Question

au + 6. The Korteweg-de Vries equation ди au + 6u 0 ( KV) at ax ar3 is an interesting model partial differential equation because two different physical effects are present: there is an expectation that solutions decay due to the third-order dispersive term; how- ever, the nonlinear term causes waves to steepen. Show that elementary traveling wave solutions of (KdV): u(x,t) = f(x - ct) yields an equation (5) corresponding to conservation of energy: 3(59)2 + fu? - bcf2 - Af = E, (KJVCE) where A is a constant, and E is the constant total energy.

Answer 1

The sau ware equation is giren by. dau(ait) - gk dru(x, t) Qt2 at2 solution o coith a cohen u(uit): Ascc.hk Caxest) agk= se equation cosich shallow water The desember elimension urcler one caves in is given by

1 q whese va is A spatial caniable x If Eco-owio and urn for all timest Ute V og hvat Our e bagh Vannas /7,6) funcfon clefining elocation na as the time variable.E. gy 9,8 m/s 6-) meam wasche dopish (m) ν 꽃 에 , t - to let h The standarche form of ledo is. Utrbuurt unnno. En ① Inicially u(x,0) = Uo(a). Therefore uenit). clenvafoves decays: rapidly if all a approaches as /u/to Consicher o and u be too solution of blu and let co-e-2 them. wote connnt ou con f 60 2x za. it as multiply by w of integrate it wing by parts welneo So we get IE(E): integration a z ware of un- 4 wa da.

coace solution fo re elementary havelling kar equation u(a,t)= fra-ct) = f(2) whose transition is - Egn @ from one constant asymptotic state as 2-20 and anorker constant asymptotic state as 270 Apply Eqn in egr 0 standard form get du of kdo equation we هادى f 64 du t es dt 223 az * Egin anch egn @ >tha thioch order ordinary differential equatan anesa fill cfle offlzo coe know fama a=fl" and order. Integrado greld hondinat oxelinary differential equation fll +37²-cf-A=0 → Egn constant multiply Egn ③ by fl and integrate with respect to & + (f) aff3 - £cf". Af=E newtons law) а. where we get re + ? 2 This equation corresponds to conservation

re of energy where e Constant total energy £ (fi) kinetic energy f3 $ cf-Af pohostial energy can graphically represent potential energy as a function of f. conitical points for potintal occur. 310-cf-Azo aceoseling to Egmo blyac = ceza Cifra so , potential ene monotonically increasing too equilibria epost 0 ad saalde foon point کره energy for CRPIRA 70 / One osher Å a center potential EPILASO f Emin mar k. 6. 4680=-(f-dmor) (A-frinje? fman : tc. & solitung were solution cefnit) C seth it fmino & fmanso 65448 1062