Propagation of Solitons Under Colored Noise
|
|
- Russell Owens
- 5 years ago
- Views:
Transcription
1 Propagation of Solitons Under Colored Noise Dr. Russell Herman Departments of Mathematics & Statistics, Physics & Physical Oceanography UNC Wilmington, Wilmington, NC January 6, 2009
2 Outline of Talk 1 Solitary Waves and Solitons 2 White Noise and Colored Noise 3 Exact Solutions of the Stochastic KdV 4 Numerical Results to date 5 Summary Dr. Herman (UNCW) Stochastic Solitons JMM / 25
3 Solitary Waves and Solitons Solitary Waves The propagation of non-dispersive energy bundles through discrete and continuous media. Example - Burgers Equation u t + αuu x + βu xx = 0, u(x, t) = c α [1 + β tanh c 2 (x ct) ]. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
4 Solitary Waves and Solitons Solitary Waves The propagation of non-dispersive energy bundles through discrete and continuous media. Example - Burgers Equation u t + αuu x + βu xx = 0, u(x, t) = c α [1 + β tanh c 2 (x ct) ]. Solitons Traveling wave solutions satisfying 1 They are of permanent form; 2 They are localised within a region; 3 They can interact with other solitons, and emerge from the collision unchanged, except for a phase shift. Example - KdV Equation u t + 6uu x + u xxx = 0, u(x, t) = 2η 2 sech 2 η(x 4η 2 t), c = 4η 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
5 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
6 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
7 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
8 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
9 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
10 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
11 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
12 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
13 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
14 Evolution of One Soliton Solution Dr. Herman (UNCW) Stochastic Solitons JMM / 25
15 The Two Soliton Solution of the KdV Equation Form of the Solution When two solitons collide, they interact elastically. The exact solution for the two soliton equation is given by u(x, t) = 2(p2 q 2 )(p 2 + q 2 sech 2 χ(x, t)sinh 2 θ(x, t)) (p coshθ(x, t) q tanhχ(x, t)sinh θ(x, t)) 2 (1) where the phases are and θ(x, t) = px 4p 3 (t t 0 ) (2) χ(x, t) = qx 4q 3 (t t 0 ). (3) In our simulation we take p = 2, q = 1.5 and t 0 = 0.5. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
16 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
17 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
18 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
19 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
20 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
21 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
22 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
23 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
24 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
25 Animation of Two Solitons Colliding Dr. Herman (UNCW) Stochastic Solitons JMM / 25
26 The Two Soliton Solution of the KdV Equation Dr. Herman (UNCW) Stochastic Solitons JMM / 25
27 KdV-Burgers Equation Special Cases The KdV-Burgers Equation is given by u t + αuu x + βu xx + su xxx = 0. Traveling wave solutions are given for u(x, t) = 2k α [1 + tanh k(x 2kt)] (4) u(x, t) = 12sk2 α sech 2 k(x 4sk 2 t) (5) u(x, t) = Asech 2 η(x vt) + 2A[1 + tanhη(x vt)], (6) where A = 3β2 25αs, v = 6β2 5s, η = β 10s. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
28 White Noise and Colored Noise Noise Types White - equal energy/cycle - constant frequency spectrum Pink - 1/f -noise - flat in log space - decreases 3 db per octave Brown - Decrease of 6 db per octave Blue - Increase 3 db per octave Purple - Increase 6 db per octave Dr. Herman (UNCW) Stochastic Solitons JMM / 25
29 White Noise and Colored Noise Noise Types White - equal energy/cycle - constant frequency spectrum Pink - 1/f -noise - flat in log space - decreases 3 db per octave Brown - Decrease of 6 db per octave Blue - Increase 3 db per octave Purple - Increase 6 db per octave Dr. Herman (UNCW) Stochastic Solitons JMM / 25
30 White Noise and Colored Noise Noise Types White - equal energy/cycle - constant frequency spectrum Pink - 1/f -noise - flat in log space - decreases 3 db per octave Brown - Decrease of 6 db per octave Blue - Increase 3 db per octave Purple - Increase 6 db per octave Dr. Herman (UNCW) Stochastic Solitons JMM / 25
31 White Noise and Colored Noise Noise Types White - equal energy/cycle - constant frequency spectrum Pink - 1/f -noise - flat in log space - decreases 3 db per octave Brown - Decrease of 6 db per octave Blue - Increase 3 db per octave Purple - Increase 6 db per octave Dr. Herman (UNCW) Stochastic Solitons JMM / 25
32 White Noise and Colored Noise Noise Types White - equal energy/cycle - constant frequency spectrum Pink - 1/f -noise - flat in log space - decreases 3 db per octave Brown - Decrease of 6 db per octave Blue - Increase 3 db per octave Purple - Increase 6 db per octave Dr. Herman (UNCW) Stochastic Solitons JMM / 25
33 Quantifying White Noise White Noise < η(t) >= 0. < η(t)η(s) >= δ(t s). η(t) is formal derivative of a Wiener process, W(t) Dr. Herman (UNCW) Stochastic Solitons JMM / 25
34 Quantifying White Noise White Noise < η(t) >= 0. < η(t)η(s) >= δ(t s). η(t) is formal derivative of a Wiener process, W(t) Brownian Motion dw dt = η(t) or dw = η(t)dt. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
35 Quantifying White Noise White Noise < η(t) >= 0. < η(t)η(s) >= δ(t s). η(t) is formal derivative of a Wiener process, W(t) Brownian Motion dw dt = η(t) or dw = η(t)dt. Stochastic ODE dx dt = ǫη(t) dx = ǫdw. x i x i 1 = ǫ(w(t i ) W(t i 1 )) ǫdw i. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
36 Quantifying Colored Noise Exponentially Correlated (Real) Noise - Ornstein-Uhlenbeck Process for dξ dt = 1 τ ξ + ǫ τ η(t) < ξ(t) >= 0, < ξ(t)ξ(s) >= ǫ2 2τ e t s /τ. Gaussian and stationary if < ξ(0) >= 0, < ξ 2 (0) >= ǫ2 2τ τ p(ξ 0 ) = ǫπ e ξ2 0 τ/ǫ Dr. Herman (UNCW) Stochastic Solitons JMM / 25
37 Quantifying Colored Noise Exponentially Correlated (Real) Noise - Ornstein-Uhlenbeck Process for dξ dt = 1 τ ξ + ǫ τ η(t) < ξ(t) >= 0, < ξ(t)ξ(s) >= ǫ2 2τ e t s /τ. Gaussian and stationary if < ξ(0) >= 0, < ξ 2 (0) >= ǫ2 2τ τ p(ξ 0 ) = ǫπ e ξ2 0 τ/ǫ 1/f β -noise, Kaulakys, et al., Physica A, 365 (2006) dx dt = Γx2σ 1 + x σ η(t), β = 2 2Γ + 1 2σ 2 Dr. Herman (UNCW) Stochastic Solitons JMM / 25
38 Exact Solution of the Stochastic KdV Equation Problem u t + 6uu x + u xxx = ζ(t), (7) Dr. Herman (UNCW) Stochastic Solitons JMM / 25
39 Exact Solution of the Stochastic KdV Equation Problem u t + 6uu x + u xxx = ζ(t), (7) Galilean transformation - Wadati 1983 u(x, t) = U(X, T) + W(T), X = x + m(t), T = t, (8) m(t) = 6 t W(t )dt, W(t) = t 0 0 transforms the stochastic KdV: U T + 6UU X + U XXX = 0. ζ(t )dt, Dr. Herman (UNCW) Stochastic Solitons JMM / 25
40 Exact Solution of the Stochastic KdV Equation Problem u t + 6uu x + u xxx = ζ(t), (7) Galilean transformation - Wadati 1983 u(x, t) = U(X, T) + W(T), X = x + m(t), T = t, (8) m(t) = 6 t 0 W(t )dt, W(t) = transforms the stochastic KdV: U T + 6UU X + U XXX = 0. The One Soliton Solution of the KdV t 0 ζ(t )dt, U(X, T) = 2η 2 sech 2 (η(x 4η 2 T X 0 )) (9) ( t )) u(x, t) = 2η 2 sech (η 2 x 4η 2 t x 0 6 W(t )dt + W(t). (10) 0 Dr. Herman (UNCW) Stochastic Solitons JMM / 25
41 Ensemble (Statistical) Average General < u(x, t) > = 4η2 π where a = 2η(x x 0 2η 2 t), = η 2 πb πk 2 sinhπk eiak bk dk. e (a s)2 /4b sech 2 s ds, (11) 2 b = 2η 2 < m 2 (t) >, t m(t) = 6 W(t )dt. 0 Dr. Herman (UNCW) Stochastic Solitons JMM / 25
42 Ensemble (Statistical) Average General < u(x, t) > = 4η2 π where a = 2η(x x 0 2η 2 t), = η 2 πb πk 2 sinhπk eiak bk dk. e (a s)2 /4b sech 2 s ds, (11) 2 b = 2η 2 < m 2 (t) >, t m(t) = 6 W(t )dt. 0 Results White Noise (Wadati 1983): b = 48η 2 ǫt 3 Exponential noise (Orlowski and Sobczyk 1989): ( 2 b = 36η 2 ǫ 2 τ 3 3 ( t τ )3 ( t τ )2 2(1 t ) τ )e t/τ + 2 Dr. Herman (UNCW) Stochastic Solitons JMM / 25
43 Averaged Soliton with Noise Dr. Herman (UNCW) Stochastic Solitons JMM / 25
44 Stochastic KdV: < u(x, t) > max vs. t Figure: 500 runs for x [ 10,90] with N = 1000, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
45 Stochastic KdV: log(e 2γt/3 < u(x, t) > max ) vs. log t Figure: One Soliton Amplitude with Damping and Noise. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
46 Stochastic KdV: < u(x, t) > max vs. t Figure: Two Soliton Solution with Damping and Noise. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
47 What About Colored Noise vs White? 2.5 KdV Soliton Under White Noise 2 Amplitude <u(x,t)> Time t Figure: 500 runs for x [ 10, 90] with N = 500, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
48 Stochastic KdV Amplitudes - Pink Noise 2.5 KdV Soliton Under Pink Noise 2 Amplitude <u(x,t)> Time t Figure: 1000 runs for x [ 10, 90] with N = 500, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
49 Stochastic KdV Amplitudes - Pink Noise 2 KdV Soliton Under Pink Noise 1.5 Amplitude <u(x,t)> Time t Figure: 2000 runs for x [ 10, 90] with N = 500, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
50 Stochastic KdV Amplitudes - Brown Noise KdV Soliton Under Brown Noise Amplitude <u(x,t)> Time t Figure: 1000 runs for x [ 10, 90] with N = 500, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
51 Stochastic KdV Amplitudes - Brown Noise 2.5 KdV Soliton Under Brown Noise 2 Amplitude <u(x,t)> Time t Figure: 2000 runs for x [ 10, 90] with N = 500, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
52 Stochastic KdV Amplitudes - Brown Noise 2.5 KdV Soliton Under Brown Noise 2 Amplitude <u(x,t)> Time t Figure: 3000 runs for x [ 10, 90] with N = 500, ǫ =.1, and η = 2. Dr. Herman (UNCW) Stochastic Solitons JMM / 25
53 Outline of Talk 1 Solitary Waves and Solitons 2 White Noise and Colored Noise 3 Exact Solutions of the Stochastic KdV 4 Numerical Results to date 5 Summary Dr. Herman (UNCW) Stochastic Solitons JMM / 25
54 Sample References Herman R L, 1990 The Stochastic, Damped KdV Equation J. Phys. A Herman R L and Rose A, 2008 Numerical Realizations of Solutions of the Stochastic KdV Equation, to appear in Mathematics and Computers in Simulation. Desmond J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review, vol. 43, no. 3, 2001, Kaulakys B et al., 2006, Nonlinear stochastic models of 1/f noise and power law distributions, Physicsa A 365, Miki Wadati, Stochastic Korteweg-de Vries Equation, Journal of the Physical Society of Japan, vol. 52, no. 8, August 1983, Miki Wadati and Yasuhiro Akutsu, Stochastic Korteweg-de Vries Equation with and without Damping, Journal of the Physical Society of Japan, vol. 53, no. 10, October 1984, Dr. Herman (UNCW) Stochastic Solitons JMM / 25
Are Solitary Waves Color Blind to Noise?
Are Solitary Waves Color Blind to Noise? Dr. Russell Herman Department of Mathematics & Statistics, UNCW March 29, 2008 Outline of Talk 1 Solitary Waves and Solitons 2 White Noise and Colored Noise? 3
More informationThe Stochastic KdV - A Review
The Stochastic KdV - A Review Russell L. Herman Wadati s Work We begin with a review of the results from Wadati s paper 983 paper [4] but with a sign change in the nonlinear term. The key is an analysis
More informationNUMERICAL SIMULATIONS OF THE STOCHASTIC KDV EQUATION
NUMERICAL SIMULATIONS OF THE STOCHASTIC KDV EQUATION Andrew Rose A Thesis Submitted to the University of North Carolina Wilmington in Partial Fulfillment Of the Requirements for the Degree of Master of
More informationSolitons : An Introduction
Solitons : An Introduction p. 1/2 Solitons : An Introduction Amit Goyal Department of Physics Panjab University Chandigarh Solitons : An Introduction p. 2/2 Contents Introduction History Applications Solitons
More informationPath integrals for classical Markov processes
Path integrals for classical Markov processes Hugo Touchette National Institute for Theoretical Physics (NITheP) Stellenbosch, South Africa Chris Engelbrecht Summer School on Non-Linear Phenomena in Field
More information2 Soliton Perturbation Theory
Revisiting Quasistationary Perturbation Theory for Equations in 1+1 Dimensions Russell L. Herman University of North Carolina at Wilmington, Wilmington, NC Abstract We revisit quasistationary perturbation
More informationA Note On Solitary Wave Solutions of the Compound Burgers-Korteweg-de Vries Equation
A Note On Solitary Wave Solutions of the Compound Burgers-Korteweg-de Vries Equation arxiv:math/6768v1 [math.ap] 6 Jul 6 Claire David, Rasika Fernando, and Zhaosheng Feng Université Pierre et Marie Curie-Paris
More informationA path integral approach to the Langevin equation
A path integral approach to the Langevin equation - Ashok Das Reference: A path integral approach to the Langevin equation, A. Das, S. Panda and J. R. L. Santos, arxiv:1411.0256 (to be published in Int.
More informationAlgebraic geometry for shallow capillary-gravity waves
Algebraic geometry for shallow capillary-gravity waves DENYS DUTYKH 1 Chargé de Recherche CNRS 1 Université de Savoie Laboratoire de Mathématiques (LAMA) 73376 Le Bourget-du-Lac France Seminar of Computational
More informationExact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable Coefficients
Contemporary Engineering Sciences, Vol. 11, 2018, no. 16, 779-784 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.8262 Exact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable
More informationAvailable online at J. Math. Comput. Sci. 2 (2012), No. 1, ISSN:
Available online at http://scik.org J. Math. Comput. Sci. 2 (2012), No. 1, 15-22 ISSN: 1927-5307 BRIGHT AND DARK SOLITON SOLUTIONS TO THE OSTROVSKY-BENJAMIN-BONA-MAHONY (OS-BBM) EQUATION MARWAN ALQURAN
More informationRelation between Periodic Soliton Resonance and Instability
Proceedings of Institute of Mathematics of NAS of Ukraine 004 Vol. 50 Part 1 486 49 Relation between Periodic Soliton Resonance and Instability Masayoshi TAJIRI Graduate School of Engineering Osaka Prefecture
More informationSoliton Solutions of a General Rosenau-Kawahara-RLW Equation
Soliton Solutions of a General Rosenau-Kawahara-RLW Equation Jin-ming Zuo 1 1 School of Science, Shandong University of Technology, Zibo 255049, PR China Journal of Mathematics Research; Vol. 7, No. 2;
More informationDispersion in Shallow Water
Seattle University in collaboration with Harvey Segur University of Colorado at Boulder David George U.S. Geological Survey Diane Henderson Penn State University Outline I. Experiments II. St. Venant equations
More informationLANGEVIN THEORY OF BROWNIAN MOTION. Contents. 1 Langevin theory. 1 Langevin theory 1. 2 The Ornstein-Uhlenbeck process 8
Contents LANGEVIN THEORY OF BROWNIAN MOTION 1 Langevin theory 1 2 The Ornstein-Uhlenbeck process 8 1 Langevin theory Einstein (as well as Smoluchowski) was well aware that the theory of Brownian motion
More informationGroup analysis, nonlinear self-adjointness, conservation laws, and soliton solutions for the mkdv systems
ISSN 139-5113 Nonlinear Analysis: Modelling Control, 017, Vol., No. 3, 334 346 https://doi.org/10.15388/na.017.3.4 Group analysis, nonlinear self-adjointness, conservation laws, soliton solutions for the
More informationMaking Waves in Vector Calculus
Making Waves in Vector Calculus J. B. Thoo Yuba College 2014 MAA MathFest, Portland, OR This presentation was produced
More informationContinuous limits and integrability for a semidiscrete system Zuo-nong Zhu Department of Mathematics, Shanghai Jiao Tong University, P R China
Continuous limits and integrability for a semidiscrete system Zuo-nong Zhu Department of Mathematics, Shanghai Jiao Tong University, P R China the 3th GCOE International Symposium, Tohoku University, 17-19
More informationThe General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method
Applied and Computational Mathematics 015; 4(5): 335-341 Published online August 16 015 (http://www.sciencepublishinggroup.com/j/acm) doi: 10.11648/j.acm.0150405.11 ISSN: 38-5605 (Print); ISSN: 38-5613
More informationPeriodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type Equation
Contemporary Engineering Sciences Vol. 11 2018 no. 16 785-791 HIKARI Ltd www.m-hikari.com https://doi.org/10.12988/ces.2018.8267 Periodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type
More information1/f Fluctuations from the Microscopic Herding Model
1/f Fluctuations from the Microscopic Herding Model Bronislovas Kaulakys with Vygintas Gontis and Julius Ruseckas Institute of Theoretical Physics and Astronomy Vilnius University, Lithuania www.itpa.lt/kaulakys
More informationPeriodic and Solitary Wave Solutions of the Davey-Stewartson Equation
Applied Mathematics & Information Sciences 4(2) (2010), 253 260 An International Journal c 2010 Dixie W Publishing Corporation, U. S. A. Periodic and Solitary Wave Solutions of the Davey-Stewartson Equation
More informationA NEW APPROACH FOR SOLITON SOLUTIONS OF RLW EQUATION AND (1+2)-DIMENSIONAL NONLINEAR SCHRÖDINGER S EQUATION
A NEW APPROACH FOR SOLITON SOLUTIONS OF RLW EQUATION AND (+2-DIMENSIONAL NONLINEAR SCHRÖDINGER S EQUATION ALI FILIZ ABDULLAH SONMEZOGLU MEHMET EKICI and DURGUN DURAN Communicated by Horia Cornean In this
More informationGeneration of undular bores and solitary wave trains in fully nonlinear shallow water theory
Generation of undular bores and solitary wave trains in fully nonlinear shallow water theory Gennady El 1, Roger Grimshaw 1 and Noel Smyth 2 1 Loughborough University, UK, 2 University of Edinburgh, UK
More informationDavydov Soliton Collisions
Davydov Soliton Collisions Benkui Tan Department of Geophysics Peking University Beijing 100871 People s Republic of China Telephone: 86-10-62755041 email:dqgchw@ibmstone.pku.edu.cn John P. Boyd Dept.
More informationThe Stochastic Burger s Equation in Ito s Sense
The Stochastic Burger s Equation in Ito s Sense By Javier Villarroel We consider the solution to Burger s equation coupled to a stochastic noise in Ito s sense. The main random properties of the wave are
More informationNUMERICAL METHODS FOR SOLVING NONLINEAR EVOLUTION EQUATIONS
NUMERICAL METHODS FOR SOLVING NONLINEAR EVOLUTION EQUATIONS Thiab R. Taha Computer Science Department University of Georgia Athens, GA 30602, USA USA email:thiab@cs.uga.edu Italy September 21, 2007 1 Abstract
More informationMaking Waves in Multivariable Calculus
Making Waves in Multivariable Calculus J. B. Thoo Yuba College 2014 CMC3 Fall Conference, Monterey, Ca This presentation
More informationOn Decompositions of KdV 2-Solitons
On Decompositions of KdV 2-Solitons Alex Kasman College of Charleston Joint work with Nick Benes and Kevin Young Journal of Nonlinear Science, Volume 16 Number 2 (2006) pages 179-200 -5-2.50 2.5 0 2.5
More informationNonlinear Wave Propagation in 1D Random Media
Nonlinear Wave Propagation in 1D Random Media J. B. Thoo, Yuba College UCI Computational and Applied Math Seminar Winter 22 Typeset by FoilTEX Background O Doherty and Anstey (Geophysical Prospecting,
More informationSymmetry Reductions of (2+1) dimensional Equal Width. Wave Equation
Authors: Symmetry Reductions of (2+1) dimensional Equal Width 1. Dr. S. Padmasekaran Wave Equation Asst. Professor, Department of Mathematics Periyar University, Salem 2. M.G. RANI Periyar University,
More informationNew approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations
Volume 28, N. 1, pp. 1 14, 2009 Copyright 2009 SBMAC ISSN 0101-8205 www.scielo.br/cam New approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations HASSAN A. ZEDAN Mathematics
More informationSolutions of differential equations using transforms
Solutions of differential equations using transforms Process: Take transform of equation and boundary/initial conditions in one variable. Derivatives are turned into multiplication operators. Solve (hopefully
More informationMath 575-Lecture 26. KdV equation. Derivation of KdV
Math 575-Lecture 26 KdV equation We look at the KdV equations and the so-called integrable systems. The KdV equation can be written as u t + 3 2 uu x + 1 6 u xxx = 0. The constants 3/2 and 1/6 are not
More informationA Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning
Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 22, 1097-1106 A Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning M. T. Darvishi a,, S.
More informationA Novel Nonlinear Evolution Equation Integrable by the Inverse Scattering Method
Proceedings of Institute of Mathematics of NAS of Ukraine 2002, Vol. 43, Part, 384 39 A Novel Nonlinear Evolution Equation Integrable by the Inverse Scattering Method Vyacheslav VAKHNENKO and John PARKES
More informationLow-Resistant Band-Passing Noise and Its Dynamical Effects
Commun. Theor. Phys. (Beijing, China) 48 (7) pp. 11 116 c International Academic Publishers Vol. 48, No. 1, July 15, 7 Low-Resistant Band-Passing Noise and Its Dynamical Effects BAI Zhan-Wu Department
More informationLecture17: Generalized Solitary Waves
Lecture17: Generalized Solitary Waves Lecturer: Roger Grimshaw. Write-up: Andrew Stewart and Yiping Ma June 24, 2009 We have seen that solitary waves, either with a pulse -like profile or as the envelope
More informationSoliton and Periodic Solutions to the Generalized Hirota-Satsuma Coupled System Using Trigonometric and Hyperbolic Function Methods.
ISSN 1749-889 (print), 1749-897 (online) International Journal of Nonlinear Science Vol.14(01) No.,pp.150-159 Soliton and Periodic Solutions to the Generalized Hirota-Satsuma Coupled System Using Trigonometric
More informationThe Whitham Equation. John D. Carter April 2, Based upon work supported by the NSF under grant DMS
April 2, 2015 Based upon work supported by the NSF under grant DMS-1107476. Collaborators Harvey Segur, University of Colorado at Boulder Diane Henderson, Penn State University David George, USGS Vancouver
More informationStochastic nonlinear Schrödinger equations and modulation of solitary waves
Stochastic nonlinear Schrödinger equations and modulation of solitary waves A. de Bouard CMAP, Ecole Polytechnique, France joint work with R. Fukuizumi (Sendai, Japan) Deterministic and stochastic front
More informationA Very General Hamilton Jacobi Theorem
University of Salerno 9th AIMS ICDSDEA Orlando (Florida) USA, July 1 5, 2012 Introduction: Generalized Hamilton-Jacobi Theorem Hamilton-Jacobi (HJ) Theorem: Consider a Hamiltonian system with Hamiltonian
More informationSymmetry reductions and exact solutions for the Vakhnenko equation
XXI Congreso de Ecuaciones Diferenciales y Aplicaciones XI Congreso de Matemática Aplicada Ciudad Real, 1-5 septiembre 009 (pp. 1 6) Symmetry reductions and exact solutions for the Vakhnenko equation M.L.
More informationOn the Whitham Equation
On the Whitham Equation Henrik Kalisch Department of Mathematics University of Bergen, Norway Joint work with: Handan Borluk, Denys Dutykh, Mats Ehrnström, Daulet Moldabayev, David Nicholls Research partially
More informationRogue periodic waves for mkdv and NLS equations
Rogue periodic waves for mkdv and NLS equations Jinbing Chen and Dmitry Pelinovsky Department of Mathematics, McMaster University, Hamilton, Ontario, Canada http://dmpeli.math.mcmaster.ca AMS Sectional
More informationLecture 11: Internal solitary waves in the ocean
Lecture 11: Internal solitary waves in the ocean Lecturer: Roger Grimshaw. Write-up: Yiping Ma. June 19, 2009 1 Introduction In Lecture 6, we sketched a derivation of the KdV equation applicable to internal
More informationShock Waves & Solitons
1 / 1 Shock Waves & Solitons PDE Waves; Oft-Left-Out; CFD to Follow Rubin H Landau Sally Haerer, Producer-Director Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from
More informationNew Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Equations
ISSN 1749-3889 print), 1749-3897 online) International Journal of Nonlinear Science Vol.008) No.1,pp.4-5 New Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Euations
More informationRecurrence in the KdV Equation? A. David Trubatch United States Military Academy/Montclair State University
Nonlinear Physics: Theory and Experiment IV Gallipoli, Lecce, Italy 27 June, 27 Recurrence in the KdV Equation? A. David Trubatch United States Military Academy/Montclair State University david.trubatch@usma.edu
More informationStochastic contraction BACS Workshop Chamonix, January 14, 2008
Stochastic contraction BACS Workshop Chamonix, January 14, 2008 Q.-C. Pham N. Tabareau J.-J. Slotine Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 1 / 19 Why stochastic contraction?
More informationStochastic Particle Methods for Rarefied Gases
CCES Seminar WS 2/3 Stochastic Particle Methods for Rarefied Gases Julian Köllermeier RWTH Aachen University Supervisor: Prof. Dr. Manuel Torrilhon Center for Computational Engineering Science Mathematics
More informationNumerical Implementation of Fourier Integral-Transform Method for the Wave Equations
Numerical Implementation of Fourier Integral-Transform Method for the Wave Equations Michail D. Todorov Faculty of Applied Mathematics and Computer Science Technical University of Sofia, Bulgaria (Work
More informationInterest Rate Models:
1/17 Interest Rate Models: from Parametric Statistics to Infinite Dimensional Stochastic Analysis René Carmona Bendheim Center for Finance ORFE & PACM, Princeton University email: rcarmna@princeton.edu
More informationThe Orchestra of Partial Differential Equations. Adam Larios
The Orchestra of Partial Differential Equations Adam Larios 19 January 2017 Landscape Seminar Outline 1 Fourier Series 2 Some Easy Differential Equations 3 Some Not-So-Easy Differential Equations Outline
More information08. Brownian Motion. University of Rhode Island. Gerhard Müller University of Rhode Island,
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 1-19-215 8. Brownian Motion Gerhard Müller University of Rhode Island, gmuller@uri.edu Follow this
More informationNotes on the Inverse Scattering Transform and Solitons. November 28, 2005 (check for updates/corrections!)
Notes on the Inverse Scattering Transform and Solitons Math 418 November 28, 2005 (check for updates/corrections!) Among the nonlinear wave equations are very special ones called integrable equations.
More informationSimulating Solitons of the Sine-Gordon Equation using Variational Approximations and Hamiltonian Principles
Simulating Solitons of the Sine-Gordon Equation using Variational Approximations and Hamiltonian Principles By Evan Foley A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND COMPUTER
More informationINTERACTIONS OF SOLITARY WAVES IN HIERARCHICAL KdV-TYPE SYSTEM Research Report Mech 291/08
INTERACTIONS OF SOLITARY WAVES IN HIERARCHICAL KdV-TYPE SYSTEM Research Report Mech 291/8 Lauri Ilison, Andrus Salupere Centre for Nonlinear Studies, Institute of Cybernetics at Tallinn University of Technology,
More informationWhite Noise Functional Solutions for Wick-type Stochastic Fractional KdV-Burgers-Kuramoto Equations
CHINESE JOURNAL OF PHYSICS VOL. 5, NO. August 1 White Noise Functional Solutions for Wick-type Stochastic Fractional KdV-Burgers-Kuramoto Equations Hossam A. Ghany 1,, and M. S. Mohammed 1,3, 1 Department
More informationChapter 3. Head-on collision of ion acoustic solitary waves in electron-positron-ion plasma with superthermal electrons and positrons.
Chapter 3 Head-on collision of ion acoustic solitary waves in electron-positron-ion plasma with superthermal electrons and positrons. 73 3.1 Introduction The study of linear and nonlinear wave propagation
More informationEXACT SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS OF THE KAUP-KUPERSCHMIDT EQUATION
Journal of Applied Analysis and Computation Volume 5, Number 3, August 015, 485 495 Website:http://jaac-online.com/ doi:10.11948/015039 EXACT SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS OF THE KAUP-KUPERSCHMIDT
More informationHead-on collisions of electrostatic solitons in nonthermal plasmas
Head-on collisions of electrostatic solitons in nonthermal plasmas Frank Verheest 1,2, Manfred A Hellberg 2 and Willy A. Hereman 3 1 Sterrenkundig Observatorium, Universiteit Gent, Belgium 2 School of
More informationTwo Loop Soliton Solutions. to the Reduced Ostrovsky Equation 1
International Mathematical Forum, 3, 008, no. 31, 159-1536 Two Loop Soliton Solutions to the Reduced Ostrovsky Equation 1 Jionghui Cai, Shaolong Xie 3 and Chengxi Yang 4 Department of Mathematics, Yuxi
More informationThe Smoluchowski-Kramers Approximation: What model describes a Brownian particle?
The Smoluchowski-Kramers Approximation: What model describes a Brownian particle? Scott Hottovy shottovy@math.arizona.edu University of Arizona Applied Mathematics October 7, 2011 Brown observes a particle
More informationAPPROXIMATE MODEL EQUATIONS FOR WATER WAVES
COMM. MATH. SCI. Vol. 3, No. 2, pp. 159 170 c 2005 International Press APPROXIMATE MODEL EQUATIONS FOR WATER WAVES RAZVAN FETECAU AND DORON LEVY Abstract. We present two new model equations for the unidirectional
More informationLocal vs. Nonlocal Diffusions A Tale of Two Laplacians
Local vs. Nonlocal Diffusions A Tale of Two Laplacians Jinqiao Duan Dept of Applied Mathematics Illinois Institute of Technology Chicago duan@iit.edu Outline 1 Einstein & Wiener: The Local diffusion 2
More informationGalerkin method for the numerical solution of the RLW equation using quintic B-splines
Journal of Computational and Applied Mathematics 19 (26) 532 547 www.elsevier.com/locate/cam Galerkin method for the numerical solution of the RLW equation using quintic B-splines İdris Dağ a,, Bülent
More informationSome Topics in Stochastic Partial Differential Equations
Some Topics in Stochastic Partial Differential Equations November 26, 2015 L Héritage de Kiyosi Itô en perspective Franco-Japonaise, Ambassade de France au Japon Plan of talk 1 Itô s SPDE 2 TDGL equation
More informationBoussineq-Type Equations and Switching Solitons
Proceedings of Institute of Mathematics of NAS of Ukraine, Vol. 3, Part 1, 3 351 Boussineq-Type Equations and Switching Solitons Allen PARKER and John Michael DYE Department of Engineering Mathematics,
More informationCNH3C3 Persamaan Diferensial Parsial (The art of Modeling PDEs) DR. PUTU HARRY GUNAWAN
CNH3C3 Persamaan Diferensial Parsial (The art of Modeling PDEs) DR. PUTU HARRY GUNAWAN Partial Differential Equations Content 1. Part II: Derivation of PDE in Brownian Motion PART II DERIVATION OF PDE
More informationHomogenization with stochastic differential equations
Homogenization with stochastic differential equations Scott Hottovy shottovy@math.arizona.edu University of Arizona Program in Applied Mathematics October 12, 2011 Modeling with SDE Use SDE to model system
More informationModel Equation, Stability and Dynamics for Wavepacket Solitary Waves
p. 1/1 Model Equation, Stability and Dynamics for Wavepacket Solitary Waves Paul Milewski Mathematics, UW-Madison Collaborator: Ben Akers, PhD student p. 2/1 Summary Localized solitary waves exist in the
More informationSmoothing Effects for Linear Partial Differential Equations
Smoothing Effects for Linear Partial Differential Equations Derek L. Smith SIAM Seminar - Winter 2015 University of California, Santa Barbara January 21, 2015 Table of Contents Preliminaries Smoothing
More informationSession 1: Probability and Markov chains
Session 1: Probability and Markov chains 1. Probability distributions and densities. 2. Relevant distributions. 3. Change of variable. 4. Stochastic processes. 5. The Markov property. 6. Markov finite
More informationState Space Representation of Gaussian Processes
State Space Representation of Gaussian Processes Simo Särkkä Department of Biomedical Engineering and Computational Science (BECS) Aalto University, Espoo, Finland June 12th, 2013 Simo Särkkä (Aalto University)
More informationDerivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations
Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations H. A. Erbay Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University, Cekmekoy 34794,
More informationJACOBI ELLIPTIC FUNCTION EXPANSION METHOD FOR THE MODIFIED KORTEWEG-DE VRIES-ZAKHAROV-KUZNETSOV AND THE HIROTA EQUATIONS
JACOBI ELLIPTIC FUNCTION EXPANSION METHOD FOR THE MODIFIED KORTEWEG-DE VRIES-ZAKHAROV-KUZNETSOV AND THE HIROTA EQUATIONS ZAI-YUN ZHANG 1,2 1 School of Mathematics, Hunan Institute of Science Technology,
More informationLecture 12: Transcritical flow over an obstacle
Lecture 12: Transcritical flow over an obstacle Lecturer: Roger Grimshaw. Write-up: Erinna Chen June 22, 2009 1 Introduction The flow of a fluid over an obstacle is a classical and fundamental problem
More informationModified Reductive Perturbation Method as Applied to Long Water-Waves:The Korteweg-de Vries Hierarchy
ISSN 749-3889 (print), 749-3897 (online) International Journal of Nonlinear Science Vol.6(28) No.,pp.-2 Modified Reductive Perturbation Method as Applied to Lon Water-Waves:The Kortewe-de Vries Hierarchy
More informationApplication of linear combination between cubic B-spline collocation methods with different basis for solving the KdV equation
Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 4, No. 3, 016, pp. 191-04 Application of linear combination between cubic B-spline collocation methods with different basis
More informationSolving Nonlinear Wave Equations and Lattices with Mathematica. Willy Hereman
Solving Nonlinear Wave Equations and Lattices with Mathematica Willy Hereman Department of Mathematical and Computer Sciences Colorado School of Mines Golden, Colorado, USA http://www.mines.edu/fs home/whereman/
More informationResearch Article Soliton Solutions for the Wick-Type Stochastic KP Equation
Abstract and Applied Analysis Volume 212, Article ID 327682, 9 pages doi:1.1155/212/327682 Research Article Soliton Solutions for the Wick-Type Stochastic KP Equation Y. F. Guo, 1, 2 L. M. Ling, 2 and
More informationLecture 1: Pragmatic Introduction to Stochastic Differential Equations
Lecture 1: Pragmatic Introduction to Stochastic Differential Equations Simo Särkkä Aalto University, Finland (visiting at Oxford University, UK) November 13, 2013 Simo Särkkä (Aalto) Lecture 1: Pragmatic
More informationLecture 7: Oceanographic Applications.
Lecture 7: Oceanographic Applications. Lecturer: Harvey Segur. Write-up: Daisuke Takagi June 18, 2009 1 Introduction Nonlinear waves can be studied by a number of models, which include the Korteweg de
More informationFractional Laplacian
Fractional Laplacian Grzegorz Karch 5ème Ecole de printemps EDP Non-linéaire Mathématiques et Interactions: modèles non locaux et applications Ecole Supérieure de Technologie d Essaouira, du 27 au 30 Avril
More information1/f noise from the nonlinear transformations of the variables
Modern Physics Letters B Vol. 9, No. 34 (015) 15503 (6 pages) c World Scientiic Publishing Company DOI: 10.114/S0179849155031 1/ noise rom the nonlinear transormations o the variables Bronislovas Kaulakys,
More informationJacobi elliptic function solutions of nonlinear wave equations via the new sinh-gordon equation expansion method
MM Research Preprints, 363 375 MMRC, AMSS, Academia Sinica, Beijing No., December 003 363 Jacobi elliptic function solutions of nonlinear wave equations via the new sinh-gordon equation expansion method
More informationNUMERICAL SOLITARY WAVE INTERACTION: THE ORDER OF THE INELASTIC EFFECT
ANZIAM J. 44(2002), 95 102 NUMERICAL SOLITARY WAVE INTERACTION: THE ORDER OF THE INELASTIC EFFECT T. R. MARCHANT 1 (Received 4 April, 2000) Abstract Solitary wave interaction is examined using an extended
More informationDispersion relations, stability and linearization
Dispersion relations, stability and linearization 1 Dispersion relations Suppose that u(x, t) is a function with domain { < x 0}, and it satisfies a linear, constant coefficient partial differential
More informationIntroduction to nonequilibrium physics
Introduction to nonequilibrium physics Jae Dong Noh December 18, 2016 Preface This is a note for the lecture given in the 2016 KIAS-SNU Physics Winter Camp which is held at KIAS in December 17 23, 2016.
More informationSolitary Wave Solutions of a Fractional Boussinesq Equation
International Journal of Mathematical Analysis Vol. 11, 2017, no. 9, 407-423 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.7346 Solitary Wave Solutions of a Fractional Boussinesq Equation
More informationExact Solutions of The Regularized Long-Wave Equation: The Hirota Direct Method Approach to Partially Integrable Equations
Thai Journal of Mathematics Volume 5(2007) Number 2 : 273 279 www.math.science.cmu.ac.th/thaijournal Exact Solutions of The Regularized Long-Wave Equation: The Hirota Direct Method Approach to Partially
More informationSoliton solutions of some nonlinear evolution equations with time-dependent coefficients
PRAMANA c Indian Academy of Sciences Vol. 80, No. 2 journal of February 2013 physics pp. 361 367 Soliton solutions of some nonlinear evolution equations with time-dependent coefficients HITENDER KUMAR,
More informationKdV equation obtained by Lie groups and Sturm-Liouville problems
KdV equation obtained by Lie groups and Sturm-Liouville problems M. Bektas Abstract In this study, we solve the Sturm-Liouville differential equation, which is obtained by using solutions of the KdV equation.
More information2. The generalized Benjamin- Bona-Mahony (BBM) equation with variable coefficients [30]
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.12(2011) No.1,pp.95-99 The Modified Sine-Cosine Method and Its Applications to the Generalized K(n,n) and BBM Equations
More informationPresenter: Noriyoshi Fukaya
Y. Martel, F. Merle, and T.-P. Tsai, Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gkdv Equations, Comm. Math. Phys. 31 (00), 347-373. Presenter: Noriyoshi
More informationThe first integral method and traveling wave solutions to Davey Stewartson equation
18 Nonlinear Analysis: Modelling Control 01 Vol. 17 No. 18 193 The first integral method traveling wave solutions to Davey Stewartson equation Hossein Jafari a1 Atefe Sooraki a Yahya Talebi a Anjan Biswas
More informationAuto-Bäcklund transformation and exact solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear terms of any order
Physics Letters A 305 (00) 377 38 www.elsevier.com/locate/pla Auto-Bäcklund transformation and exact solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear terms of any
More informationENVELOPE SOLITONS, PERIODIC WAVES AND OTHER SOLUTIONS TO BOUSSINESQ-BURGERS EQUATION
Romanian Reports in Physics, Vol. 64, No. 4, P. 95 9, ENVELOPE SOLITONS, PERIODIC WAVES AND OTHER SOLUTIONS TO BOUSSINESQ-BURGERS EQUATION GHODRAT EBADI, NAZILA YOUSEFZADEH, HOURIA TRIKI, AHMET YILDIRIM,4,
More information