Quadratic Nonlinear Klein Gordon Equation Research Articles

Bright, Dark, and Rogue Wave Soliton Solutions of the Quadratic Nonlinear Klein–Gordon Equation

This article reflects on the Klein–Gordon model, which frequently arises in the fields of solid-state physics and quantum field theories. We analytically delve into solitons and composite rogue-type wave propagation solutions of the model via the generalized Kudryashov and the extended Sinh Gordon expansion approaches. We obtain a class of analytically exact solutions in the forms of exponential and hyperbolic functions involving some arbitrary parameters with the help of Maple, which included comparing symmetric and non-symmetric solutions with other methods. After analyzing the dynamical behaviors, we caught distinct conditions on the accessible parameters of the solutions for the model. By applying conditions to the existing parameters, we obtained various types of rogue waves, bright and dark bells, combing bright–dark, combined dark–bright bells, kink and anti-kink solitons, and multi-soliton solutions. The nature of the solitons is geometrically explained for particular choices of the arbitrary parameters. It is indicated that the nonlinear rogue-type wave packets are restricted in two dimensions that characterized the rogue-type wave envelopes.

Modified scattering for the quadratic nonlinear Klein–Gordon equation in two dimensions

In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein–Gordon equation (NLKG) in two space dimensions: $(\Box +1)u=\lambda |u|u$, $t\in \mathbb {R}$, $x\in \mathbb {R}^{2}$, where $\Box =\partial _{t}^{2}-\Delta$ is d’Alembertian. For a given asymptotic profile $u_{\mathrm {ap}}$, we construct a solution $u$ to (NLKG) which converges to $u_{\mathrm {ap}}$ as $t\to \infty$. Here the asymptotic profile $u_{\mathrm {ap}}$ is given by the leading term of the solution to the linear Klein–Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on Fourier series expansion of the nonlinearity.

A bilinear estimate and its application to a quadratic nonlinear Klein–Gordon equation in two space dimensions

We study the existence of the wave operators for the nonlinear Klein–Gordon equation with quadratic nonlinearity in two space dimensions ( ∂ t 2 − Δ + 1 ) u = λ u 2 , ( t , x ) ∈ R × R 2 . We prove existence of wave operators in lower order Sobolev spaces by using estimates of bilinear operators associated with Klein–Gordon equation.

A Generalized Auxiliary Equation Method for the Quadratic Nonlinear KG Equation

A generalized auxiliary equation method with symbolic computation is used to construct more general exact solutions of the quadratic nonlinear Klein-Gordon (KG) equation. As a result, new and more general solutions are obtained. It is shown that the generalized auxiliary equation method provides a more powerful mathematical tool for solving nonlinear partial differential equations arising in the fields of nonlinear sciences.

Singular solitons and bifurcation analysis of quadratic nonlinear Klein-Gordon equation

This paper studies the Klein-Gordon equation with quadratic nonlinearity. The ansatz approach is used to first obtain the singular soliton solution of the equation along with the corresponding domain restriction. The bifurcation analysis is also carried out. By this analysis, a few more traveling wave solutions are retrieved. The bifurcation phase portraits are also given.

Quadratic nonlinear Klein-Gordon equation in one dimension

We study the initial value problem for the quadratic nonlinear Klein-Gordon equation vtt + v − vxx = λv2, t ∈ R, x ∈ R, with initial conditions v(0, x) = v0(x), vt(0, x) = v1(x), x ∈ R, where v0 and v1 are real-valued functions, λ ∈ R. Using the method of normal forms of Shatah [“Normal forms and quadratic nonlinear Klein-Gordon equations,” Commun. Pure Appl. Math. 38, 685–696 (1985)], we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data, which was assumed in the previous work of J.-M. Delort [“Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi-linéaire á données petites en dimension 1,” Ann. Sci. Ec. Normale Super. 34(4), 1–61 (2001)].

Wave operators to a quadratic nonlinear Klein–Gordon equation in two space dimensions revisited

We continue to study the existence of the wave operators for the nonlinear Klein–Gordon equation with quadratic nonlinearity in two space dimensions \({\left(\partial_{t}^{2}-\Delta+m^{2}\right) u=\lambda u^{2},\left( t,x\right) \in\mathbf{R}\times\mathbf{R}^{2}}\). We prove that if $$u_{1}^{+}\in\mathbf{H}^{\frac{3}{2}+3\gamma,1}\left( \mathbf{R}^{2}\right),\text{ }u_{2}^{+}\in\mathbf{H}^{\frac{1}{2}+3\gamma,1}\left( \mathbf{R} ^{2}\right),$$ where \({\gamma\in\left( 0,\frac{1}{4}\right)}\) and the norm \({\left\Vert u_{1}^{+}\right\Vert_{\mathbf{H}_{1}^{\frac{3}{2}+\gamma}}+\left\Vert u_{2}^{+}\right\Vert_{\mathbf{H}_{1}^{\frac{1}{2}+\gamma}}\leq\rho,}\) then there exist ρ > 0 and T > 1 such that the nonlinear Klein–Gordon equation has a unique global solution \({u\in\mathbf{C}\left( \left[ T,\infty\right) ;\mathbf{H}^{\frac{1}{2}}\left( \mathbf{R}^{2}\right) \right) }\) satisfying the asymptotics $$\left\Vert u\left( t\right) -u_{0} \left( t\right) \right\Vert _{\mathbf{H}^{\frac{1}{2}}} \leq Ct^{-\frac{1}{2}-\gamma}$$ for all t > T, where u0 denotes the solution of the free Klein–Gordon equation.

The Initial Value Problem for the Quadratic Nonlinear Klein‐Gordon Equation

We study the initial value problem for the quadratic nonlinear Klein‐Gordon equation , (t, x) ∈ R × R, u(0, x) = u0(x), x ∈ R, where ℒ = ∂t + i〈i∂x〉 and . Using the Shatah normal forms method, we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data which was assumed in the previous works.

Wave operators to a quadratic nonlinear Klein–Gordon equation in two space dimensions

We study asymptotics around the final states of solutions to the nonlinear Klein–Gordon equations with quadratic nonlinearities in two space dimensions ( ∂ t 2 − Δ + m 2 ) u = λ u 2 , ( t , x ) ∈ R × R 2 , where λ ∈ C . We prove that if the final states u 1 + ∈ H q q − 1 4 − 4 q ( R 2 ) ∩ H 5 2 , 1 ( R 2 ) ∩ H 1 2 ( R 2 ) , u 2 + ∈ H q q − 1 3 − 4 q ( R 2 ) ∩ H 3 2 , 1 ( R 2 ) ∩ H 1 1 ( R 2 ) , and ‖ u 1 + ‖ H 1 2 + ‖ u 2 + ‖ H 1 1 is sufficiently small, where 4 < q ≤ ∞ , then there exists a unique global solution u ∈ C ( [ T , ∞ ) ; L 2 ( R 2 ) ) to the nonlinear Klein–Gordon equations such that u ( t ) tends as t → ∞ in the L 2 sense to the solution u 0 ( t ) = u 1 + cos ( 〈 i ∇ 〉 m t ) + ( 〈 i ∇ 〉 m − 1 u 2 + ) sin ( 〈 i ∇ 〉 m t ) of the free Klein–Gordon equation.

Soliton perturbation theory for the quadratic nonlinear Klein–Gordon equation

This paper studies the adiabatic dynamics of soliton velocity due to the quadratic Klein–Gordon equations, in presence of perturbation terms. The soliton perturbation theory is exploited for 1-soliton solution of this equation.

A new auxiliary equation and exact travelling wave solutions of nonlinear equations

A new auxiliary ordinary differential equation and its solutions are used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the auxiliary equation which has more new exact solutions. More new exact travelling wave solutions are obtained for the quadratic nonlinear Klein–Gordon equation, the combined KdV and mKdV equation, the sine-Gordon equation and the Whitham–Broer–Kaup equations.