Showing papers in "Journal of Mathematical Physics in 2018"
TL;DR: In this paper, the inverse scattering transform for the nonlocal NLS equation with nonzero boundary conditions at infinity is presented in four different cases when the data at infinity have constant amplitudes.
Abstract: In 2013, a new nonlocal symmetry reduction of the well-known AKNS (an integrable system of partial differential equations, introduced by and named after Mark J. Ablowitz, David J. Kaup, and Alan C. Newell et al. (1974)) scattering problem was found. It was shown to give rise to a new nonlocal PT symmetric and integrable Hamiltonian nonlinear Schrodinger (NLS) equation. Subsequently, the inverse scattering transform was constructed for the case of rapidly decaying initial data and a family of spatially localized, time periodic one-soliton solutions was found. In this paper, the inverse scattering transform for the nonlocal NLS equation with nonzero boundary conditions at infinity is presented in four different cases when the data at infinity have constant amplitudes. The direct and inverse scattering problems are analyzed. Specifically, the direct problem is formulated, the analytic properties of the eigenfunctions and scattering data and their symmetries are obtained. The inverse scattering problem, which...
156 citations
TL;DR: In this article, the standard and non-local nonlinear Schrodinger (NLS) equations obtained from the coupled NLS system of equations (AKNS) were studied by using the Hirota bilinear method.
Abstract: We study standard and nonlocal nonlinear Schrodinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions, respectively. By using the Hirota bilinear method, we first find soliton solutions of the coupled NLS system of equations; then using the reduction formulas, we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function |q(t, x)|2 for the standard and nonlocal NLS equations.
154 citations
TL;DR: In this paper, the parity (P), time reversal (T), charge conjugation (Ĉ), and their possible combinations such as P^T^, P^Ĉ, and P^ T^, etc., can be successively applied.
Abstract: To describe two-place physical problems, many possible models named Alice-Bob (AB) systems are proposed. To find and to solve these systems, the parity (P^), time reversal (T^), charge conjugation (Ĉ), and their possible combinations such as P^T^, P^Ĉ, and P^T^Ĉ, etc., can be successively applied. Especially, some special types of P^-T^-Ĉ group invariant multi-soliton solutions for the KdV-KP-Toda type, mKdV-sG type, and nonlinear Schrodinger equations (NLS) type AB systems are explicitly constructed. The possible P^T^ symmetry breaking solutions of two special ABKdV systems are explicitly given. Applying the P^-T^-Ĉ symmetries to coupled Ablowitz-Kaup-Newell-Segur systems, some four-place nonlocal NLS systems are also derived.
96 citations
TL;DR: In this paper, the 3-loop master integrals for heavy quark correlators and the three-loop quantum chromodynamics corrections to the ρ-parameter were derived in terms of 2F1 Gaus hypergeometric functions at rational argument.
Abstract: We calculate 3-loop master integrals for heavy quark correlators and the 3-loop quantum chromodynamics corrections to the ρ-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-N space either. The solution of the homogeneous equations is possible in terms of 2F1 Gaus hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using q-product and series representations implied by Jacobi’s ϑi functions and Dedekind’s η-function. The corresponding representations can be traced back to polynomials out of Lambert–Eisenstein series, having representations also as elliptic polylogarithms, a q-factorial 1/ηk(τ), logarithms, and polylogarithms of q and their q-integrals. Due to the specific form of the physical variable x(q) for different processes, different representations do usually appear. Numerical results are also presented.
94 citations
TL;DR: In this article, an effective and straightforward method is presented to succinctly construct the bilinear representation of the (4+1)-dimensional nonlinear Fokas equation, which is an important physics model.
Abstract: Under investigation in this paper is the (4+1)-dimensional nonlinear Fokas equation, which is an important physics model. With the aid of Bell’s polynomials, an effective and straightforward method is presented to succinctly construct the bilinear representation of the equation. By using the resulting bilinear formalism, the soliton solutions and Riemann theta function periodic wave solutions of the equation are well constructed. Furthermore, the extended homoclinic test method is employed to construct the breather wave solutions and rogue wave solutions of the equation. Finally, a connection between periodic wave solutions and soliton solutions is systematically established. The results show that the periodic waves tend to solitary waves under a limiting procedure.
72 citations
TL;DR: In this paper, the authors present a new approach to construct the separate variables basis leading to the full characterization of the transfer matrix spectrum of quantum integrable lattice models, which is a key feature of the Liouville-Arnold classical integrability framework where the complete set of conserved charges defines both the level manifold and the flows on it leading to construction of actionangle variables.
Abstract: We present a new approach to construct the separate variables basis leading to the full characterization of the transfer matrix spectrum of quantum integrable lattice models. The basis is generated by the repeated action of the transfer matrix itself on a generically chosen state of the Hilbert space. The fusion relations for the transfer matrix, stemming from the Yang-Baxter algebra properties, provide the necessary closure relations to define the action of the transfer matrix on such a basis in terms of elementary local shifts, leading to a separate transfer matrix spectral problem. Hence our scheme extends to the quantum case a key feature of the Liouville-Arnold classical integrability framework where the complete set of conserved charges defines both the level manifold and the flows on it leading to the construction of action-angle variables. We work in the framework of the quantum inverse scattering method. As a first example of our approach, we give the construction of such a basis for models associated with Y(gln) and argue how it extends to their trigonometric and elliptic versions. Then we show how our general scheme applies concretely to fundamental models associated with the Y(gl2) and Y(gl3) R-matrices leading to the full characterization of their spectrum. For Y(gl2) and its trigonometric deformation, a particular case of our method reproduces Sklyanin’s construction of separate variables. For Y(gl3), it gives new results, in particular, through the proper identification of the shifts acting on the separate basis. We stress that our method also leads to the full characterization of the spectrum of other known quantum integrable lattice models, including, in particular, trigonometric and elliptic spin chains, open chains with general integrable boundaries, and further higher rank cases that we will describe in forthcoming publications.
67 citations
TL;DR: In this article, an identity that relates the q-Laplace transform of the height function of a higher spin inhomogeneous stochastic six vertex model in a quadrant on one side and a multiplicative functional of a Macdonald measure on the other was proved.
Abstract: We prove an identity that relates the q-Laplace transform of the height function of a (higher spin inhomogeneous) stochastic six vertex model in a quadrant on one side and a multiplicative functional of a Macdonald measure on the other. The identity is used to prove the GUE Tracy-Widom asymptotics for two instances of the stochastic six vertex model via asymptotic analysis of the corresponding Schur measures.
57 citations
TL;DR: In this paper, the authors present a global construction of a so-called D-bracket appearing in the physics literature of Double Field Theory (DFT) and show that if certain integrability criteria are satisfied, it can be seen as a sum of two Courant algebroid brackets.
Abstract: We present a global construction of a so-called D-bracket appearing in the physics literature of Double Field Theory (DFT) and show that if certain integrability criteria are satisfied, it can be seen as a sum of two Courant algebroid brackets. In particular, we show that the local picture of the extended space-time used in DFT fits naturally in the geometrical framework of para-Hermitian manifolds and that the data of an (almost) para-Hermitian manifold is sufficient to construct the D-bracket. Moreover, the twists of the bracket appearing in DFT can be interpreted in this framework geometrically as a consequence of certain deformations of the underlying para-Hermitian structure.
55 citations
TL;DR: In this article, the distinction between interacting and non-interacting integrable systems is characterized by the Onsager matrix and zero is the defining property of a noninteracting system.
Abstract: We propose that the distinction between interacting and noninteracting integrable systems is characterized by the Onsager matrix. It being zero is the defining property of a noninteracting integrable system. To support our view, various classical and quantum integrable chains are discussed.We propose that the distinction between interacting and noninteracting integrable systems is characterized by the Onsager matrix. It being zero is the defining property of a noninteracting integrable system. To support our view, various classical and quantum integrable chains are discussed.
54 citations
TL;DR: In this article, a generalized index theorem for topological free-fermion systems on a hypercubic lattice in spatial dimensions d ≥ 1 has been proposed, which is a noncommutative analog of the Atiyah-Singer index theorem.
Abstract: We study a wide class of topological free-fermion systems on a hypercubic lattice in spatial dimensions d ≥ 1. When the Fermi level lies in a spectral gap or a mobility gap, the topological properties, e.g., the integral quantization of the topological invariant, are protected by certain symmetries of the Hamiltonian against disorder. This generic feature is characterized by a generalized index theorem which is a noncommutative analog of the Atiyah-Singer index theorem. The noncommutative index defined in terms of a pair of projections gives a precise formula for the topological invariant in each symmetry class in any dimension (d ≥ 1). Under the assumption on the nonvanishing spectral or mobility gap, we prove that the index formula reproduces Bott periodicity and all of the possible values of topological invariants in the classification table of topological insulators and superconductors. We also prove that the indices are robust against perturbations that do not break the symmetry of the unperturbed Hamiltonian.
52 citations
TL;DR: In this article, for a large class of cell kinetics including sub-logistic sources, an explicit condition involving the chemotactic strength, the asymptotic damping rate, and the initial mass of cells was found to ensure uniform-in-time boundedness for the corresponding 2D Neumann initial-boundary value problem.
Abstract: It is well known that the Neumann initial-boundary value problem for the minimal Keller-Segel chemotaxis system in a 2D bounded smooth domain has no blow-ups for any presence of logistic source of cell kinetics. Here, for a large class of cell kinetics including sub-logistic sources, we find an explicit condition involving the chemotactic strength, the asymptotic “damping” rate, and the initial mass of cells to ensure the uniform-in-time boundedness for the corresponding 2D Neumann initial-boundary value problem. Our finding in particular shows that sub-logistic source can prevent blow-up in 2D, indicating that logistic damping is not the weakest damping to guarantee boundedness for the 2D Keller-Segel minimal chemotaxis model.
TL;DR: In this article, the leaves of the null foliation are endowed with a non-relativistic structure dual to the Newtonian one, dubbed Carrollian spacetime, and a new non-Lorentzian ambient metric structure of which they study the geometry.
Abstract: Connections compatible with degenerate metric structures are known to possess peculiar features: on the one hand, the compatibility conditions involve restrictions on the torsion; on the other hand, torsionfree compatible connections are not unique, the arbitrariness being encoded in a tensor field whose type depends on the metric structure. Nonrelativistic structures typically fall under this scheme, the paradigmatic example being a contravariant degenerate metric whose kernel is spanned by a one-form. Torsionfree compatible (i.e., Galilean) connections are characterised by the gift of a two-form (the force field). Whenever the two-form is closed, the connection is said Newtonian. Such a nonrelativistic spacetime is known to admit an ambient description as the orbit space of a gravitational wave with parallel rays. The leaves of the null foliation are endowed with a nonrelativistic structure dual to the Newtonian one, dubbed Carrollian spacetime. We propose a generalisation of this unifying framework by introducing a new non-Lorentzian ambient metric structure of which we study the geometry. We characterise the space of (torsional) connections preserving such a metric structure which is shown to project to (respectively, embed) the most general class of (torsional) Galilean (respectively, Carrollian) connections.
TL;DR: In this paper, a Fredholm determinant and short-distance series representation of the Painleve V tau function τt associated with generic monodromy data are derived.
Abstract: We prove a Fredholm determinant and short-distance series representation of the Painleve V tau function τt associated with generic monodromy data. Using a relation of τt to two different types of irregular c = 1 Virasoro conformal blocks and the confluence from Painleve VI equation, connection formulas between the parameters of asymptotic expansions at 0 and i∞ are conjectured. Explicit evaluations of the connection constants relating the tau function asymptotics as t → 0, +∞, i∞ are obtained. We also show that irregular conformal blocks of rank 1, for arbitrary central charge, are obtained as confluent limits of the regular conformal blocks.
TL;DR: In this article, a compositional framework for finite quantum set theory based on a 2-category of quantum sets and quantum functions is introduced, and dualizable and invertible 1-morphisms in these 2-categories are analyzed.
Abstract: We introduce a notion of quantum function and develop a compositional framework for finite quantum set theory based on a 2-category of quantum sets and quantum functions. We use this framework to formulate a 2-categorical theory of quantum graphs, which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero-error communication and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups. We show that the 2-categories of quantum sets and quantum graphs are semisimple. We analyze dualisable and invertible 1-morphisms in these 2-categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs.
TL;DR: In this paper, the authors present an action for a six-dimensional superconformal field theory containing a non-abelian tensor multiplet, which is a key step towards a potential construction of the (2, 0)-theory.
Abstract: We present an action for a six-dimensional superconformal field theory containing a non-abelian tensor multiplet. All of the ingredients of this action have been available in the literature. We bring these pieces together by choosing the string Lie 2-algebra as a gauge structure, which we motivated in previous work. The kinematical data contains a connection on a categorified principal bundle, which is the appropriate mathematical description of the parallel transport of self-dual strings. Our action can be written down for each of the simply laced Dynkin diagrams, and each case reduces to a four-dimensional supersymmetric Yang–Mills theory with corresponding gauge Lie algebra. Our action also reduces nicely to an M2-brane model which is a deformation of the Aharony-Bergman-Jafferis-Maldacena (ABJM) model. While this action is certainly not the desired M5-brane model, we regard it as a key stepping stone towards a potential construction of the (2, 0)-theory.
TL;DR: In this paper, a deformation theory approach is presented to the classification of kinematical Lie algebras in 3 + 1 dimensions and calculations leading to the classifications of all deformations of the static kinematic Lie algebra and of its universal central extension, up to isomorphism.
Abstract: We present a deformation theory approach to the classification of kinematical Lie algebras in 3 + 1 dimensions and present calculations leading to the classifications of all deformations of the static kinematical Lie algebra and of its universal central extension, up to isomorphism. In addition, we determine which of these Lie algebras admit an invariant symmetric inner product. Among the new results, we find some deformations of the centrally extended static kinematical Lie algebra which are extensions (but not central) of deformations of the static kinematical Lie algebra. This paper lays the groundwork for two companion papers which present similar classifications in dimension D + 1 for all D⩾4 and in dimension 2 + 1.
TL;DR: In this paper, the authors show that the dynamics of the C-field and its dual are encoded in a unified way in the 4-sphere, which suggests the corresponding spectrum as the generalized cohomology theory describing the fields.
Abstract: We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over Q, Z and Z2, the Kervaire invariant, the f-invariant, and the ν-invariant. This requires either a framing or a corner structure. The canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies. We characterize the flux quantization condition on the C-field and the topological action of the M5-brane via the Hopf invariant, and the dual of the C-field as (a refinement of) an element of Hopf invariant two. In the signature formulation, the contribution to the M-brane effective action is given by the Maslov index of the corner. The Kervaire invariant implies that the effective action of the M5-brane is quadratic. Our study leads to viewing the self-dual string, which is the boundary of the M2-brane on the M5-brane worldvolume, as a string theory in the sense of cobordism of manifolds with corners. We show that the dynamics of the C-field and its dual are encoded in a unified way in the 4-sphere, which suggests the corresponding spectrum as the generalized cohomology theory describing the fields. The effective action of the corner is captured by the f-invariant, which is an invariant at chromatic level two. Finally, considering M-theory on manifolds with G2 holonomy, we show that the canonical G2 structure minimizes the topological part of the M5-brane action. This is done via the ν-invariant and a variant that we introduce related to the one-loop polynomial.We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over Q, Z and Z2, the Kervaire invariant, the f-invariant, and the ν-invariant. This requires either a framing or a corner structure. The canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies. We characterize the flux quantization condition on the C-field and the topological action of the M5-brane via the Hopf invariant, and the dual of the C-field as (a refinement of) an element of Hopf invariant two. In the signature formulation, the contribution to the M-brane effective action is given by the Maslov index of the corner. The Kervaire invariant implies that the effective action of the M5-brane is quadratic. Our study leads to viewing the self-dual string, which is the boundary of the M2-brane on the M5-brane worldvolume, as a string theory in the sense of cobordism of manifolds with corners. ...
TL;DR: In this paper, the higher order discrete rogue wave (RWs) of the integrable discrete Ablowitz-Ladik equation are reported using a novel discrete version of generalized perturbation Darboux transformation.
Abstract: The higher order discrete rogue waves (RWs) of the integrable discrete Ablowitz-Ladik equation are reported using a novel discrete version of generalized perturbation Darboux transformation. The dynamical behaviors of strong and weak interactions of these RWs are analytically and numerically discussed, which exhibit the abundant wave structures. We numerically show that a small noise has the weaker effect on strong-interaction RWs than weak-interaction RWs, whose main reason may be related to main energy distributions of RWs. The interaction of two first-order RWs is shown to be non-elastic. Moreover, we find that the maximal number (Smax) of the possibly split first-order ones of higher order RWs is related to the number (Pmax) of peak points of their strongest-interaction cases, that is, Smax = (Pmax + 1)/2. The results will excite to further understand the discrete RW phenomena in nonlinear optics and relevant fields.
TL;DR: In this article, a canonical transformation (x, p)→(x,p,p^q) was proposed to lead a position-dependent mass particle in usual space to another Hamiltonian of a particle with constant mass in a conservative force field of the deformed space.
Abstract: We consider canonically conjugated generalized space and linear momentum operators x^q and p^q in quantum mechanics, associated with a generalized translation operator which produces infinitesimal deformed displacements controlled by a deformation parameter q. A canonical transformation (x^,p^)→(x^q,p^q) leads the Hamiltonian of a position-dependent mass particle in usual space to another Hamiltonian of a particle with constant mass in a conservative force field of the deformed space. The equation of motion for the classical phase space (x, p) may be expressed in terms of the deformed (dual) q-derivative. We revisit the problem of a q-deformed oscillator in both classical and quantum formalisms. Particularly, this canonical transformation leads a particle with position-dependent mass in a harmonic potential to a particle with constant mass in a Morse potential. The trajectories in phase spaces (x, p) and (xq, pq) are analyzed for different values of the deformation parameter. Finally, we compare the resul...
TL;DR: In this article, the authors apply deformation theory to the classification of kinematical Lie algebras in arbitrary dimensions up to isomorphism and determine which of them admit an invariant inner product.
Abstract: We classify kinematical Lie algebras in dimension D + 1 for D > 3 up to isomorphism. This is part of a series of papers applying deformation theory to the classification of kinematical Lie algebras in arbitrary dimension. This is approached via the classification of deformations of the relevant static kinematical Lie algebra. We also classify the deformations of the universal central extension of the static kinematical Lie algebra in dimension D + 1 for D > 3. In addition, we determine which of these Lie algebras admit an invariant inner product.
TL;DR: In this paper, it was shown that every unital positive partial transpose (PPT) channel has a finite index of separability and that the class of unital channels that have a finite-indexed linear map becomes separable in the tensor product space.
Abstract: We analyze linear maps on matrix algebras that become entanglement breaking after composing a finite or infinite number of times with themselves. This means that the Choi matrix of the iterated linear map becomes separable in the tensor product space. If a linear map becomes entanglement breaking after finitely many iterations, we say that the map has a finite index of separability. In particular, we show that every unital positive partial transpose (PPT) channel has a finite index of separability and that the class of unital channels that have a finite index of separability is a dense subset of the unital channels. We construct concrete examples of maps which are not PPT but have a finite index of separability. We prove that there is a large class of unital channels that are asymptotically entanglement breaking. Our work is motivated by Christandl’s PPT-squared conjecture. This conjecture states that every PPT channel, when composed with itself, becomes entanglement breaking.
TL;DR: In this paper, a unified structure of rogue wave and multiple rogue wave solutions for all equations of the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and their mixed and deformed versions is described.
Abstract: We describe a unified structure of rogue wave and multiple rogue wave solutions for all equations of the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and their mixed and deformed versions. The definition of the AKNS hierarchy and its deformed versions is given in the Sec. II. We also consider the continuous symmetries of the related equations and the related spectral curves. This work continues and summarises some of our previous studies dedicated to the rogue wave-like solutions associated with AKNS, nonlinear Schrodinger, and KP hierarchies. The general scheme is illustrated by the examples of small rank n, n ⩽ 7, rational or quasi-rational solutions. In particular, we consider rank-2 and rank-3 quasi-rational solutions that can be used for prediction and modeling of the rogue wave events in fiber optics, hydrodynamics, and many other branches of science.We describe a unified structure of rogue wave and multiple rogue wave solutions for all equations of the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and their mixed and deformed versions. The definition of the AKNS hierarchy and its deformed versions is given in the Sec. II. We also consider the continuous symmetries of the related equations and the related spectral curves. This work continues and summarises some of our previous studies dedicated to the rogue wave-like solutions associated with AKNS, nonlinear Schrodinger, and KP hierarchies. The general scheme is illustrated by the examples of small rank n, n ⩽ 7, rational or quasi-rational solutions. In particular, we consider rank-2 and rank-3 quasi-rational solutions that can be used for prediction and modeling of the rogue wave events in fiber optics, hydrodynamics, and many other branches of science.
TL;DR: In this paper, a thorough mathematical analysis was developed to deduce conditions for the accuracy and convergence of different approximations of the memory integral in the Mori-Zwanzig (MZ) equation.
Abstract: We develop a thorough mathematical analysis to deduce conditions for the accuracy and convergence of different approximations of the memory integral in the Mori-Zwanzig (MZ) equation. In particular, we derive error bounds and sufficient convergence conditions for short-memory approximations, the t-model, and hierarchical (finite-memory) approximations. In addition, we derive useful upper bounds for the MZ memory integral, which allow us to estimate a priori the contribution of the MZ memory to the dynamics. Such upper bounds are easily computable for systems with finite-rank projections. Numerical examples are presented and discussed for linear and nonlinear dynamical systems evolving from random initial states.We develop a thorough mathematical analysis to deduce conditions for the accuracy and convergence of different approximations of the memory integral in the Mori-Zwanzig (MZ) equation. In particular, we derive error bounds and sufficient convergence conditions for short-memory approximations, the t-model, and hierarchical (finite-memory) approximations. In addition, we derive useful upper bounds for the MZ memory integral, which allow us to estimate a priori the contribution of the MZ memory to the dynamics. Such upper bounds are easily computable for systems with finite-rank projections. Numerical examples are presented and discussed for linear and nonlinear dynamical systems evolving from random initial states.
TL;DR: In this paper, it was shown that the set of synchronous quantum correlations is not closed, which implies Slofstra's result, by giving an example of a synchronous game that has a perfect quantum approximate strategy but no perfect quantum strategy.
Abstract: Recently, Slofstra proved that the set of quantum correlations is not closed. We prove that the set of synchronous quantum correlations is not closed, which implies his result, by giving an example of a synchronous game that has a perfect quantum approximate strategy but no perfect quantum strategy. We also exhibit a graph for which the quantum independence number and the quantum approximate independence number are different. We prove new characterisations of synchronous quantum approximate correlations and synchronous quantum spatial correlations. We solve the synchronous approximation problem of Dykema and the second author, which yields a new equivalence of Connes’ embedding problem in terms of synchronous correlations.
TL;DR: In this paper, the stability and instability of standing waves for the fractional nonlinear Schrodinger equation i∂tu = (−Δ)su − |u|2σu, where (t,x)∈R × RN, 12
Abstract: In this paper, we study the stability and instability of standing waves for the fractional nonlinear Schrodinger equation i∂tu = (−Δ)su − |u|2σu, where (t,x)∈R × RN, 12
TL;DR: In this paper, the Riccati-Bernoulli sub-ordinary differential equation and modified tanh-coth methods are used to reach soliton solutions of the nonlinear evolution equation.
Abstract: In this work, the Riccati-Bernoulli sub-ordinary differential equation and modified tanh-coth methods are used to reach soliton solutions of the nonlinear evolution equation We acquire new types of traveling wave solutions for the governing equation We show that the equation is nonlinear self-adjoint by obtaining suitable substitution Therefore, we construct conservation laws for the equation using new conservation theorem The obtained solutions in this work may be used to explain and understand the physical nature of the wave spreads in the most dispersive medium The constraint condition for the existence of solitons is stated Some three dimensional figures for some of the acquired results are illustrated
TL;DR: In this article, a graphical interpretation of the weighting is given in terms of constellations mapped onto the Riemann sphere, with the Baker function at t = 0 satisfying the quantum spectral curve equation, whose classical limit is rational.
Abstract: Multiparametric families of hypergeometric τ-functions of KP or Toda type serve as generating functions for weighted Hurwitz numbers, providing weighted enumerations of branched covers of the Riemann sphere. A graphical interpretation of the weighting is given in terms of constellations mapped onto the covering surface. The theory is placed within the framework of topological recursion, with the Baker function at t = 0 shown to satisfy the quantum spectral curve equation, whose classical limit is rational. A basis for the space of formal power series in the spectral variable is generated that is adapted to the Grassmannian element associated with the τ-function. Multicurrent correlators are defined in terms of the τ-function and shown to provide an alternative generating function for weighted Hurwitz numbers. Fermionic vacuum state expectation value representations are provided for the adapted bases, pair correlators, and multicurrent correlators. Choosing the weight generating function as a polynomial and restricting the number of nonzero “second” KP flow parameters in the Toda τ-function to be finite implies a finite rank covariant derivative equation with rational coefficients satisfied by a finite “window” of adapted basis elements. The pair correlator is shown to provide a Christoffel-Darboux type finite rank integrable kernel, and the WKB series coefficients of the associated adjoint system are computed recursively, leading to topological recursion relations for the generators of the weighted Hurwitz numbers.
TL;DR: Using the variational method, this paper obtained three ground state solutions (one positive, one negative, and one sign-changing) for the double phase problem and proved a strong maximum principle for the problem.
Abstract: Using the variational method, we obtain three ground state solutions (one positive, one negative, and one sign-changing) for the double phase problem. In particular, a strong maximum principle for the double phase problem will be proved.
TL;DR: In this paper, the authors considered a laminated beam model with two identical uniform layers on top of each other, taking into account that an adhesive of small thickness is bonding the two surfaces and produces an interfacial slip.
Abstract: In this paper, we consider a laminated beam model. This structure is given by two identical uniform layers on top of each other, taking into account that an adhesive of small thickness is bonding the two surfaces and produces an interfacial slip. We use boundary feedback control and establish an exponential energy decay result. Our result improves the earlier related results in the literature.
TL;DR: In this article, a nonlinear Schrodinger equation in two spatial dimensions subject to a periodic honeycomb lattice potential is considered and an effective model of nonlinear Dirac type is derived using a multi-scale expansion together with rigorous error estimates.
Abstract: We consider a nonlinear Schrodinger equation in two spatial dimensions subject to a periodic honeycomb lattice potential. Using a multi-scale expansion together with rigorous error estimates, we derive an effective model of nonlinear Dirac type. The latter describes the propagation of slowly modulated, weakly nonlinear waves spectrally localized near a Dirac point.