We report rogue-wave and lump patterns associated with Umemura polynomials, which arise in rational solutions of the third Painlevé equation. We first show that in many integrable equations such as the nonlinear Schrödinger equation and the Boussinesq equation, when internal parameters of their rogue wave solutions are large and of certain form, then their rogue patterns in the spatial-temporal plane can be asymptotically predicted by root distributions of Umemura polynomials (or equivalently, pole distributions of rational solutions to the third Painlevé equation). Specifically, every simple root of the Umemura polynomial would induce a fundamental rogue wave whose spatial-temporal location is linearly related to that simple root, while a multiple root of the Umemura polynomial would induce a non-fundamental rogue wave in the $O(1)$ neighborhood of the spatial-temporal origin. Next, we show that in a certain class of higher-order lump solutions of the Kadomtsev-Petviashvili-I (KPI) equation, when their internal parameters are large and of certain form, then their lump patterns at $O(1)$ time can also be predicted asymptotically by root distributions of Umemura polynomials, where s
Breast cancer is the leading cause of mortality among women. Inspection of breasts by palpation is the key to early detection. We aim to create a wearable tactile glove that could localize the lump in breasts using deep learning (DL). In this work, we present our flexible fabric-based and soft wearable tactile glove for detecting the lumps within custom-made silicone breast prototypes (SBPs). SBPs are made of soft silicone that imitates the human skin and the inner part of the breast. Ball-shaped silicone tumors of 1.5-, 1.75- and 2.0-cm diameters are embedded inside to create another set with lumps. Our approach is based on the InceptionTime DL architecture with transfer learning between experienced and non-experienced users. We collected a dataset from 10 naive participants and one oncologist-mammologist palpating SBPs. We demonstrated that the DL model can classify lump presence, size and location with an accuracy of 82.22%, 67.08% and 62.63%, respectively. In addition, we showed that the model adapted to unseen experienced users with an accuracy of 95.01%, 88.54% and 82.98% for lump presence, size and location classification, respectively. This technology can assist inexperienc
Large-time patterns of general higher-order lump solutions in the KP-I equation are investigated. It is shown that when the index vector of the general lump solution is a sequence of consecutive odd integers starting from one, the large-time pattern in the spatial $(x, y)$ plane generically would comprise fundamental lumps uniformly distributed on concentric rings. For other index vectors, the large-time pattern would comprise fundamental lumps in the outer region as described analytically by the nonzero-root structure of the associated Wronskian-Hermit polynomial, together with possible fundamental lumps in the inner region that are uniformly distributed on concentric rings generically. Leading-order predictions of fundamental lumps in these solution patterns are also derived. The predicted patterns at large times are compared to true solutions, and good agreement is observed.
Remote palpation enables noninvasive tissue examination in telemedicine, yet current tactile displays often lack the fidelity to convey both large-scale forces and fine spatial details. This study introduces a hybrid fingertip display comprising a rigid platform and a $4\times4$ soft pneumatic tactile display (4.93 mm displacement and 1.175 N per single pneumatic chamber) to render a hard lump beneath soft tissue. This study compares three rendering strategies: a Platform-Only baseline that renders the total interaction force; a Hybrid A (Position + Force Feedback) strategy that adds a dynamic, real-time soft spatial cue; and a Hybrid B (Position + Preloaded Stiffness Feedback) strategy that provides a constant, pre-calculated soft spatial cue. In a 12-participant lump detection study, both hybrid methods dramatically improved accuracy over the Platform-Only baseline (from 50\% to over 95\%). While the Hybrid B was highlighted qualitatively for realism, its event-based averaging is expected to increase interaction latency in real-time operation. This suggests a trade-off between perceived lump realism and real-time responsiveness, such that rendering choices that enhance realism ma
We investigate the interaction characteristics of nonlinear coherent structures in the couple Boussinesq (CB) system using the Hirota bilinear approach. First, we derive the lump solutions using a positive quadratic polynomial within the Hirota perturbation technique. Next, We study the explicit interactions between lumps and one or two-kink waves, and observe double lump patterns. Furthermore, we discover the interactions of breathers, revealing a diffusion-like behavior. We notice that breather waves can interact with periodic, kink, and bright solitons for specific parameter sets in the CB system. Interactions between two chains of double breathers are also found. We analyze our results using a combination of symbolic computations and graphical representations, providing a deeper understanding of their behavior. This study reveals previously unreported nonlinear dynamics in the CB system.
In this paper, we study in detail the nonlinear propagation of magnetic soliton in a ferromagnetic film. The sample is magnetized to saturation by an external field perpendicular to film plane. A new generalized (2+1)-dimensional short-wave asymptotic model is derived. The bilinear-like forms of this equation are constructed, and exact magnetic line soliton solutions are exhibited. It is observed that a series of stable lumps can be generated by an unstable magnetic soliton under Gaussian disturbance. Such magnetic lumps are highly stable and can maintain their shapes and velocities during evolution or collision. The interaction between lump and magnetic soliton, as well as interaction between two lumps, are numerically investigated. We further discuss the nonlinear motion of lumps in ferrites with Gilbert-damping and inhomogeneous exchange effects. The results show that the Gilbert-damping effects make the amplitude and velocity of the magnetic lump decay exponentially during propagation. And the shock waves are generated from a lump when quenching the strength of inhomogeneous exchange.
In this paper, a generalized (3 + 1)-dimensional variable-coefficient nonlinear-wave equation is studied in liquid with gas bubbles. Based on the Hirota's bilinear form and symbolic computation, lump and interaction solutions between lump and solitary wave are obtained. Their interaction phenomena is shown in some 3d graphs and contour plots, which include a periodic-shape lump solution, a parabolic-shape lump solution, a cubic-shape lump solution,interaction solutions between lump and one solitary wave, and between lump and two solitary waves. The spatial structures called the bright lump wave and the bright-dark lump wave are discussed. Interaction behaviors of two bright-dark lump waves and a periodic-shape bright lump wave are also presented.
The three condensate wavefunctions of a spinor BEC on a spherical shell can map the real space to the order-parameter space that also has a spherical geometry, giving rise to topological excitations called lump solitons. The homotopy of the mapping endows the lump solitons with quantized winding numbers counting the wrapping between the two spaces. We present several lump-soliton solutions to the nonlinear coupled equations minimizing the energy functional. The energies of the lump solitons with different winding numbers indicate coexistence of lumps with different winding numbers and a lack of advantage to break a higher-winding lump soliton into multiple lower-winding ones. Possible implications are discussed since the predictions are testable in cold-atom experiments.
Of concern are lump solutions for the fractional Kadomtsev--Petviashvili (fKP) equation. As in the classical Kadomtsev--Petviashvili equation, the fKP equation comes in two versions: fKP-I (strong surface tension case) and fKP-II (weak surface tension case). We prove the existence of nontrivial lump solutions for the fKP-I equation in the energy subcritical case $α>\frac{4}{5}$ by means of variational methods. It is already known that there exist neither nontrivial lump solutions belonging to the energy space for the fKP-II equation nor for the fKP-I when $α\leq \frac{4}{5}$. Furthermore, we show that for any $α>\frac{4}{5}$ lump solutions for the fKP-I equation are smooth and decay quadratically at infinity. Numerical experiments are performed for the existence of lump solutions and their decay. Moreover, numerically, we observe cross-sectional symmetry of lump solutions for the fKP-I equation.
The magnetic Skyrmion is described by one control parameter and one length scale. We study the two extreme limits of the control parameter - infinitely large and vanishing - and find that the magnetic Skyrmion becomes a "restricted" magnetic Skyrmion and an O(3) sigma model lump, respectively. Depending on the potential under consideration, the restricted limit manifests differently. In the case of the Zeeman term, the restricted magnetic Skyrmion becomes a "supercompacton" that develops a discontinuity, whereas for the Zeeman term to the power 3/2 it becomes a normal compacton. In both the lump and the restricted limit the solution is given in exact explicit form. We observe that the case of the Zeeman term squared, which can also be understood as a special combination of the Zeeman term and the easy-plane potential - realizable in the laboratory, the analytically exact solution for all values of the coupling - including the Bogomol'nyi-Prasad-Sommerfield (BPS) case - is also of the lump type. Finally, we notice that certain materials (e.g. Fe$_{1-x}$Co$_x$Si or Mn$_{1-x}$Fe$_x$Ge) have a rather large control parameter $ε$ of order 100, making the restricted limit a suitable rough
In this paper, we use a very prominent technique, Hirota Bilinear Method (HBM) to survey the lump structures of the Kadomtsev-Petviashvili (KP) equation in the frame of a collisionless magnetized plasma system composed of dust grains, ions, and nonextensive electrons. Nonlinearity has worldwide applications, and soliton theory is a powerful appliance to illustrate its qualitative behaviors. So, lump solitons are very significant and also interesting. We have observed that lump structures differ due to the correlated parameters of the plasma system. It has also been found that the nonextensive parameter crucially changes the lump features.
The KP-I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution. This is a traveling wave, and the KP-I equation in this case reduces to the Boussinesq equation. In this paper we classify the lump type solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman-Miller, we show that the lump type solutions are rational and their tau function has to be a polynomial of degree $k(k+1)$. In particular, this implies that the lump solution is the unique ground state of the KP-I equation (as conjectured by Klein and Saut in \cite{Klein0}). Our result generalizes a theorem by Airault-McKean-Moser on the classification of rational solutions for the KdV equation.
A family of higher-order rational lumps on non-zero constant background of Davey-Stewartson (DS) II equation are investigated. These solutions have multiple peaks whose heights and trajectories are approximately given by asymptotical analysis. It is found that the heights are time-dependent and for large time they approach the same constant height value of the first-order fundamental lump. The resulting trajectories are considered and it is found that the scattering angle can assume arbitrary values in the interval of $(\fracπ{2}, π)$ which is markedly distinct from the necessary orthogonal scattering for the higher-order lumps on zero background. Additionally, it is illustrated that the higher-order lumps containing multi-peaked $n$-lumps can be regarded as a nonlinear superposition of $n$ first-order ones as $|t|\rightarrow\infty$.
Harmonic maps that minimise the Dirichlet energy in their homotopy classes are known as lumps. Lump solutions on real projective space are explicitly given by rational maps subject to a certain symmetry requirement. This has consequences for the behaviour of lumps and their symmetries. An interesting feature is that the moduli space of charge three lumps is a $7$-dimensional manifold of cohomogeneity one which can be described as a one-parameter family of symmetry orbits of $D_2$ symmetric maps. In this paper, we discuss the charge three moduli spaces of lumps from two perspectives: discrete symmetries of lumps and the Riemann-Hurwitz formula. We then calculate the metric and find explicit formulas for various geometric quantities. We also discuss the implications for lump decay.
Accreting binary black holes (BBHs) are multi-messenger sources, emitting copious electromagnetic (EM) and gravitational waves. One of their most promising EM signatures is the lightcurve modulation caused by a strong, unique and extended azimuthal overdensity structure orbiting at the inner edge of the circumbinary disc (CBD), dubbed "lump". In this paper, we investigate the origin of this structure using 2D general-relativistic (GR) hydrodynamical simulations of a CBD in an approximate BBH spacetime. First, we use the symmetric mass-ratio case to study the transition from the natural m = 2 mode to m = 1. The asymmetry with respect to m = 2 grows exponentially, pointing to an instability origin. We indeed find that the CBD edge is prone to a (magneto-)hydrodynamical instability owing to the disc edge density sharpness: the Rossby Wave Instability (RWI). The RWI criterion is naturally fullfilled at the CBD edge and we report the presence of vortices, which are typical structures of the RWI. The RWI is also at work in the asymmetric mass-ratio cases (from 0.1 to 0.5). However, the CBD edge sharpness decreases with a decreasing mass ratio, and so the lump. By proposing a scenario for
In this work we consider the scalar field model with a false vacuum proposed by A. T. Avelar, D. Bazeia, L. Losano and R. Menezes, Eur. Phys. J. C 55, 133-143 (2008). The model depends on a parameter $s>0$. The model has unstable nontopological lump solutions with a bell shape for small $s$, acquiring a flat plateau around the maximum for large $s$. For $s\to\infty$ the $φ^4$ model is recovered. We show that for $s\gtrsim 2$ the lump is metastable with the only negative mode very close to zero. Metastable lumps can propagate and survive long enough to produce dynamical effects. Due to their simplicity, they can be an alternative to the procedure of stabilization which requires, for instance, a complex scalar field to construct nontopological solitons. We study lump-lump collisions in this model, describing the main characteristics of the scattering products at their dependence on $s$ and the initial velocity modulus of each lump.
We analyse the detail of interactions of two-dimensional solitary waves called lumps and one-dimensional line solitons within the framework of the Kadomtsev-Petviashvili equation describing wave processes in media with positive dispersion. We show that line solitons can emit or absorb lumps or periodic chains of lumps; they can interact with each other by means of lumps. Within a certain time interval, lumps or lump chains can emerge between two line solitons and disappear then due to absorption by one of the solitons. This phenomenon resembles the appearance of rogue waves in the oceans. The results obtained are graphically illustrated and can be applicable to the description of physical processes occurring in plasma, fluids, solids, nonlinear optical media, and other fields.
Lump solutions are spatially rationally localized solutions which usually arise as solutions to higher dimensional nonlinear partial differential equations often possessing Hirota bilinear forms. Under some parameter constraint, these solutions may lead to rogue wave solutions. In this article, we study lump and rogue wave solutions of a new nonlinear non-evolutionary equation in 2+1 dimensions with the aid of a computer algebra system. We present illustrative examples and analyze the dynamical behavior of the solutions using graphical representations
We study some properties of the tachyonic lumps in the level truncation scheme of bosonic cubic string field theory. We find that several gauges work well and that the size of the lump as well as its tension is approximately independent of these gauge choices at level (2,4). We also examine the fluctuation spectrum around the lump solution, and find that a tachyon with m^2=-0.96 and some massive scalars appear on the lump world-volume. This result strongly supports the conjecture that a codimension 1 lump solution is identified with a D-brane of one lower dimension within the framework of bosonic cubic string field theory.
This paper studies false vacuum lumps surrounded by the true vacuum in a real scalar field potential in flat spacetime. Fermions reside in the core of the lump, which are coupled with the scalar field via Yukawa interaction. Such lumps are stable against spherical collapse and deformation from spherical shape based on energetics considerations. The fermions inside the lump are treated as a uniform Fermi gas. We consider the Fermi gas in both ultrarelativistic and nonrelativistic limits. The mass and size of these lumps depend on the scale characterizing the scalar field potential as well as the mass density of the fermions.