By Louis De Broglie

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**Extra info for An introduction to the study of wave mechanics**

**Example text**

12. 86 celllls- l . 13. lS-l Ls cell VrnLs2 = 63. 14. Width L s (ceII) of the soliton as a function of the square root of the velocity v (celllls- 1t 1/2 dots: experiments; continuous line: theory. 13. Onee the veloeity is determined one ean express the width L s in eell units. 4 . 15. Amplitude V m (Volts) of the soliton as a function of its velocity v (celllls- l ). Dots : experiments; continuous line: theory. l, Zabusky and Kruskal discovered numerically that two solitons which are solutions of the KdV equation conserve their identities after interacting nonlinearly with each other.

As time increases, the initial parabolic waveform is progressively deformed. 11). 3b. 3. 8 the wave breaking is not observed on a real transmission line because the wave of voltage cannot be multivalued (b) In the linearized case b=O, the pulse moves with constant velocity and without change of form. 1b) with respect to t, and eliminating the current, one obtains the wave equation 02 v = t ~ [c(v) ~Vt ]. 7) contains the linear terms. The right-hand side accounts for the nonlinearity. 3). Remark Note that in the above transmission line, instead of using a nonlinear capacitance c(v) and a linear inductance t, we can use a nonlinear inductance t(i) with a linear capacitance c.

5) for the KdV model. When they propagate, the solitons separate from each other, each traveling with its own velocity and a soliton of large amplitude traveling faster than a soliton with low amplitude. _. _.. _......... :........... _ ....... _.. .... _... ' ~ .... ~........ ,.... j ~ a . i . ·-:. ·. ·. ·;_·. ·,·. ···. ·:-·,·_·,··'s.... _... ···_·_~.. · .. ·_ .. " ...... _..... ,.... -.. : ··i·····_·.. -f·· .. :..... ; . ,...... ,....... L..... :. ,1...... ;-_·_--1··.. _.. . . i ; ·. {-·.