What we do understand by the notion of soliton or rather solitary wave is a wave prole which is very stable in the following sense

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1 Introduction to Waves and Solitons What we do understand by the notion of soliton or rather solitary wave is a wave prole which is very stable in the following sense i) It is localized, which means it decays as ±. ii) It is a traveling wave. That is, the wave prole u at any time t 0 and R is determined by u(t, ) = f( ct), where f is a function on R and c denotes the speed of propagation. iii) iv) It satises some kind of dispersion relation (taller waves travel faster than smaller ones). It obtains a sort of superposition principle. Like linear waves, if two solitary waves with dierent speed (the one with higher speed behind the one with slower speed) travel together, the taller one will catch up after some nite time, an interaction of both waves takes place and afterwards they separate again, both keep speed and shape. But now the taller wave is in front of the smaller one. This kind of wave was rst discovered by an English civil engineer called John Scott Russel in 834. One day he was passing by a canal where a boat was drawn by two horses along the canal. Russel observed that the boat was pushing a wave and when the boat suddenly stopped, the wave traveled forward along the canal without loosing its shape or speed. He was following the wave for some miles before he lost it in the windings of the canal. Motivated by this observation J.S. Russel built a canal and tried successfully to recreate this wave, which he called the great wave of translation. This wave satised all the properties we claim for a wave being called a solitary wave. Up to then no mathematician took Russel and his discoveries for serious and the great wave of translation became forgotten for about 60 years. In 895 two dutch mathematicians (Korteweg and de Vries) studied shallow water waves and derived a model for describing them. This equation is the famous KdV equation given by u t 6uu + u = 0. By chance they found out that this equation permits localized wave solutions, which satisfy all the properties that where predicted by Russel 60 years ago. Finally an equation was found, that admits solitary wave solutions. A wave packet consisting of n solitary waves propagating with dierent speeds is called an nsoliton and the eistence of nsolitons as solutions to KdV was rstly discovered by Kruskal and Zabusky in 965. Because of its application in physics and computer science the study of solitons became very popular in the last 40 years. So far only discovered by computer animations (numerical methods), there eists the conjecture that for

2 2 any initial data, the KdV admits a solution which splits for t into an nsoliton plus small perturbation terms. What makes the KdV equation so special is, that it contains a linear and a nonlinear part, which on their own fail to admit solitary wave solutions, but the balance of both eects (dispersion for the linear part and localization for the nonlinear part) allow for solitary waves. In the following we will see why both equations u t + u = 0 (linear KdV equation) and u t + uu = 0 (Burgers equation) do not permit solitary wave solutions, but the KdV equation does. At the end a connection between traveling solutions of KdV and elliptic curves is shown. Linear KdV equation (Standard eample of dispersive waves). Dispersive means that there eists a relation between wavenumber k and speed c of a wave. For eample, waves with high wavenumber travel faster. The linear KdV equation is given by u t + u = 0. It allows for traveling waves, which means there eist solutions of the form u(t, ) = f( ct). If we assume to have a crest at the origin, the complete set of traveling solutions is given by u(t, ) = h + a cos(k ωt), h, a, k, ω R, where the relation ω = k 3 needs to be satised. The parameter ω denotes the frequency, hence the speed is given by c = ω k. In other words the dispersion relation for the linear KdV equation is given by c = ω k = k2. We see the following: First, the traveling waves are left propagating and second, the dispersion relation yields that waves with high wavenumber travel faster. We conclude: Properties (Linear KdV equation). i) Admits traveling wave solutions. ii) Traveling wave solutions are periodic; no localization. iii) Traveling wave solutions satisfy a dispersion relation. Properties i) and iii) are something we like to see when searching for solitary waves. Unfortunately, the linear KdV equation does not permit traveling wave solutions which are not periodic but localized.

3 Burgers equation (Standard eample of wave breaking). The Burgers equation is a nonlinear equation given by 3 u t + uu = 0. () It does not admit any traveling solutions. Assume there eists a traveling solution, which is not the trivial solution (u is constant). That means u(t, ) = f( ct) with f nonconstant. Then we obtain from () cf + ff = 0, which is the case if (f c)f = 0. Since we suppose f to be not a constant function, f can not be zero everywhere, thus f = c. But this again contradicts the assumption that f is nonconstant. Solutions of () are implicitly (uniquely) determined by u(t, ) = f( ut). A proof by Methods of Characteristics can be found in [, Chapter 4]. The Burgers equation allows for localized solutions. Unfortunately, for all localized solutions there appears wave breaking after nite time. Take for eample the initial condition 0, 2 + 2, ( 2, ) u(0, ) = f() :=, [, ] 2, (, 2) 0, 2. f t = 0 Since we know that the solution is given by u(t, ) = f( ut), we obtain 0, 2 ut + 2, ut ( 2, ) u(t, ) = f( ut) =, u [, ] 2 + ut, ut (, 2) 0, 2.

4 4 We can calculate the implicitly given terms and nd 0, t, +2 +t t ( 2, ) u(t, ) =, u [, ] 2 t, 0, 2. 2 t t (, 2) Note that at time t = the solution is not continuous anymore. wavebreak. At this point we obtain a u(0, ) u( 2, ) u(, ) t = 0 t = 2 t = Properties (Burgers equation). i) Admits localized solutions. ii) No traveling wave solutions; wave breaking. Hence, the nonlinear Burgers equation satises the condition of a localized wave with decay, but, unfortunately, it is not stable in shape, but the wave will break after nite time. KdV equation (Standard eample of solitary wave solutions). The KdV equation is one besides many other equations like for eample the SineGordon equation or the nonlinear Schrödinger equation, which allow solitary wave solutions. It became the standard eample for this kind of solution because the eistence of solitons was rstly discovered via the KdV equation, which is given by u t 6uu + u = 0. Note that the KdV equation serves as a combination of the linear KdV equation and the nonlinear Burgers equation. Back in 895 it was a complete surprise that these two eects (dispersion and localization) in some sense balance each other such that it permits for very stable traveling waves. Searching for traveling waves, we assume that u(t, ) = f( ct), which leads to cf 6ff + f = 0. By integrating this equation, multiplying it by 2f and again integration, we obtain cf 2 2f 3 + (f ) 2 = 2af + b, (2)

5 for integration constants a, b R. Since we claim solitary waves, which decay as ±, we have f() = f () = f () = 0 as ±. This implies a = b = 0 in (2). The solution to the ordinary dierential equation cf 2 2f 3 + (f ) 2 = 0 is given by (separation of variables) f() = [ ] 2 c 2 sech2 c( 0 ), where 0 is a constant we can always choose to be zero and sech() = 2(e + e ). Thus, the KdV equation admits localized traveling wave solutions with a dispersion relation in that sense that taller waves travel faster (eactly how Russel described them to be) u(t, ) = [ ] 2 c 2 sech2 c( ct 0 ). (3) 5 Remark. i) The KdV equation also admits periodic traveling waves, socalled cnoidal waves, by solving (2) for arbitrary a, b R. ii) One can show (see [] and references therein) that two solitary waves of dierent speed (when the taller one is behind the smaller one) do collide, interact and after some time separate again with same speed and shape as before the interaction. This behavior was so far only known for linear equations and it was a big surprise that something like that can be possible also for nonlinear equations. In total, the KdV equation admits the following solutions KdV equation traveling wave solutions other solutions solitary solutions periodic solutions Connection between traveling wave solutions to KdV and elliptic curves. Let Λ be a lattice in C. The Weierstraÿ function is given by p(z) := z 2 + ( (z ω) 2 + ) ω 2 ω Λ\{0} The function p satises the following properties: for all z C. (4)

6 6 i) p is periodic, that is p(z + ω) = p(z) for z C and ω Λ. ii) p is even, that is p( z) = p(z) for all z C. iii) p has innitely many poles (for all ω Λ). iv) p solves p = 2pp. (5) Note, that p solves (5) not in the classical sense, since p is not even continuous in ω Λ and thus, far away from being dierentiable. But still, at least formal or in a weak sense, p satises (5). It is now easy to check, that u(t, ) := 2 cp ( 2 ) c( ct) c, t 0, R, (6) 6 is a family (in c) of solutions to the KdV equation. The parameter 0 R serves again as an integration constant and might be set to zero. These are the solutions corresponding to the traveling wave solutions that are periodic. On the other hand, ing the lattice by claiming that the lattice is generated by ω R and ω 2 ir, the Weierstraÿ function (and therefore the family of solutions to KdV, dependent on the choice of Λ) corresponds to the elliptic curve C /Λ. Considering the limit case, where ω tends to innity, one notices by onepoint compactication, that the elliptic curve deforms to a degenerate noded curve. These curves (which are not elliptic anymore) correspond to the case of solitary waves. In fact, the Weierstaÿ function (4) converges (in the limit case ω ) in some weak sence to the sech 2 solution (3), the solitary wave solution. Thus, we get the following connection u Λ (t, ; c) {family of per. solutions to KdV} C/Λ {elliptic curve} and in the case ω {solitary wave solution to KdV} {degenerate noded curve} References [] W.A. Strauss: Partielle Dierentialgleichungen, Vieweg und Sohn Verlagsgesellschaft mbh, Braunschweig/Wiesbaden, 995. [2] P.G. Drazin, R.S. Johnson: Solitons: An Introduction, Cambridge University Press, 989. [3] E. Arbarello: Sketches of KdV, in Symposium in Honor of C.H. Clemens (Salt Lake City, UT, 2000), Contemp. Math., Amer. Math. Soc., Providence, RI, pp. 969, 2002.