In general a wave can have components of both, and the displacement itself becomes a vector quantity, u(x, t). Sabir, Z., Wahab, H.A.: Evolutionary heuristic with Gudermannian neural networks for the nonlinear singular models of third kind. Equation 9.2.8 describes a wave characterized by a one-dimensional displacement (either longitudinal or transverse) in three dimensions. Rezazadeh, H., Ullah, N., Akinyemi, L., Shah, A., Mirhosseini-Alizamin, S.M., Chu, Y.M., Ahmad, H.: Optical soliton solutions of the generalized non-autonomous nonlinear Schrödinger equations by the new Kudryashov’s method. Radha, R., Kumar, C.S.: Localized excitations and their collisional dynamics in (2+1)-dimensional Broer–Kaup–Kupershmidt equation. Mirzazadeh, M., Eslami, M., Zerrad, E., Mahmood, M.F., Biswas, A., Belic, M.: Optical solitons in nonlinear directional couplers by sine-cosine function method and Bernoulli’s equation approach. Then, the reduced ODE was solved with the help of two methods which are called the modified \((G^/G)\)-expansion method and traveling wave solutions of nonlinear the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity. Firstly, the given system was reduced to an ordinary differential equation (ODE) with the help of the wave transformations.
Examples are provided in Chapter 4, Activity 2, and Chapter 5, Activity 1.The exact solutions of the (2 + 1) dimensional Broer–Kaup–Kupershmidt (BKK) system which has been recommended to model the nonlinear and dispersive long gravity waves traveling along with the two horizontal directions in the shallow water of uniform depth were obtained. In this situation, the oscillatory time dependence does not cancel out in calculations, but rather accounts for the time dependence of physical observables. When a system is not is a stationary state, the wavefunction can be represented by a sum of eigenfunctions like those above. A wavefunction with this oscillatory time dependence e-iωt therefore is called a stationary-state function. We will see that all observable properties of a molecule in an eigenstate are constant or independent of time because the calculation of the properties from the eigenfunction is not affected by the time dependence of the eigenfunction. When molecules are described by such an eigenfunction, they are said to be in an eigenstate of the time-independent Hamiltonian operator.
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