Integrability of nonlinear wave equations and solvability of their initial value problem

Abstract

We investigate one particular prototypical system of evolution equations in 1+1 dimensions, which models the (nonlocal) interaction of two waves in quadratic nonlinear media. These equations are integrable as they possess a Lax pair. We consider the spectral method of solving the associated initial value problem and, as recently pointed out, we show that generically the associated initial value problem cannot be linearized. We further elaborate on the integrability issue for this model by displaying the corresponding conservation laws and the motion of a particular soliton.

Rational solitons of wave resonant-interaction models

“…In particular, for the system (2), if x is the evolution (f.i. time) variable, then this system is the the well known 3 wave resonant interaction (3WRI) equation [4] where the three characteristic velocities are c 1 , c 2 , 0; otherwise, if the evolution variable is t, this system models the nonlocal interaction of two waves (NL2W) [5,6]. Here rescaling transformations have been used to give the equations (1) and (2) a neat form in terms of their coefficients.…”

General splitting methods for abstract semilinear evolution equations

“…Nonlinear wave interaction model. Consider the system of evolution equations modelling wave-wave interaction in quadratic nonlinear media (see [2] and references therein). This model describes the nonlinear and nonlocal cross-interaction of two waves in 1 + 1 dimensions.…”

“…(T) is a conserved quantity, this condition holds for any t. The potential V can be calculated by V = −G * , where = D − |u|2 and G is the Green potential defined as the 1-periodic function such that G(x) = x(1 − x)/2 on [0,1]. We consider H 0 = L 2 (T), H 1 = H 2 (T),and defining the self-adjoint operator A = −∂ xx and B (u) = −|u| 2 u + (G * )u, we can write (5.1) in the form (1.1) and from Lemma 4.2, B satisfies (3.4).The linear flow Φ A can be written as Φ A (t)u (x) = p∈Z ûp e −i4π 2 p 2 t e i2πpx , where ûp = T u(x)e −i2πpx dx.…”

N -soliton solutions for the nonlocal two-wave interaction system via the Riemann–Hilbert method

“…For example, multi-soliton solutions, collisions, algebro-geometric solutions, asymptotic behaviour of solutions, and other properties have been widely investigated for the coupled nonlinear Schrödinger equation of Manakov type. [5][6][7][8][9][10][11][12][13] The aim of this paper is to study the Cauchy problem of the nonlocal two-wave interaction system from the Manakov hierarchy [14][15][16][17]…”