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The linear transport equation allows to advect level-set functions to represent moving sharp interfaces in multiphase flows as zero level-sets. A recent development in computational fluid dynamics is to modify the linear transport equation by introducing a nonlinear term to preserve certain geometrical features of the level-set function, where the zero level-set must stay invariant under the modification. The present work establishes mathematical justification for a specific class of modified level-set equations on a bounded domain, generated by a given smooth velocity field in the framework of the initial/boundary value problem of Hamilton–Jacobi equations. The first main result is the existence of smooth solutions defined in a time-global tubular neighborhood of the zero level-set, where an infinite iteration of the method of characteristics within a fixed small time interval is demonstrated; the smooth solution is shown to possess the desired geometrical feature. The second main result is the existence of time-global viscosity solutions defined in the whole domain, where standard Perron's method and the comparison principle are exploited. In the first and second main results, the zero level-set is shown to be identical with the original one. The third main result is that the viscosity solution coincides with the local-in-space smooth solution in a time-global tubular neighborhood of the zero level-set, where a new aspect of localized doubling the number of variables is utilized. [ABSTRACT FROM AUTHOR]

In this paper, we extend the results of Brézis and Willem (J Funct Anal 255:2286–2298, 2008) to the equation with single or double weighted critical exponents, including Hardy–Sobolev, Sobolev and Hénon–Sobolev exponents. More precisely, we establish the existence or nonexistence of equation with different coefficient which has an important impact. [ABSTRACT FROM AUTHOR]

This study presents the problem of spreading non fatal disease in a population by using the Morgan-Voyce collocation method. The main aim of this paper is to find the exact solutions of the SIR model with vaccination. The problem may be modelled mathematically with a nonlinear system of ordinary differential equations. The presented method reduces the problem into a nonlinear algebraic system of equations by using unknown coefficient Morgan-Voyce polynomials and expanding approximate solutions. Morgan-Voyce polynomials are used. These unknown coefficients are calculated via the collocation method and matrix operation derivations. Two examples are given to show the feasibility of the method. To calculate the solutions, MATLAB R2021a is used. Additionally, comparing our method to the Homotopy perturbation method (HPM) and the Laplace Adomian decomposition method (LADM) proves the accuracy of the solution. The method studied can be seen as effective from these comparisons. So, it is essential to find solutions for the governing model. The study will contribute to literature since we also discuss the vaccination situation. The results of this study are valuable for controlling an epidemic. [ABSTRACT FROM AUTHOR]

This work studies the first equation of the Kadomtsev-Petviashvili (KP) hierarchy. The sine-Gordon expansion method (SGEM) and the rational SGEM (RSGEM) are applied to the governing model. RSGEM is the developed version of SGEM. New complex travelling wave solutions, logarithmic and complex function properties are obtained. Several simulations such as 2D, 3D and contour surfaces of the obtained results are plotted. Physical meanings of these solutions are also reported. Strain conditions are also extracted. [ABSTRACT FROM AUTHOR]

In this paper, we shall discuss singular solutions of semilinear elliptic equations with general supercritical growth on spherically symmetric Riemannian manifolds. More precisely, we shall prove the existence, uniqueness and asymptotic behavior of the singular radial solution, and also show that regular radial solutions converges to the singular solution. In particular, we shall provide these properties on spherically symmetric Riemannian manifolds including the hyperbolic space as well as the sphere. [ABSTRACT FROM AUTHOR]

This paper explores the nonuniqueness of solutions to the L p chord Minkowski problem for negative p. The L p chord Minkowski problem was recently posed by Lutwak, Xi, Yang and Zhang, which seeks to determine the necessary and sufficient conditions for a given finite Borel measure such that it is the L p chord measure of a convex body, and it includes the chord Minkowski problem and the L p Minkowski problem. [ABSTRACT FROM AUTHOR]

This article discusses a novel idea from cell therapy in which nanoparticles (NPs) are adsorbed on red blood cells (RBCs). RBCs serve as a drug carrier for NPs or nanodrugs adsorbed on the cell membrane of RBC. For the purpose of examination, Fe3O4{{\rm{Fe}}}_{3}{{\rm{O}}}_{4} NPs are adsorbed on RBCs, collectively called NP-RBC complex. RBCs being a natural vascular carrier, have high transfusion rates and biocompatibility. This mathematical study provides a basis to attempt nanodrug delivery via RBCs, as carriers for nanodrugs, to the stenosed sites in an artery. The mathematical model is developed for an artery with stenosis and a catheter that regards the temperature and velocity of the NP-RBC complex. Catheter coated with the NP-RBC complex is inserted into the lumen of the stenosed artery. The mathematical problem is solved numerically using Bernstein polynomials. The physical features were discussed through graphs plotted using MATLAB. The influence of parameters such as volume fraction, radius of the NP-RBC complex in blood, and the thickness of the nanolayer on RBCs was studied. A noticeable outcome states that the nanolayer of optimum thickness about 50–40 nm is suitable for this purpose. Thus, this is an attempt to study the delivery of NPs adsorbed on the surface of RBCs to develop newfangled strategies in nanomedicine bearing high precision and efficiency.

We study general problems modeling electrostatic microelectromechanical systems devices (Pλ )φ(r,−u′(r))=λ∫0rf(s)g(u(s))ds,r∈(0,1),00\lambda \gt 0 is a parameter. We obtain results on the existence and regularity of a touchdown solution to (Pλ{P}_{\lambda }) and find upper and lower bounds on the respective pull-in voltage. In the particular case, when φ(r,v)=rα∣v∣βv\varphi \left(r,v)={r}^{\alpha }{| v| }^{\beta }v, i.e., when the associated differential equation involves the operator r−γ(rα∣u′∣βu′)′{r}^{-\gamma }\left({r}^{\alpha }{| u^{\prime} | }^{\beta }u^{\prime} )^{\prime} , we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.

In this article, we investigate the Cauchy problem for Klein-Gordon equations with combined power-type nonlinearities. Coefficients in the nonlinearities depend on the space variable. They are sign preserving functions except one of the coefficients, which may change its sign. We study completely the structure of the Nehari manifold. By using the potential well method, we give necessary and sufficient conditions for nonexistence of global solution for subcritical initial energy by means of the sign of the Nehari functional. When the energy is positive, we propose new sufficient conditions for finite time blow up of the weak solutions. One of these conditions is independent of the sign of the scalar product of the initial data. We also prove uniqueness of the weak solutions under slightly more restrictive assumptions for the powers of the nonlinearities.

Solutions such as symmetric, bright soliton and periodic solutions play a prominent role in the field of differential equations, and they can be used to investigate several phenomena in nonlinear sciences. Some waves such as ion and magneto-sound waves in plasma are investigated by using some partial differential equations (PDEs) such as Novikov-Veselov (NV) equations. In this work, the improved exp ⁡ (− Υ (η)) -expansion approach is utilized to extract numerous soliton solutions for NV system. Hamiltonian system is invoked to analyze the stability of some solutions. The finite difference approach is successfully applied to achieve the numerical simulations of the proposed equations. We also introduce the stability and the accuracy of the numerical scheme. In order to validate the correctness of the accomplished results, we compare the exact solutions with the numerical solutions analytically and graphically. The presented techniques are very convenient and adequate and can be employed to other types of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This paper presents a new strategy to deal with the excessive diffusion that standard finite volume methods for compressible Euler equations display in the limit of low Mach number. The strategy can be understood as using centered discretizations for the acoustic part of the Euler equations and stabilizing them with a leap-frog-type (“sequential explicit”) time integration, a fully explicit method. This time integration takes inspiration from time-explicit staggered grid numerical methods. In this way, advantages of staggered methods carry over to collocated methods. The paper provides a number of new collocated schemes for linear acoustic/Maxwell equations that are inspired by the Yee scheme. They are then extended to an all-speed method for the full Euler equations on Cartesian grids. By taking the opposite view and taking inspiration from collocated methods, the paper also suggests a new way of staggering the variables which increases the stability as compared to the traditional Yee scheme. [ABSTRACT FROM AUTHOR]

We apply the theory of Lie symmetries for the analysis of the group properties for a recent proposed 2+1 extended Boiti-Leon-Manna-Pempinelli equation. The equation admits infinity Lie symmetries which corresponds to the infinity number of solutions. We show that D'Alember-type wave solutions follow from the application of Lie invariants, while new periodic solutions are determined. [ABSTRACT FROM AUTHOR]

Based on the theory of ordinary differential equations, this paper proposes a stable and discriminative method for piano tone tuning. We perform discriminative training on the hidden Markov tone model according to ordinary differential equations’ feature extraction parameters and model parameters. The model can improve the recognition rate of piano tones. This paper trains and tests the MAPS universal dataset for musical transcription of automatic piano tones. The model is uniformly trained on the synthetic part and tested on the real recording part. The experimental study found that the proposed model has high recognition stability and accuracy in piano tone recognition.

This article combines the fractional differential theory with the total variation method and applies it to defining cultural industry image color matching. At the same time, we propose a new image color matching denoising model based on fractional partial differential equations. The model achieves simultaneous denoising in the time direction and the spaceplane. Experiments have proved that the fractional partial differential equations method has more advantages than integer-order partial differential equations in denoising and reducing step effects. It can effectively improve the contrast and clarity of image color matching in the cultural industry.

In this paper, we obtain the numerical solution of two-dimensional inverse time-fractional diffusion problem with non-local boundary condition using a-polynomials. These polynomials are equipped with an unknown parameter a that is obtained by collocation and least squares methods. In fact, the optimal parameter a is replaced in the a-polynomials, and then with these polynomials that no longer have the a parameter, the numerical solution is approximated. Time discretization of the equation is performed by L 1 method. The convergence theorem for a-polynomials has been proved in this article. In three examples, four types of measurement errors have been used to confirm the accuracy of the present method and compare the results with other methods. The stability of the present method has been investigated in all examples when the input data is contaminated with noise. Considering that in each example, the optimum value of parameter a is obtained, the results are stable in noise mode. The results in the examples also show the accuracy and advantage of the present method in comparison with the other method. [ABSTRACT FROM AUTHOR]

In this paper, we consider exponentially fitted peer methods for the numerical solution of first order differential equations and we investigate how the frequencies can be tuned in order to obtain the maximal benefit. We will show that the key is analyzing the error's behavior. Formulae for optimal frequencies are computed. Numerical experiments show the properties of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We prove the existence of non- H 0 1 (B 1 (0)) solutions of a class of singular elliptic problem in two dimensions. [ABSTRACT FROM AUTHOR]

A positive radial singular solution for Δ u + f (u) = 0 with a general supercritical growth is constructed. An exact asymptotic expansion as well as its uniqueness in the space of radial functions are also established. These results can be applied to the bifurcation problem Δ u + λ f (u) = 0 on a ball. Our method can treat a wide class of nonlinearities in a unified way, e.g., u p log u , exp (u p) and exp (⋯ exp (u) ⋯) as well as u p and e u . Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques. [ABSTRACT FROM AUTHOR]

In this paper, the (3+1)-dimensional variable-coefficient generalized nonlinear wave equation arising in a liquid with gas bubbles is studied. The Hirota bilinear technique and binary bell polynomials are considered. Some new analytic solutions containing one-lump soliton, two-lump soliton, and three-lump soliton, and also 1-breather and 1-lump, the interaction 1-bell-shaped soliton with 1-lump or 1-breather, and other soliton solutions to the mentioned equation are obtained and analyzed. To create the solutions of Maple symbolic package is used. Using suitable mathematical assumptions and Hirota’s bilinear technique, the new types of M-soliton and N-soliton solutions are derived and established in view of the hyperbolic, trigonometric, and rational functions of the governing equation. To achieve this, an illustrative example of the nonlinear wave equation in a liquid with gas bubbles is provided to demonstrate the feasibility and reliability of the procedure used in this study. The trajectory solutions of the N-soliton are shown explicitly and graphically (2D, density, and 3D graphs). The effect of parameters on the behavior of acquired solutions for N = 3 , 4 , 5 orders are also discussed. By comparing the proposed method with the other existing methods, the results show that the execution of this method is concise, simple, and straightforward. The results are useful for obtaining and explaining some new soliton phenomena. [ABSTRACT FROM AUTHOR]

In this paper we study the stability of homogeneous states in a continuous one spatial variable of conservation equations in Lagrangian coordinates, including terms of dissipation and relaxation. Local existence is proved applying Kawashima’s theorem for hyperbolic-parabolic systems [7]. We establish that when the subcharactertistic condition is satisfied, the structure of the system is not of regularity-loss type, but of the standard type, even though the linear term associated with relaxation is not symmetric nor positive semidefinite. [ABSTRACT FROM AUTHOR]

To systematically understand the ability and technical mastery of football training of ethnic college students in colleges and universities, we use fractional differential equations to analyze the passing technique of college ethnic football. Research has found that establishing differential equation models in football can help us use mathematical ideas, methods, and knowledge to solve practical problems.

College students are prone to psychological changes and personality distortions under heavy pressure. It will lead to catastrophe if it is not promptly guided or improperly used to reduce stress and relax. Now we integrate holistic education into the construction of the mental health education system in colleges and universities. This is conducive to solving practical problems and improving the effectiveness of system construction. We set up a differential equation model (DCE) based on the discrete choice to study mental health due to practice. At the same time, we adopt the mutual influence model of penetrating dissemination of thoughts and public opinion and human behaviour, and establish the SEIR DCE with constant input and bilinear incidence rate. The experiment verifies the algorithm’s effectiveness on the model of whole-person mental health education activities in colleges and universities.

User consumption behaviour is a subject worthy of study. Because consumers’ consumption behaviours are dynamic and with individual differences, various factors need to be considered when establishing a fractional differential equation model of users’ online consumption behaviour. The two elements, namely advertising and price are more evident in influencing consumer behaviour. Therefore, the paper establishes a product diffusion fractional differential equation model of price and advertising presence or absence to study the impact of these two factors on consumer behaviour. It turns out that ignoring the advertisement and the cost of the product is related to the characteristics of the consumer network. When there are advertising and price factors, product awareness is related to price constraints.

In this paper, we introduce a theoretical framework for semi-discrete optimization using ideas from optimal transport. Our primary motivation is in the field of deep learning, and specifically in the task of neural architecture search. With this aim in mind, we discuss the geometric and theoretical motivation for new techniques for neural architecture search [in the companion work (García-Trillos et al. in Traditional and accelerated gradient descent for neural architecture search, 2021); we show that algorithms inspired by our framework are competitive with contemporaneous methods]. We introduce a Riemannian like metric on the space of probability measures over a semi-discrete space R d × G where G is a finite weighted graph. With such Riemannian structure in hand, we derive formal expressions for the gradient flow of a relative entropy functional, as well as second-order dynamics for the optimization of said energy. Then, with the aim of providing a rigorous motivation for the gradient flow equations derived formally we also consider an iterative procedure known as minimizing movement scheme (i.e., Implicit Euler scheme, or JKO scheme) and apply it to the relative entropy with respect to a suitable cost function. For some specific choices of metric and cost, we rigorously show that the minimizing movement scheme of the relative entropy functional converges to the gradient flow process provided by the formal Riemannian structure. This flow coincides with a system of reaction–diffusion equations on R d . [ABSTRACT FROM AUTHOR]

Many indefinite integrals are derived for Bessel functions and associated Legendre functions from particular transformations of their differential equations which are closely linked to Wronskians. A large portion of the results for Bessel functions is known, but all the results for associated Legendre functions appear to be new. The method can be applied to many other special functions. All results have been checked by differentiation using Mathematica. [ABSTRACT FROM AUTHOR]

To uncover the effective interventions during the pandemic period, a novel mathematical model, which incorporates separate compartments for incubation and asymptomatic individuals, has been developed in this paper. On the basis of a general mixing, final size relation and next-generation matrix are derived for a meta-population model by introducing the matrix blocking. The final size (F ) and the basic reproduction number ( R 0 ) are no longer a simple monotonous relationship. The analytical results of heterogeneity illustrate that activity is more sensitive than the others. And the proportion of asymptomatic individuals is a key factor for final epidemic size compared to the regulatory factor. Furthermore, the impact of preferential contact level on F and R 0 is comparatively complex. The isolation can effectively reduce the final size, which further verifies its effectiveness. When vaccination is considered, the mixing methods maybe influence the doses of vaccination used and its effective. Moreover, using the present predictive model, we can provide the valuable reference about identifying the ideal strategies to curb the pandemic disease. [ABSTRACT FROM AUTHOR]

The numerical simulation of hot and low density plasmas using the Vlasov–Poisson model is necessary for many practical applications such as the characterization of laboratory and astrophysical plasmas. The numerical treatment of the Vlasov equation is addressed using Eulerian methods when high precision and low noise are required. Among these methods, we highlight those based on finite-volumes without splitting, which have shown to be a good option for capturing small structures in phase space with low dissipation while preserving positivity. The problem is that kinetic simulations usually require the discretization of 3–6-dimensional phase-spaces which results in a huge number of ordinary differential equations (ODEs). This stands out the importance of using efficient schemes for the time integration. In this article, linear multistep methods are implemented for the time stepping of the resulting equations, and compared against traditional Runge–Kutta ones. Schemes with built-in error estimation are also implemented in an attempt to perform adaptive stepsize control. Their accuracy, stability and computational cost are compared through the simulation of classical benchmark problems for the two-dimensional Vlasov–Poisson system. [ABSTRACT FROM AUTHOR]

In this article, we applied the improved and generalized method to some nonlinear problems such as: the (2+1) dimensional Bogoyavliskii equation and the Ablowitz-Kuap-Newell-Segur equation (AKNSE), but distinctively and simple, we called it the Ξ -expansion method. This method is characterized by the addition of an arbitrary constant d to the general solution, which leads to the presentation of new exact solutions. This method is simple but powerful for extracting exact solutions of partial differential equation, which habitually occur in mathematical physics, engineering sciences and applied mathematics. [ABSTRACT FROM AUTHOR]

In this study, a nonlinear fractional logistic equations is proposed in the context of a modified form of the conformable fractional-order derivative. The analytical and approximate-analytical solutions are obtained for the proposed mathematical model via the method of conformable fractional differential transform method (CFDTM) and all obtained results have been analyzed and compared with homotopy perturbation method (HPM). With plotting the figures, we compared the CFDTM, HPM, and exact solution together and conclude that the CFDTM solution is closer than the exact solution in comparison with the HPM. The methods are illustrated by two examples of FOLE and solution is obtained. All obtained solutions have been analyzed and compared graphically to validate the effiency and applicability of all results. [ABSTRACT FROM AUTHOR]

We study explicit solutions to the 2 dimensional Euler equations in the Lagrangian framework. All known solutions have been of the separation of variables type, where time and space dependence are treated separately. The first such solutions were known already in the 19th century. We show that all the solutions known previously belong to two families of solutions and introduce three new families of solutions. It seems likely that these are all the solutions that are of the separation of variables type. [ABSTRACT FROM AUTHOR]

In the current paper, we carry out an investigation into the exact solutions of the (3+1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov (fractional mKdV–ZK) equation. Based on the conformable fractional derivative and its properties, the fractional mKdV–ZK equation is reduced into an ordinary differential equation which has been solved analytically by the variable separated ODE method. Various types of analytic solutions in terms of hyperbolic functions, trigonometric functions and Jacobi elliptic functions are derived. All conditions for the validity of all obtained solutions are given.

Abstract Linearization criteria for two-dimensional systems of second-order ordinary differential equations (ODEs) have been derived earlier using complex symmetry analysis. For such systems, the linearizable form, linearization criteria and symmetry group classification are presented. In this paper, we extend the complex approach to obtain a complex-linearizable form of two-dimensional systems of third-order ODEs. This form leads to a linearizable class and linearization criteria of these systems of ODEs.

We consider a one-dimensional discrete particle system of two species coupled through nonlocal interactions driven by the Newtonian potential, with repulsive self-interaction and attractive cross-interaction. After providing a suitable existence theory in a finite-dimensional framework, we explore the behaviour of the particle system in case of collisions and analyse the behaviour of the solutions with initial data featuring particle clusters. Subsequently, we prove that the empirical measure associated to the particle system converges to the unique 2-Wasserstein gradient flow solution of a system of two partial differential equations with nonlocal interaction terms in a proper measure sense. The latter result uses uniform estimates of the L m -norms of a piecewise constant reconstruction of the density using the particle trajectories. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and uniqueness of the mild solutions for a class of fractional non-autonomous evolution equations with delay and Caputo's fractional derivatives. By using the measure of noncompactness, β-resolvent family, fixed point theorems and Banach contraction mapping principle, we improve and generalizes some related results on this topic. At last, we give an example to illustrate the application of the main results of this paper. [ABSTRACT FROM AUTHOR]

The time-ordered exponential of a time-dependent matrix A(t) is defined as the function of A(t) that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in A(t). The authors have recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by *. Yet, the existence of such inverses, crucial to avoid algorithmic breakdowns, still needed to be proved. Here we constructively prove that *-inverses exist for all non-identically null, smooth, separable functions of two variables. As a corollary, we partially solve the Green's function inverse problem which, given a distribution G, asks for the differential operator whose fundamental solution is G. Our results are abundantly illustrated by examples. [ABSTRACT FROM AUTHOR]

We study singular radial solutions of the semilinear elliptic equation Δ u + f (u) = 0 on finite balls in R N with N ≥ 3 . We assume that f satisfies either f (u) = u p + o (u p) with p > (N + 2) / (N - 2) or f (u) = e u + o (e u) as u → ∞ . We provide the existence and uniqueness of the singular radial solution, and show the convergence of regular radial solutions to the singular solution. Some applications to the bifurcation diagram of an elliptic Dirichlet problem are also given. Our results generalize and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form Δ u + g (u) = 0 . Our result applies in particular to the double power non-linearity where g (u) = u q - u p - μ u for p > q > 1 and μ > 0 , which we discuss with more details. In this case, the non-degeneracy of the unique solution u μ allows us to derive its behavior in the two limits μ → 0 and μ → μ ∗ where μ ∗ is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a conjecture about the variations of the L 2 mass of u μ in terms of μ , which we illustrate with numerical simulations. If valid, this conjecture would imply the uniqueness of energy minimizers in all cases and also give some important information about the orbital stability of u μ . [ABSTRACT FROM AUTHOR]

The quantum calculus, q-calculus, is a relatively new branch in which the derivative of a real function can be calculated without limits. In this paper, the falling body problem in a resisting medium is revisited in view of the q-calculus to the first time. The q-differential equations describing the vertical velocity and distance of the body are obtained. Accordingly, exact expressions for the vertical velocity and the vertical distance are provided. The solutions are expressed in terms of the small q-exponential function which is an elementary function in the q-calculus. The dimensionality of the obtained formulae of the velocity and the distance are also analyzed. In addition, the present exact solutions reduce to the corresponding solutions in classical Newtonian mechanics when the quantum parameter q tends to one.

In this paper, we consider the analogous of the Hénon equation for the prescribed mean curvature problem in ℝN{{\mathbb{R}^{N}}}, both in the Euclidean and in the Minkowski spaces. Motivated by the studies of Ni and Serrin [W. M. Ni and J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Att. Convegni Lincei 77 1985, 231–257], we have been interested in finding the relations between the growth of the potential and that of the local nonlinearity in order to prove the nonexistence of a radial ground state. We also present a partial result on the existence of a ground state solution in the Minkowski space.

We prove structure results for the radial solutions of the semilinear problem

In this paper, we study a class of nonlinear time fractional partial integro-differential equations. The main results for the problem are obtained by the measure of noncompactness, solution operator, convex-power condensing operator and general Banach contraction mapping principle. [ABSTRACT FROM AUTHOR]

Let 1 < p < +∞ and let Ω ⊂ ℝ N be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type $-\Delta _pu = f(u),\quad u > 0\,{\rm in }\,\Omega ,\quad \partial _\nu u = 0\,{\rm on }\,\partial \Omega.$ We suppose that f (0) = f (1) = 0 and that f is negative between the two zeros and positive after. In case Ω is a ball, we also require that f grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focussing in particular on solutions which start above 1. As an application, we use the shooting technique to get existence, multiplicity and oscillatory behaviour (around 1) of non-constant radial solutions. [ABSTRACT FROM AUTHOR]

Bäcklund transformations are applied to study the Gross–Pitaevskii equation. Supported by previous results, a class of Bäcklund transformations admitted by this equation are constructed. Schwarzian derivative as well as its invariance properties turn out to represent a key tool in the present investigation. Examples and explicit solutions of the Gross–Pitaevskii equation are obtained. [ABSTRACT FROM AUTHOR]

Linearization criteria for two-dimensional systems of second-order ordinary differential equations (ODEs) have been derived earlier using complex symmetry analysis. For such systems, the linearizable form, linearization criteria and symmetry group classification are presented. In this paper, we extend the complex approach to obtain a complex-linearizable form of two-dimensional systems of third-order ODEs. This form leads to a linearizable class and linearization criteria of these systems of ODEs. [ABSTRACT FROM AUTHOR]

In recent collaborative work we studied existence and uniqueness of a Lagrangian description for absolutely continuous curves in spaces of Borel probabilities on the real line with finite moments of given order. Of course, a measurable velocity driving the evolution in Eulerian coordinates is necessary to define the Eulerian and Lagrangian descriptions of fluid flow; here we prove that in this case it is also sufficient for a Lagrangian description. More precisely, we argue that the existence of the integrable velocity along an absolutely continuous curve in the set of Borel probabilities on the line is enough to produce a canonical Lagrangian description for the curve; this is given by the family of optimal maps between the uniform distribution on the unit interval and the measures on the curve. Moreover, we identify a necessary and sufficient condition on said family of optimal maps which ensures that the measurable velocity along the curve exists. [ABSTRACT FROM AUTHOR]

In this study, we theoretically prove the presence of homoclinic and heteroclinic motions in the dynamics of economic models perturbed with exogenous shocks. An illustrative example based on the Kaldor model of the aggregate economy is presented.