Nonlocal Nonlinear Equation Research Articles

Long-time behavior of solutions to the general class of coupled nonlocal nonlinear wave equations

Long-time behavior of solutions to the general class of coupled nonlocal nonlinear wave equations

ANALYTICAL SOLUTIONS OF THE NONLOCAL NONLINEAR SCHRÖDINGER-TYPE EQUATIONS

In physics, nonlinear equations are applіed to characterize the varied phenomena. Usually, the nonlinear equations are presented by nonlinear partial differential equations, that can be received as conditions for the compatibility of two linear differentіal equations, named the Lax pairs. The presence of the Lax pair determines integrability for the nonlinear partial differentіal equation. Linked to this development was the realization that certаіn coherent structures, known as solіtons, which play a fundamental role in nonlinear phenomena as lattice dynamics, nonlinear optіcs, and fluіd mechanics. One of the famous equations is the nonlinear Schrödinger equation which is associated with various physical phenomena in nonlinear optics and Bose-Einstein condensates. This equation allows the Lax pair thus it is integrable. This work investigates nonlocal nonlinear Schrödinger-type equations with PT symmetry. Nonlocal nonlinear equations arise in various physical contexts as fluid dynamics, condensed matter physics, optics, and so on. We introduce the Lax pair formulation for the nonlocal nonlinear Schrödinger-type equations. The method of the Darboux transformation is applied to receive analytical solutions.

Maximal $$L_p$$-regularity for x-dependent fractional heat equations with Dirichlet conditions

AbstractWe prove optimal regularity results in $$L_p$$ L p -based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2a ($$0<a<1$$ 0 < a < 1 ) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian $$(-\Delta )^a$$ ( - Δ ) a , as well as elliptic operators $$(-\nabla \cdot A(x)\nabla +b(x))^a$$ ( - ∇ · A ( x ) ∇ + b ( x ) ) a . The proofs are based on general results on maximal $$L_p$$ L p -regularity and its relation to $$\mathcal {R}$$ R -boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.

Global Well-Posedness and Long-Time Asymptotics of a General Nonlinear Non-local Burgers Equation

Global Well-Posedness and Long-Time Asymptotics of a General Nonlinear Non-local Burgers Equation

Global regularity for critical SQG in bounded domains

AbstractWe prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in . We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system.

Regularity estimates for fully nonlinear integro-differential equations with nonhomogeneous degeneracy

We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class Cloc1,α , for some constant α∈(0,1) . In addition, under suitable conditions on degree of the operator σ, we prove regularity estimates in Hölder spaces for any viscosity solution. We also examine the singular setting and prove Hölder regularity estimates for the gradient of the solutions.

The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation

The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation

A Conservative Finite Element Scheme for the Kirchhoff Equation

This article presents an implicit two-layer finite element scheme for solving the Kirchhoff equation, a nonlinear nonlocal equation of hyperbolic type with the Dirichlet integral. The discrete scheme was designed considering the solution of the problem and its derivative for the time variable. It ensures total energy conservation at a discrete level. The use of the Newton method was proven to be effective for solving the scheme on the time layer despite the nonlocality of the equation. The test problems with smooth solutions showed that the scheme can define both the solution of the problem and its time derivative with an error of O(h2+τ2) in the root-mean-square norm, where τ and h are the grid steps in time and space, respectively.

Soliton solutions and a bi-Hamiltonian structure of the fifth-order nonlocal reverse-spacetime Sasa-Satsuma-type hierarchy via the Riemann-Hilbert approach

<p>Our objective is to explore the intricacies of a nonlinear nonlocal fifth-order scalar Sasa-Satsuma equation in reverse spacetime which is rooted in a nonlocal $ 5 \times 5 $ matrix AKNS spectral problem. Starting with this spectral problem, we derive both local and nonlocal symmetry relations through rotations within a defined group. We then formulate a specific type of Riemann-Hilbert problem, facilitating the generation of soliton solutions. These solutions are generated by utilizing vectors that reside in the kernel of the matrix Jost solutions. Under the condition where reflection coefficients are null, the jump matrix reduces to the identity, leading to soliton solutions via the corresponding Riemann-Hilbert problem. The explicit formulas of these soliton solutions enable a comprehensive exploration of their dynamics.</p>

Nonlinear nonlocal equations involving subcritical or power nonlinearities and measure data

<abstract><p>Let $ s\in(0, 1), $ $ 1 < p < \frac{N}{s} $ and $ \Omega\subset{\mathbb R}^N $ be an open bounded set. In this work we study the existence of solutions to problems ($ E_\pm $) $ Lu\pm g(u) = \mu $ and $ u = 0 $ a.e. in $ {\mathbb R}^N\setminus \Omega, $ where $ g\in C({\mathbb R}) $ is a nondecreasing function, $ \mu $ is a bounded Radon measure on $ \Omega $ and $ L $ is an integro-differential operator with order of differentiability $ s\in(0, 1) $ and summability $ p\in(1, \frac{N}{s}). $ More precisely, $ L $ is a fractional $ p $-Laplace type operator. We establish sufficient conditions for the solvability of problems ($ E_\pm $). In the particular case $ g(t) = |t|^{ \kappa-1}t; $ $ \kappa > p-1, $ these conditions are expressed in terms of Bessel capacities.</p></abstract>

Christov expansion method for nonlocal nonlinear evolution equations

Christov functions are a complete orthonormal set of functions on L 2(-∞,∞) that allow us to expand derivatives, nonlinear products, and nonlocal (integro-differential) terms back into the same basis. These properties are beneficial when solving nonlinear evolution equations using Galerkin spectral methods. In this work, we demonstrate such a “Christov expansion method” for the Benjamin–Ono (BO) equation. In the BO equation, the dispersion term is nonlocal, given by the Hilbert transform of the second spatial derivative of the unknown function. The Hilbert transform of the Christov functions can be computed using complex integration and Cauchy’s residue theorem to obtain simple relations. Then, a Galerkin spectral expansion can be used to the solve the BO equation. Time integration is performed using a Crank–Nicolson-type scheme. Importantly, the Christov expansion method yields a banded matrix for the spatial discretization, even though the spatial terms are nonlocal. To demonstrate the approach and its implementation, we perform numerical experiments showing the steady propagation of single and the overtaking interaction of multiple BO solitary waves.

Smooth approximations for fully nonlinear nonlocal elliptic equations

We show that any viscosity solution to a general fully nonlinear nonlocal elliptic equation can be approximated by smooth ( C ∞ C^\infty ) solutions.

Global existence and blow-up of solutions of nonlinear nonlocal parabolic equation with absorption under nonlinear nonlocal boundary condition

Global existence and blow-up of solutions of nonlinear nonlocal parabolic equation with absorption under nonlinear nonlocal boundary condition

Soliton solutions of the space–time fractional nonlocal nonlinear Schrödinger equation

In this paper, the space–time fractional nonlocal nonlinear Schrödinger equations under Jumarie modified Riemann–Liouville derivative is studied. The soliton solutions are formally derived by using the extended tanh method of fractional Riccati subequation and the ansatz method combined with the Hermite–Gaussian function and Mittag-Leffler function. Different types of solutions are obtained and the necessary conditions for the existence of solutions are given. With the help of Maple, we plot figures exhibit that solitons are forming as fractional order λ increasing, and also, we noticed that at different order n of the Hermite–Gaussian function Hn(y), mainly affects the amplitude of soliton. Furthermore, we have discussed the new soliton solutions in the fractional case. It is of great significance to understand complex physical phenomena described by fractional nonlocal nonlinear equations.

Global Existence and Finite-Time Blow-Up for a Nonlinear Nonlocal Evolution Equation

We study the Cauchy problem for a nonlinear nonlocal evolution equation arising in the study of oceanic flows in equatorial regions. We present a well-posedness result and show that while some initial data develop into solutions that exist for all times, others lead to blow-up in finite time.

Nonlocal Harnack inequality for fractional elliptic equations with Orlicz growth

AbstractWe prove Harnack inequality for weak solutions to nonlinear nonlocal equations of fractional ‐Laplace type, under natural assumptions on the ‐function .

A Hopf type lemma for nonlocal pseudo-relativistic equations and its applications

In this paper, we consider the nonlinear equation involving the nonlocal pseudo-relativistic operators ( − Δ + m 2 ) s u ( x ) = f ( x , u ( x ) ) , where 0<s<1 and mass m>0. The nonlocal pseudo-relativistic operator includes the pseudo-relativistic Schrödinger operator − Δ + m 2 . When m → 0 + , the nonlocal pseudo-relativistic operator ( − Δ + m 2 ) s is also closely related to the fractional Laplacian operator ( − Δ ) s . But these two operators are quite different. We first establish a Hopf type lemma for anti-symmetric functions to nonlocal pseudo-relativistic operators, which play a key role in the method of moving planes. The main difficulty is to construct a suitable sub-solution to nonlocal pseudo-relativistic operators. Then we prove a pointwise estimate to nonlocal pseudo-relativistic operators. As an application, combined with the Hopf type lemma and the pointwise estimate, we obtain the radial symmetry and monotonicity of positive solutions to the above nonlinear nonlocal pseudo-relativistic equation in the whole space. We believe that the Hopf type lemma will become a powerful tool in applying the method of moving planes on nonlocal pseudo-relativistic equations to obtain qualitative properties of solutions.

Monotonicity and uniqueness of positive solutions to elliptic fractional p-equations

In this paper, we first obtain a maximum principle for parabolic fractional p-Laplacians. Then combining the maximum principle for parabolic fractional p-Laplacians with the parabolic inequalities, we give a new proof for monotonicity and uniqueness of solutions to elliptic fractional p-equations with gradient terms by the sliding method. In order to overcome the difficulties caused by the gradient term, we introduce some new tricks when applying the sliding method.We believe that parabolic inequalities and the maximum principle introduced here can be applied to many nonlinear nonlocal elliptic equations with gradient terms.

Nonlinear waves and the Inverse Scattering Transform

Solitons are a class of nonlinear stable, localized waves. They arise widely in physical problems; applications include water waves, plasma physics, Bose–Einstein condensation and nonlinear optics. Such localized water waves can be traced back to research in the 1800s. In fiber optics ‘bright and dark’ solitons were discovered in 1973 by Hasegawa and Tappert. In the 1970s a general theory emerged which allows one to linearize and explicitly find soliton solutions to a class of nonlinear wave equations including physically significant equations such as the Korteweg–deVries, nonlinear Schrödinger (NLS) and sine-Gordon equations. Solitons are connected to eigenvalues/bound states of underlying linear scattering equations. The theory, termed the Inverse Scattering Transform (IST) by Ablowitz, Kaup, Newell, Segur, leads to linearization/solutions to broad classes of nonlinear wave equations. In 2013 the theory was extended to novel classes of nonlocal nonlinear wave equations including PT symmetric and reverse-space–time nonlinear equations. In 2022 the theory was shown to encompass fractional integrable nonlinear systems; the fractional KdV and NLS equations are paradigms.

Some general bright soliton solutions and interactions for a (2＋1)-dimensional nonlocal nonlinear Schrödinger equation

In this paper, a novel application of Darboux transformation is presented for (2＋1)-dimensional nonlocal nonlinear Schrödinger equation. The Darboux transformation is a power method to solve some (1＋1)-dimensional classical nonlinear Schrödinger (NLS) equations, however, there is less work of (2＋1)-dimensional ((2＋1)-D) nonlocal nonlinear Schrödinger(NNLS) equation with Darboux transformation. With the development of science, the NNLS equation gradually appears, where the nonlocality is the reverse spatial field or reverse time field. We focus on how to solve the (2＋1)-D NNLS equation with reverse time field q(x,y,−t). Using the Darboux transformation, some novel (2＋1)-D nonlocal soliton solutions are derived on a background of kink waves, including 1-soliton solution, 2-soliton solution, bright soliton and soliton interaction on kink wave backgrounds. This method is a novel application of Darboux transformation, which can be extend to some other nonlocal nonlinear or higher-dimensional soliton equations on kink wave backgrounds.