solitary waves and bions

John Boyd

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.

Keywords: Perturbation theory, nonlinear waves, solitary wave, bion, integral equation.

Mathematics Subject Classification: Primary: 34K26, 74J35; Secondary: 65H20, 65N35.

Citation: John Boyd. Strongly nonlinear perturbation theory for solitary waves and bions. Evolution Equations and 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.
[2]	——, Computation of time-periodic solutions of the Benjamin-Ono equation, J. Nonlinear Sci., 20 (2010), 375-378.
[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.
[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.
[5]	D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, Numerically Solving Polynomial Systems with Bertini, SIAM, Philadelphia, 2013.
[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.
[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.
[8]	C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. 594
[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.
[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.
[11]	T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.
[12]	H. Blasius, Grenzschichten in flüssigkeiten mit kleiner reibung, Zeitschrift fur Mathematik und Physik, 56 (1908), 1-37.
[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.
[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.
[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.
[16]	——, Solitons from sine waves: analytical and numerical methods for non-integrable solitary and cnoidal waves, Physica D, 21 (1986), 227-246.
[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.
[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.
[19]	——, The Blasius function in the complex plane, J. Experimental Math., 8 (1999), 381–394. doi: 10.1080/10586458.1999.10504626.
[20]	——, Chebyshev and Fourier Spectral Methods, Dover, New York, 2001.
[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.
[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.
[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.
[24]	——, Solving Transcendental Equations: The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation Series and Oracles, SIAM, Philadelphia, 2014. doi: 10.1137/1.9781611973525.
[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.
[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.
[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.
[28]	R. E. Davis and A. Acrivos, Solitary internal waves in deep water, J. Fluid Mech., 29 (1967), 593.
[29]	K. Dutta and M. K. Nandy, Perturbative Evaluation of Universal Numbers in Homogeneous Shear Turbulence, (19).
[30]	D. Z. Goodson and D. R. Herschbach, Summation methods for dimensional perturbation-theory, Phys. Rev. A, 46 (1992), 5428-5436.
[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.
[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.
[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.
[34]	K. A. Gorshkov and V. V. Papko, Dynamic and stochastic oscillations of soliton lattices, Soviet Physics JETP, 46 (1977), 92-97.
[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.
[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.
[37]	——, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26 (2005), 695–700.
[38]	——, New interpretation of homotopy perturbation method, Inter. J. Modern Phys. B, 20 (2006), 2561-2568. doi: 10.1142/S0217979206034819.
[39]	——, Some asymptotic methods for strongly nonlinear equations, Inter. J. Modern Phys. B, 20 (2006), 1141-1199. doi: 10.1142/S0217979206033796.
[40]	D. Herschbach, J. Avery and O. Goscinkski, eds., Dimensional Scaling in Chemical Physics, Kluwer, Dordrecht, The Netherlands, 1992.
[41]	D. R. Herschbach, Dimensional interpolation for two-electron atoms, J. Chem. Phys., 84 (1986), 838-851.
[42]	——, Dimensional scaling and renormalization, Int. J. Quantum Chem., 57 (1996), 295-308.
[43]	——, Fifty years in physical chemistry: Homage to mentors, methods, and molecules, Ann. Rev. Phys. Chem., 51 (2000), 1-39.
[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.
[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.
[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.
[47]	R. H. Kraichnan, An interpretation of the Yakhot-Orszag turbulence theory, Phys. Fluids, 30 (1987), 2400-2405.
[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.
[49]	R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Springer-Verlag, Heidelberg, 2d ed., 1994.
[50]	V. E. Shamanskii, A modification of Newton's method, Ukr. Mat. Zh., 19 (1967), 133-138.
[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.
[52]	P. M. Stevenson, Optimized perturbation theory, Phys. Rev. D, 23 (1981), 2916-2944.
[53]	D. Swade, The Cogwheel Brain: Charles Babbage and the Quest to Build the First Computer, Viking, New York, 2001.
[54]	G. 't Hooft, QCD perturbation theory, Phys. B, 72 (1974), 461.
[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.
[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.
[57]	——, Optimized perturbation theory, Annals Phys., 246 (1996), 133-165.
[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.
[59]	G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974.
[60]	K. G. Wilson, Renormalization group and critical phenomena. 1. Renormalization group and Kadanoff scaling picture, Phys. Rev. B, 4 (1971), 3174.
[61]	K. G. Wilson, Critical exponents in 3.99 dimensions, Physica, 73 (1974), 119-128.
[62]	K. G. Wilson and M. E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett., 28 (1972), 240-243.
[63]	E. Witten, Quarks, atoms, and the 1/N expansion, Phys. Today, 33 (1980), 38-43.
[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.
[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.
[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.

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.
[2]	——, Computation of time-periodic solutions of the Benjamin-Ono equation, J. Nonlinear Sci., 20 (2010), 375-378.
[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.
[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.
[5]	D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, Numerically Solving Polynomial Systems with Bertini, SIAM, Philadelphia, 2013.
[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.
[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.
[8]	C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. 594
[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.
[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.
[11]	T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.
[12]	H. Blasius, Grenzschichten in flüssigkeiten mit kleiner reibung, Zeitschrift fur Mathematik und Physik, 56 (1908), 1-37.
[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.
[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.
[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.
[16]	——, Solitons from sine waves: analytical and numerical methods for non-integrable solitary and cnoidal waves, Physica D, 21 (1986), 227-246.
[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.
[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.
[19]	——, The Blasius function in the complex plane, J. Experimental Math., 8 (1999), 381–394. doi: 10.1080/10586458.1999.10504626.
[20]	——, Chebyshev and Fourier Spectral Methods, Dover, New York, 2001.
[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.
[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.
[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.
[24]	——, Solving Transcendental Equations: The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation Series and Oracles, SIAM, Philadelphia, 2014. doi: 10.1137/1.9781611973525.
[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.
[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.
[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.
[28]	R. E. Davis and A. Acrivos, Solitary internal waves in deep water, J. Fluid Mech., 29 (1967), 593.
[29]	K. Dutta and M. K. Nandy, Perturbative Evaluation of Universal Numbers in Homogeneous Shear Turbulence, (19).
[30]	D. Z. Goodson and D. R. Herschbach, Summation methods for dimensional perturbation-theory, Phys. Rev. A, 46 (1992), 5428-5436.
[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.
[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.
[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.
[34]	K. A. Gorshkov and V. V. Papko, Dynamic and stochastic oscillations of soliton lattices, Soviet Physics JETP, 46 (1977), 92-97.
[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.
[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.
[37]	——, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26 (2005), 695–700.
[38]	——, New interpretation of homotopy perturbation method, Inter. J. Modern Phys. B, 20 (2006), 2561-2568. doi: 10.1142/S0217979206034819.
[39]	——, Some asymptotic methods for strongly nonlinear equations, Inter. J. Modern Phys. B, 20 (2006), 1141-1199. doi: 10.1142/S0217979206033796.
[40]	D. Herschbach, J. Avery and O. Goscinkski, eds., Dimensional Scaling in Chemical Physics, Kluwer, Dordrecht, The Netherlands, 1992.
[41]	D. R. Herschbach, Dimensional interpolation for two-electron atoms, J. Chem. Phys., 84 (1986), 838-851.
[42]	——, Dimensional scaling and renormalization, Int. J. Quantum Chem., 57 (1996), 295-308.
[43]	——, Fifty years in physical chemistry: Homage to mentors, methods, and molecules, Ann. Rev. Phys. Chem., 51 (2000), 1-39.
[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.
[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.
[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.
[47]	R. H. Kraichnan, An interpretation of the Yakhot-Orszag turbulence theory, Phys. Fluids, 30 (1987), 2400-2405.
[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.
[49]	R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Springer-Verlag, Heidelberg, 2d ed., 1994.
[50]	V. E. Shamanskii, A modification of Newton's method, Ukr. Mat. Zh., 19 (1967), 133-138.
[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.
[52]	P. M. Stevenson, Optimized perturbation theory, Phys. Rev. D, 23 (1981), 2916-2944.
[53]	D. Swade, The Cogwheel Brain: Charles Babbage and the Quest to Build the First Computer, Viking, New York, 2001.
[54]	G. 't Hooft, QCD perturbation theory, Phys. B, 72 (1974), 461.
[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.
[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.
[57]	——, Optimized perturbation theory, Annals Phys., 246 (1996), 133-165.
[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.
[59]	G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974.
[60]	K. G. Wilson, Renormalization group and critical phenomena. 1. Renormalization group and Kadanoff scaling picture, Phys. Rev. B, 4 (1971), 3174.
[61]	K. G. Wilson, Critical exponents in 3.99 dimensions, Physica, 73 (1974), 119-128.
[62]	K. G. Wilson and M. E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett., 28 (1972), 240-243.
[63]	E. Witten, Quarks, atoms, and the 1/N expansion, Phys. Today, 33 (1980), 38-43.
[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.
[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.
[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.

Figure 1. 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

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Figure 2. 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

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Figure 3. 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

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Figure 4. $ 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

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Figure 5. Padé approximations formed from the perturbation series in $ \delta $ for the quadratically-to-cubically-nonlinear Benjamin-Ono homotopy

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Figure 6. 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

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Figure 7. 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.

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Figure 8. Solitons of unit phase speed. Generalized Benjamin-Ono.

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Figure 9. 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

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Figure 10. 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 $

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Table 1. 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]

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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]

Table 2. 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

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$ 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

Table 3. 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

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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

Table 4. 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 parameter
t=acot(X/L); U=0;
for j=1:90, U=U + a(j)* (cos(2*j*t)-1); end
end

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Table 5. 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

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American Institute of Mathematical Sciences

Strongly nonlinear perturbation theory for solitary waves and bions

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American Institute of Mathematical Sciences

Strongly nonlinear perturbation theory for solitary waves and bions

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