Nonlinear Waves Equation: Integral System Problem, Solution & Analysis

This document presents a comprehensive solution to a Nonlinear Waves Equation Integral System Problem. It begins by addressing the invariance of the equation under specific transformations, detailing the derivation of the solution and the application of scaling transformations. The solution explores the Korteweg-de Vries equation and nonlinear Schrödinger equations. The document includes a demonstration of how a given function remains invariant under the transformations and provides an analysis of the problem, including the use of scaling transformations and the resulting constraint equations. The document concludes with a list of relevant references, including key publications on the topic, such as works by Miura and De Bouard, Hayashi, and Kato, providing a valuable resource for students studying nonlinear wave phenomena.

Nonlinear Waves Equation Integral System Problem 1
NONLINEAR WAVES EQUATION INTEGRAL SYSTEM PROBLEM
By(Name)
The Name of the Class(Course)
Professor(Tutor)
The Name of the School(University)
The City and State Where it is located
The Date

Solution 3(a)
(a) Solution of equation (2) which remains invariant
until S, it satisfies (2) and constraint xu2 + 2 tu
Xu2 + 2 tu = 0
Given that S2= e 2n
Since ex ent, e n u = e2n
Then 2 tu = e2n
Xu2+ e2n = 0 = x, tu
eu xu
u + e2n= t
t= e2n + u
The equation (2) is invariant under the
transformation since it is a quadratic equation.
When we equate s2(x, t, u) it will give us a resulting
notation of e2n which is an invariant.

Solution (b)
(B) Scaling transformations
S1 (x, t, u) = (en x, e2 t, enu)
S2 (cx, tu) = (ex, en t, enu)
Cx = ex
S2 (tu = ent en u)
tu tu
S2= e2n
To satisfy equation (2) there has to be a constraint
equation to e2n + u therefore t value is constant from
solution (a S2 = e2n hence 2tu = e2n

Solution (c)
(c) Show that the solution U (x,t) = 3x + 9t is invariant
under S2
S2 = e 2n
Therefore
t= u + e2n
U (x, u +e2n) = 3x + 9 (u +e2n)
(Ux, u + e2n) = 3x + (9u +9 e2n)
U,x e2 u, e 2x2
U + n = 21 + n
n n
u = 21
From the computation it is clearly visible that
equating u(x, u + e2n) to 3x + 9 (u+e2n) giving us a
constant invariant = 21 for