New Type Of Solution Research Articles

New type of solutions for the modified Korteweg–de Vries equation

In this letter we report a new type of multi-soliton solutions for the modified Korteweg–de Vries (mKdV) equation. These solutions contain τ functions of the trigonometric solitons and classical solitons simultaneously. A new bilinear form of the mKdV equation is introduced to derive these solutions. The obtained solutions display as solitons living on a periodic background, which are analyzed and illustrated.

New type of solutions for Schrödinger equations with critical growth

We consider the following nonlinear Schrödinger equations with critical growth: −Δu+V(|y|)u=uN+2N−2,u>0inRN, where V(|y|) is a bounded positive radial function in C1, N ≥ 5. By using a finite reduction argument, we show that if r2V(r) has either an isolated local maximum or an isolated local minimum at r0 > 0 with V(r0) > 0, there exists infinitely many non-radial large energy solutions which are invariant under some sub-groups of O(3).

A New Solution of the Pulsar Equation

We present the first new type of solution of the pulsar equation since 1999. In it, the whole magnetosphere is confined inside the light cylinder and an electrically charged layer wraps around it and holds it together. The reason this new solution has never been obtained before is that all current time-dependent simulations are initialized with a vacuum dipole configuration that extends to infinity; thus, their final steady-state solution also extends to infinity. Under special conditions, such a confined configuration may be attained when the neutron star first forms in the interior of a collapsing star during a supernova explosion, or when it accretes from an external wind or disk from a donor star. It is shown that this new maximally closed non-decelerating solution is the limit of a continuous sequence of standard magnetospheres with open and closed field lines when the amount of open field lines gradually drops to zero. The minimum energy solution in this sequence is a standard magnetosphere in which the closed field line region extends up to about 80% of the light cylinder. We estimate that the released energy when the new solution transitions to the minimum energy one is enough to power a fast radio burst.

Numerical simulation of nonlinear fractional delay differential equations with Mittag-Leffler kernels

This study is concerned with finding numerical solutions of nonlinear delay differential equations involving extended Mittag-Leffler fractional derivatives of the Caputo-type. The main benefit of the used extension is to address the complexity resulting from the limitations of using fractional derivatives with non-singular Mittag-Leffler kernels. We discussed the existence and uniqueness of solutions for the studied delay models. Next, we modified an Adams-type method to numerically solve fractional delay differential equations combined with Mittag-Leffler kernels. A new type of solution belonging to the L1 space is presented for the studied models using the proposed scheme.

Numerical solutions of linear time-fractional advection-diffusion equations with modified Mittag-Leffler operator in a bounded domain

Fractional advection-diffusion equations have demonstrated to be a powerful tool in modeling complex anomalous diffusion in applied science. In this paper, we studied novel linear time-fractional advection-diffusion equations associated with an extension of Mittag-Leffler fractional derivative operator. A useful feature of the used extension is to address the limitations of the Mittag-Leffler fractional derivative model. We, mainly, proposed a numerical approach to provide approximate solutions to linear time-fractional advection-diffusion equations with the studied extended fractional derivative operator. The suggested approach is based on discretizing the studied models with respect to spatio-temporal domain using uniform meshes. A new type of solutions for the studied models was generated numerically using the proposed approach. Besides, a comparative study was conducted to verify the accuracy and feasibility of the proposed approach.

Multibump positive solutions for Choquard equation with double potentials in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$

In this paper, we construct a new family of multibump solutions to the following Choquard equation where and are radial positive functions and both of them satisfy some decay conditions at infinity. The centers of these bumps are located in points on two circles. More precisely, the new type of solutions has polygonal symmetry in the ‐plane and even symmetry about the ‐axis.

Even and Odd Self-Similar Solutions of the Diffusion Equation for Infinite Horizon

In the description of transport phenomena, diffusion represents an important aspect. In certain cases, the diffusion may appear together with convection. In this paper, we study the diffusion equation with the self-similar Ansatz. With an appropriate change of variables, we have found an original new type of solution of the diffusion equation for infinite horizon. We derive novel even solutions of diffusion equation for the boundary conditions presented. For completeness, the odd solutions are also mentioned as well, as part of the previous works. We have found a countable set of even and odd solutions, of which linear combinations also fulfill the diffusion equation. Finally, the diffusion equation with a constant source term is discussed, which also has even and odd solutions.

Vaidya and Generalized Vaidya Solutions by Gravitational Decoupling

In this paper, we apply the gravitational decoupling method for dynamical systems in order to obtain a new type of solution that can describe a hairy dynamical black hole. We consider three cases of decoupling. The first one is the simplest and most well known when the mass function is the function only of space coordinate r. The second case is a Vaidya spacetime case when the mass function depends on time v. Finally, the third case represents the generalization of these two cases: the mass function is the function of both r and v. We also calculate the apparent horizon and singularity locations for all three cases.

The two-component complex coupled integrable dispersionless equations: Darboux transformation and soliton solutions

The complex coupled integrable dispersionless (CID)system is an important system. It has been shown that complex CID equation is gauge equivalent to the Pohlmeyer–Lund–Regge model (a complex sine-Gordon equation) and can also be transformed to the complex short pulse equation. CID system can describe a current-fed string interacting with an external magnetic field. In this paper, we investigate the two-component complex coupled integrable dispersionless system. By using the Darboux transformation, we obtain a new type of solution (the breather-like soliton solution) with zero boundary condition and a large number of solutions (the breather-II type solution, bright–dark soliton solution, the resonance solutions and the semi-rational solutions) with non-zero boundary condition. In particular, these solutions are not available for the scalar complex integrable dispersionless system.

A new type of solutions for a nonlinear Schrödinger system with chi ^{(2)} nonlinearities

We are concerned with the question of constructing a new type of solution to the problem with chi ^{(2)} nonlinearities 0.1{−Δu+P(x)u=αuv,inRN,−Δv+Q(x)v=α2u2+βv2,inRN, where P(x)=P(|x|) and Q(x)=Q(|x|) are positive bounded radial potentials, 3leq N<6, alpha >0 and alpha >beta . Assuming that the potentials P(x) and Q(x) satisfy certain conditions, the existence of a new type of solutions is proved.

New type of solutions for the nonlinear Schrödinger equation in [formula omitted

We construct a new family of entire solutions for the nonlinear Schrödinger equation{−Δu+V(y)u=up,u>0,inRN,u∈H1(RN), where p∈(1,N+2N−2) and N≥3, and V(y)=V(|y|) is a positive bounded radial potential satisfyingV(|y|)=V0+a|y|m+O(1|y|m+σ),as|y|→∞, for some fixed constants V0,a,σ>0, and m>max⁡{4p−1,2}.

Boson stars and black holes with wavy scalar hair

In this paper, we follow up on the discovery of a new type of solution in the Einstein-Maxwell system coupled minimally to a self-interacting complex scalar field. For sufficiently large gravitational coupling and sufficiently small electromagnetic coupling we demonstrate that boson stars as well as black holes can carry scalar hair that shows a distinct new feature: a number of spatial oscillations in the scalar field away from the core or horizon, respectively. These spatial oscillations appear also in the curvature invariants and hence should be a detectable feature of the space-time. As a first hint that this is true, we show that the effective potential for null geodesics in this space-time possesses a local minimum indicating that in the spatial region where oscillations occur a new stable photon sphere should be possible. We also study the interior of the black holes with scalar hair and show that the curvature singularity appears at a finite value of the radius and that black holes with wavy scalar hair have this singularity very close to the center.

Trigonometric shock waves in the Kaup–Boussinesq system

We consider the modulationally stable version of the Kaup-Boussinesq system which models propagation of nonlinear waves in various physical systems. It is shown that the Whitham modulation equations for this model have a new type of solutions which describe trigonometric shock waves. In the Gurevich-Pitaevskii problem of evolution of an initial discontinuity, these solutions correspond to a non-zero wave excitation on one of the sides of the discontinuity. Our analytical results are confirmed by numerical calculations.

Construction of solutions for the nonlinear magnetic Schrödinger equation in RN

In this paper, we construct a new type of solution like the one proposed in the work of Duan and Musso (arXiv:2006.16125v1) for the magnetic Schrödinger equation. Our argument requires new estimates in order to overcome the appearance of the magnetic potential. The approach here can be seen as an extension of that mentioned in the work of Duan and Musso (arXiv:2006.16125v1).

A square-integrable spinor solution to non-interacting Dirac equations

We consider the Dirac equation written in polar form, without any external potential but equipped with a non-zero tensorial connection, and we find a new type of solution that is localized around the origin with a decreasing exponential behavior in the radial coordinate.

Robotic Sorting Systems: Performance Estimation and Operating Policies Analysis

Many distribution centers use expensive, conveyor-based sorting systems that require large buildings to house them. In areas with tight space, robotic sorting systems offer a new type of solution to sort parcels by destination. Such systems are highly flexible in throughput capacity and are now gradually being introduced, particularly in express companies. This paper studies robotic sorting system with two layouts. The first layout has two tiers: robots drive on the top tier and sort parcels by destination on spiral conveyors connected to roll containers at the lower tier. The second layout has a single tier with input and output points located at the perimeter, connected by robots. For each layout, we consider both the shortest path topology via dual-lane aisles and the detour path topology via single-lane aisles. We build closed queueing networks for performance estimation, design an iterative procedure to investigate robot congestion in the two-tier layout, and use a traffic flow function to estimate robot congestion in the single-tier layout. Random, closest, dedicated, and shortest-queue robot-to-loading-station assignment rules are examined. We validate analytical models by both simulation and a real case of Deppon Express and analyze the optimal system size and operating policies for throughput capacity and operating cost. The results show that the system throughput capacity is significantly affected by robot congestion in the single-tier layout with the detour path topology, but it is only slightly affected in the other systems. A square layout fits the shortest path and a rectangular layout fits the detour path. Both the random assignment rule and the shortest-queue assignment rule are superior for a large number of robots, whereas the dedicated assignment rule is superior for a small number of robots. We apply these insights at Deppon Express for different allocations in peak and off-peak hours. Our analysis shows that a robotic sorting system typically has lower overall annual cost than a traditional cross-belt sorting system when the required throughput capacity is not too large.

A new finite difference method for computing approximate solutions of boundary value problems including transition conditions

This article is aimed at computing numerical solutions of new type of boundary value problems (BVPs) for two-linked ordinary differential equations. The problem studied here differs from the classical BVPs such that it contains additional conditions at the point of interaction, so-called transition conditions. Naturally, such type of problems is much more complicated to solve than classical problems. It is not clear how to apply the classical numerical methods to such type of boundary value transition problems (BVTPs). Based on the finite difference method (FDM) we have developed a new numerical algorithm for computing numerical solution of BVTPs for two-linked ordinary differential equations. To demonstrate the reliability and efficiency of the presented algorithm we obtained numerical solution of one BVTP and the results are compared with the corresponding exact solution. The maximum absolute errors (MAEs) are presented in a table.

Partly clustering solutions of nonlinear Schrödinger systems with mixed interactions

In this paper, we prove a partly clustering phenomenon for nonlinear Schrödinger systems with large mixed couplings of attractive and repulsive forces, which arise from the models in Bose-Einstein condensates and nonlinear optics. More precisely, we consider a system with three components where the interaction between the first two components and the third component is repulsive, and the interaction between the first two components is attractive. Recent studies [10–13] in this case show that for large interaction forces, the first two components are localized in a region with a small energy and the third component is close to a solution of a single equation. Especially, the results in the works [12,13] say that the region of localization for a (locally) least energy vector solution on a ball in the class of radially symmetric functions is the origin or the whole boundary depending on the space dimension 1≤n≤3. In this paper we construct a new type of solutions with a region of localization different from the origin or the whole boundary. In fact, we show that there exist radially symmetric positive vector solutions with clustering multi-bumps for the first two components near the maximum point of rn−1U3, where U is the limit of the third component and the maximum point is the only critical point different from the origin and the boundary.

On the new type of solutions to Benney-Luke equation

In this study, the Benney-Luke equation is considered. In order to derive new type of solutions, the sn-ns method is applied to this equation. Then, we introduce trigonometric and elliptic functions solutions in addition to the hyperbolic ones which are gained by tanh-coth. Three types of solutions are derived at the same time with the help of this method. Therefore, it can be said that this method is convenient to obtain more solutions to many kinds of nonlinear partial differential equations.

A Statistical Robust Approach to Design Parallel Transmit Radiofrequency Excitations in MRI

In MRI, at ultrahigh field, the use of parallel transmit radiofrequency (RF) arrays is very beneficial to better control spin excitation spatially. In that framework, the so-called “universal pulse” technique, proposed recently for head imaging at 7 tesla, gives access to “plug-and-play” nonadiabatic solutions exhibiting good robustness against intersubject variations in the resonant transmit fields. This new type of solution has been defined so far as the result of numerical pulse optimizations performed across a collection of RF field maps acquired on a small population sample (pulse design database). In this work, we investigate an alternative universal pulse design approach in the linear small tip angle regime whereby the database of RF field maps is first transformed into a second-order statistical model and which then exploits a statistical robust design formalism for the optimization of the RF and magnetic field gradient waveforms. Experimental validation with an eightfold transmit RF coil for 7 tesla brain imaging shows that this new approach brings some benefit in terms of computational efficiency. Hence, for a design database composed of 35 maps, the computation time initially of 50 min could be reduced down to 3 min. The proposed statistical approach thus enables integration of large databases, presumably necessary to ensure robust solutions. Finally, it provides means to compute flip angle statistics and, along with it, simple performance metrics for quality assurance (RF pulse performance) or guidance in the optimization of TX array architectures.