The system of nonlinear 2nd order O.D.E.s are given by,
y"(t)-ay(t)={b(y(t))^2+c(x(t))^2}y(t),
x"(t)-ex(t)={b(x(t))^2+c(y(t))^2}x(t);
a,b,c,e are real constants, y"(t) and x"(t) indicate the second order derivatives of y(t) and x(t) respectively with respect to t.
you can decouple the equations into 2 o.d.e.s using generalized coordinates, and solve each equation separately, using different customary methods. In vibration analysis of M.D.o.F systems, this method is used very often. For more, You can check Roa Mechanical Vibration, Chapter 6, named 2 degrees of freedom systems.
- Hello Shibam, solution 1: indeed, as said above, using generalized coordintates with each equation, the Euler/Lagrange equation should work, (ref. ISBN: 978-613-7-37853-3, page 172, Chap: 14.5+... d/dt (dF/dx')-dF/dx = 0 and (dG/dy')-dG/dy = 0 (no constraints), where F= F(x, x', x'',t), G= G(y, y', y'',t). Same equations will be find using generallized form and H(x,y,x',y',x'',y"",t). You should obtain two equations that seems to lead to x, y belonging to a circle, hyperbola or else, base on certains conditions on a,b,c,d and e (I did not complete the calcs but it seems), from there, guess the solutions and input in the initial equations, etc. Seems to be sin and cos. Solutio2: U-substitution by u=dx/dt or dy/dt, replace x"=d2x/dt2 by (du/dx)(dx/dt), then isolate all x terms to the right and u terms to the left, you end up with integrals to play with. Solution 3: use of infinite series sum of an.t^n and sum of bn.t^n, this will lead to terms of the series in bn-2 = function of bn and an-2 function an. There is multinomial and multinomial coefficient to play with, will be tricky. Try also these solutions that may lead to Laplace transform and Generalized Integral Transform (GN) of compositions, which is tricky to evaluate as you may have x2, y2 or x3, y3 to transform: Laplace or Double Laplace Transform of Derivatives, GN, maybe Fourrier series tooo, I did not try that one.
When b=c and a=e you can use the transformation
x=R(t)cos(\theta(t)), y=R(t)sin(\theta(t)) to reduce the system to a set of two equations in R(t) and \theta(t) that are easily decoupled as
R''-cR^{-3}-a R=bR^3 and R^2 \theta'=c
or something similar (I am just doing the computations on the back of an envelope). For the first equation it is simple to obtain the energy integral and to perform a qualitative analysis. When b \neq c you can try with the transformation
c x=R(t)cos(\theta(t)), b y=R(t)sin(\theta(t))
maybe you find something similar with some more general conditions on the parameters with respect above.
Thank you Prof. Giuseppe Saccomandi and other researchers for your reply, actually I have taken similar types of transformations for the problem and was able to decouple those equations a few days ago. Currently I am trying to complete the mathematical calculations and related computations. But, I would like to mention that b can't be equal to c as they are fixed real constants; a and e can be freely assigned to any nonzero real constant. Also, I would like to mention that the aforesaid equations were obtained as an interim step to develop the breather type analytical solution of 3rd order coupled nonlinear evolution equation for rogue waves. I am looking for a suitable journal where this type of study can be published.
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