o Projects for Chapter 12 A Solitons and Korteweg-de Vries Equation The Korteweg

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o Projects for Chapter 12 A Solitons and Korteweg-de Vries Equation The Korteweg-de Vries (Kedv) equation ou au a at 2 where u r) is the vertical displacement at position r and time t is used to model long water waves. Of particular interest are traveling wave solutions, which are solutions of the form u(u, t (r ct), where c is the constant wave speed (a) Show that v(x ct) is a solution to the Kdv equation provided u(z), z x ct. satisfies du div 1 d dz dz 2 dz (b) of the traveling waves, we are most interested in solitons, traveling waves that are positive and decay to zero at oo. Therefore, integrate once and set the constant of integration equal to zero to obtain (c) Sketch the phase plane diagram for this conservative system. (d) Use the results of parts (a) and (b) to argue that there exist solitons for the Kdv equation. [Hint: The existence of a positive function v (z) that decays to zero as z too corre- sponds to a trajectory in the vv'-plane that begins at the origin and returns to the origin.l e) Does increasing the speed of the soliton produce a shorter or taller wave? Burger's Equation le model for diffusive waves in fluid mechanics that incorporates nonlinearity and diffu Burger's equation: ou ou u at an

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