Homoclinic Solutions Research Articles

Homoclinic solutions for perturbed discrete Schrödinger systems with concave–convex terms and bounded potentials

By using variational methods, we study a class of discrete nonlinear Schrödinger systems, where the potentials are bounded and the nonlinearities are composed of perturbed and concave–convex terms. The main novelties of this paper are as follows: (1) some perturbed terms and concave–convex terms are added to the systems, (2) the weight functions can be sign‐changing, and (3) the potentials are bounded, which is essentially different from the unbounded potentials studied before.

Multiple fast homoclinic solutions for a class of second-order differential systems with p -Laplacian

UDC 517.9 We study the existence of multiple fast homoclinic solutions for a class of second-order differential systems with p -Laplacian by using the minimax methods in the critical-point theory.

Existence of Homoclinic Solutions for a Class of Nonlinear Second-order Problems

Existence of Homoclinic Solutions for a Class of Nonlinear Second-order Problems

Homoclinic solutions for discrete fractional p-Laplacian equation via the Nehari manifold method

Homoclinic solutions for discrete fractional p-Laplacian equation via the Nehari manifold method

Solitons, lump and interactional solutions of the (3+1)-dimensional BLMP equation in incompressible fluid

Based on the Hirota bilinear method, we systematically investigate the (3+1)-dimensional Boiti-Leon-Manana-Pempinelli (BLMP) equation in incompressible fluids and main results include: (1) the formulas of N-kink-soliton solutions and the bound states of multi solitons are all presented, (2) the lump solution is derived by the positive quadratic function method, (3) the interactional solutions are given, i.e., one lump interacts with one- and two-kink-soliton, (4) some special periodic solutions are discussed, i.e., lump-periodic solutions and homoclinic breather solutions.

Travelling pulses on three spatial scales in a Klausmeier-type vegetation-autotoxicity model

Reaction-diffusion models describing interactions between vegetation and water reveal the emergence of several types of patterns and travelling wave solutions corresponding to structures observed in real-life. Increasing their accuracy by also considering the ecological factor known as autotoxicity has lead to more involved models supporting the existence of complex dynamic patterns. In this work, we include an additional carrying capacity for the biomass in a Klausmeier-type vegetation-water-autotoxicity model, which induces the presence of two asymptotically small parameters: ɛ, representing the usual scale separation in vegetation-water models, and δ, directly linked to autotoxicity. We construct three separate types of homoclinic travelling pulse solutions based on two different scaling regimes involving ɛ and δ, with and without a so-called superslow plateau. The relative ordering of the small parameters significantly influences the phase space geometry underlying the construction of the pulse solutions. We complement the analysis by numerical continuation of the constructed pulse solutions, and demonstrate their existence (and stability) by direct numerical simulation of the full partial differential equation model.

Ground State Homoclinic Solutions for Damped Vibration Systems with Periodicity

Ground State Homoclinic Solutions for Damped Vibration Systems with Periodicity

Existence of multi-hump generalized homoclinic solutions for a class of reversible systems

Existence of multi-hump generalized homoclinic solutions for a class of reversible systems

Homoclinic and N-soliton solutions to variable-coefficient KP equation arising two-temperature ions in dusty plasma

Homoclinic and N-soliton solutions to variable-coefficient KP equation arising two-temperature ions in dusty plasma

Bifurcations of degenerate homoclinic solutions in discontinuous systems under non-autonomous perturbations.

The main aim of this paper is to study bifurcations of bounded solutions from a degenerate homoclinic solution for discontinuous systems under non-autonomous perturbations. We use Lyapunov-Schmidt reduction to give bifurcation equations and prove that a single parameter is enough to unfold two distinct homoclinic solutions bifurcated from the unperturbed degenerate homoclinic solution. Furthermore, we give an example of a periodically perturbed piecewise smooth differential equation in R4 to support our conclusions.

Traveling waves in a model of calcium ions influx via mechanically stimulated membrane channels

We consider the problem of existence and properties of pulse solutions to a system of equations modeling fast calcium waves in long cells. These waves have the speed up to 1000 m/s. They propagate via the inflow of calcium ions from the extracellular space through the mechanically stimulated membrane channels. The channels open due to mechano‐chemical interaction, in which stretching of the cell's membrane at a point opens the calcium channels at neighboring points due to the forces exerted by the actomyosin network. The existence of homoclinic solutions is based on the celebrated exchange lemma, which cannot be applied straightforwardly due to some specificities of the model equations.

Normalized homoclinic solutions of discrete nonlocal double phase problems

The aim of this paper is to discuss the existence of normalized solutions to the following nonlocal double phase problems driving by the discrete fractional Laplacian: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] if [Formula: see text], [Formula: see text] if [Formula: see text], and [Formula: see text]([Formula: see text] or [Formula: see text], [Formula: see text] or [Formula: see text]) is the discrete fractional [Formula: see text]-Laplacian. By variational methods, we discuss the existence of non-negative normalized homoclinic solutions under the conditions that the nonlinear term satisfies sublinear growth or superlinear growth conditions. In particular, we establish the compactness of the associated energy functional of the problem without weights. Our paper is the first time to deal with the existence of normalized solutions for discrete double phase problems.

Discrete Schrödinger equations and systems with mixed and concave-convex nonlinearities

In this paper, we obtain the existence of at least two standing waves (and homoclinic solutions) for a class of time-dependent (and time-independent) discrete nonlinear Schrödinger systems or equations. The novelties of the paper are as follows. (1) Our nonlinearities are composed of three mixed growth terms, i.e., the nonlinearities are composed of sub-linear, asymptotically-linear and super-linear terms. (2) Our nonlinearities may be sign-changing. (3) Our results can also be applied to the cases of concave-convex nonlinear terms. (4) Our results can be applied to a wide range of mathematical models.

Bifurcations and Exact Solutions of a Cantilever Beam Vibration Model Without Damping and Forced Terms

For the cantilever beam vibration model without damping and forced terms, the corresponding differential system is a planar dynamical system with some singular straight lines. In this paper, by using the techniques from dynamical systems and singular traveling wave theory developed by [ Li & Chen, 2007 ] to analyze its corresponding differential system, the bifurcations and the dynamical behaviors of the corresponding phase portraits are identified and analyzed. Under different parameter conditions, exact homoclinic and heteroclinic solutions, periodic solutions, compacton solutions, as well as peakons and periodic peakons, are all found explicitly.

Multiple homoclinic solutions for nonsmooth second-order differential systems

In the present paper, we obtain infinitely many pairs of homoclinic solutions for a class of nonsmooth second-order differential systems when the energy functional associated is not continuously differentiable and does not satisfy the Palais–Smale condition.

Ground-state sign-changing homoclinic solutions for a discrete nonlinear p-Laplacian equation with logarithmic nonlinearity

By using a direct non-Nehari manifold method from (Tang and Cheng in J. Differ. Equ. 261:2384–2402, 2016), we obtain an existence result of ground-state sign-changing homoclinic solutions that only changes sign once and ground-state homoclinic solutions for a class of discrete nonlinear p-Laplacian equations with logarithmic nonlinearity. Moreover, we prove that the sign-changing ground-state energy is larger than twice the ground-state energy.

Homoclinic solutions for subquadratic Hamiltonian systems with competition potentials

Homoclinic solutions for subquadratic Hamiltonian systems with competition potentials

Homoclinic Solutions for a Class of Perturbed Fractional Hamiltonian Systems with Subquadratic Conditions

Homoclinic Solutions for a Class of Perturbed Fractional Hamiltonian Systems with Subquadratic Conditions

Homoclinic and heteroclinic solutions for non-autonomous Minkowski-curvature equations

We deal with the non-autonomous parameter-dependent second-order differential equation δ(v′1−(v′)2)′+q(t)f(v)=0,t∈R,driven by a Minkowski-curvature operator. Here, δ>0, q∈L∞(R), f:[0,1]→R is a continuous function with f(0)=f(1)=0=f(α) for some α∈]0,1[, f(s)<0 for all s∈]0,α[ and f(s)>0 for all s∈]α,1[. Based on a careful phase-plane analysis, under suitable assumptions on q we prove the existence of strictly increasing heteroclinic solutions and of homoclinic solutions with a unique change of monotonicity. Then, we analyze the asymptotic behavior of such solutions both for δ→0+ and for δ→+∞. Some numerical examples illustrate the stated results.

Square waves and Bykov T-points in a delay algebraic model for the Kerr-Gires-Tournois interferometer.

We study theoretically the mechanisms of square wave formation of a vertically emitting micro-cavity operated in the Gires-Tournois regime that contains a Kerr medium and that is subjected to strong time-delayed optical feedback and detuned optical injection. We show that in the limit of large delay, square wave solutions of the time-delayed system can be treated as relative homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of square wave solutions. In particular, we unveil the mechanisms behind the collapsed snaking scenario of square waves and explain the formation of complex-shaped multistable square wave solutions through a Bykov T-point. Finally, we relate the position of the T-point to the position of the Maxwell point in the original time-delayed system.