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MIT OpenCourseWarehttp://ocw.mit.edu 18.306 Advanced Partial Differential Equations with Applications Fall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Weakly Nonlinear Expansions for Breathers. Rodolfo R. Rosales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139 October 10, 2003 Abstract Solitary waves are localized traveling steady profile solutions for dispersive nonlinear dynamical systems — usually modeled by a pde, or a system of pde’s. Thus, at least in 1+1 dimensions, they are relatively easy to characterize analytically — since they correspond to solutions of the ode’s to which the pde’s reduce in a coordinate moving with the wave. Breathers are also localized traveling waves, but their profile is not steady, but changes periodically in time. A possible mathematical definition of what, exactly, a breather is could go as follows: It is a localized solution of the equations such that, in an appropriately selected moving coordinate frame, the solution is periodic in time. This definition does not capture all the features of the available known exact breather solutions. For these examples, in a coordinate system moving with the wave, the wave profile is itself a moving periodic steady traveling wave, contained within an amplitude envelope that keeps it localized in space. Thus, these breathers are wave-package solutions, with a localized envelope that is itself a traveling steady profile. Unfortunately, the notions of wave-package, and envelope, are not ones for which precise, and sufficiently general, definitions can be provided — at least not for fully non-linear systems. Breathers are hard to characterize analytically, even in 1+1 dimensions, since they are solutions for which the governing pde’s cannot be reduced to a lower order system. In these notes we show how to produce expansions for breather solutions, in the weakly nonlinear limit, where the breathers are of small amplitude. In this limit one can look for breathers with very “shallow” envelopes, in which case separation of scales allows a reduction of the governing equations to a lower order system. Contents 1 The Sine Gordon Equation. Lorentz invariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Limit on the signal propagation speed. . . . . . . . . . . . . . . . . . . . . . . . . . 3 1Rosales Weakly Nonlinear Expansions for Breathers. 2 1.1 Kink and anti-kink solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 PROBLEM 1: Write pseudo-spectral code. Do kinks & anti-kinks interacting. . . 4 Hint: Description of pseudo-spectral code. . . . . . . . . . . . . . . . . . . . . . . . 4 How to compute spectral derivatives for a mod-2ω periodic function. . . . . . 4 Kink asymptotic behavior as x � ±∼. . . . . . . . . . . . . . . . . . . . . . . . . . 4 Kinks direct relation to linearized (real) decaying exponential solutions. . . . . . . . 5 1.2 Breather solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Breathers parameterized by envelope speed and amplitude. . . . . . . . . . . . . . . 5 Breather envelope and phase properties. . . . . . . . . . . . . . . . . . . . . . . . . 6 Breathers parameterized by the wave-number of the linearization at ∼. . . . . . . . 6 Parameter expansions when the envelope decays slowly. . . . . . . . . . . . . . . . . 7 Breather asymptotic behavior as x � ±∼. . . . . . . . . . . . . . . . . . . . . . . . 7 Breather direct relation to linearized (complex) decaying exponential solutions. . . . 7 PROBLEM 2: Compute breather, kink and anti-kink interactions. . . . . . . . . . 8 2 Breather Expansion. 8 Nonlinear Klein-Gordon equation in 1+1 dimensions. . . . . . . . . . . . . . . . . . 8 Lorentz invariance and maximum signal propagation speed. . . . . . . . . . . . . . . 8 Relate breather to linearized exponential solutions. . . . . . . . . . . . . . . . . . . 9 Use Lorentz invariance to simplify the problem. . . . . . . . . . . . . . . . . . . . . 9 Perturbation equations to be solved. . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Fredholm alternative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Expansion in powers of the small parameter. . . . . . . . . . . . . . . . . . . . . . . 9 Condition for the existence of a breather. . . . . . . . . . . . . . . . . . . . . . . . . 10 Approximate expression for the breather solution. . . . . . . . . . . . . . . . . . . . 10 Questions regarding the validity of the expansion. . . . . . . . . . . . . . . . . . . . 10 Computing breathers numerically. Bifurcation approach. . . . . . . . . . . . . . . . 11 PROBLEM 3: Breathers for KdV-type equations. . . . . . . . . . . . . . . . . . . 12 Hint: for problem 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Rosales Weakly Nonlinear Expansions for Breathers. 3 1 The Sine Gordon Equation. Here we give examples of solitary waves and breathers. These for the Sine-Gordon equation, where exact analytical expressions are known. The Sine-Gordon equation is given by utt − uxx + sin u = 0, (1.1) where u is an angle. In another set of notes (actually, a series of problems) we show how this equation can be used …


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