Identification of solitary-wave solutions as an inverse problem: Application to shapes with oscillatory tails

Kawahara solitons in Boussinesq equations using a robust Christov–Galerkin spectral method

“…Table 2 Best fit curve formulas for spectral coefficients of initial condition of 4BTE. For the 4BTE there is no exact solution available and the comparison is done with the computational solutions obtained in [3]. In [3] the stationary shapes of the 4BTE (21) are computed using the method of variational imbedding (MVI).…”

“…For the 4BTE there is no exact solution available and the comparison is done with the computational solutions obtained in [3]. In [3] the stationary shapes of the 4BTE (21) are computed using the method of variational imbedding (MVI). The latter was introduced in [23] and applied to finding the homoclinic solution of the Lorentz equations.…”

“…However, the linear term representing the restoring force, qualitatively changes the equation for stationary waves. The problem in the moving frame (stationary case), was treated successfully with the aid the of method of variational imbedding in [3]. Here, using the aforementioned Christov-Galerkin spectral method, we rediscover the results obtained in [3] and extend our analysis to the time-dependent problem where we study the propagation and interaction of the solitary waves.…”

“…Here, we consider the sixth-order Boussinesq equation and a fourth-order Boussinesq type equation with a linear term, which have been studied using different numerical techniques (see, e.g. [1][2][3]). …”

Numerical Identification of Solitary Wave-Like solutions in Boussinesq Equation as Inverse Problem

“…Due to the nature of asymptotic boundary conditions for decay of the amplitude of the wave at infinity,the system under consideration always possesses a trivial solution. The above outline ddifficulties are much harder when the solutions have nonmomotone tails and when the intervals where they are well separated from zero, are large (see [3]). In many instances,the problem of finding the nontrivial amplitude of the bifurcating solution is akin to the problem of coefficient identification.…”