arXiv subjects

S. Y. Lou

Publications and source records attributed to S. Y. Lou.

At least 37 records · Page 2Linked to original sources

arXiv2020

Searching for missing D'Alembert waves in nonlinear system: Nizhnik-Novikov-Veselov equation

In linear science, the wave motion equation with general D'Alembert wave solutions is one of the fundamental models. The D'Alembert wave is an arbitrary travelling wave moving along one direction under a fixed model (material) dependent velocity. However, the D'Alembert waves are missed when nonlinear effects are introduced to wave motions. In this paper, we study the possible travelling wave solutions, multiple soliton solutions and soliton molecules for a special (2+1)-dimensional Koteweg-de Vries (KdV) equation, the so-called Nizhnik-Novikov-Veselov (NNV) equation. The missed D'Alembert wave is re-discovered from the NNV equation. By using the velocity resonance mechanism, the soliton molecules are found to be closely related to D'Alembert waves. In fact, the soliton molecules of the NNV equation can be viewed as special D'Alembert waves. The interaction solutions among special D'Alembert type waves ($n$-soliton molecules and soliton-solitoff molecules) and solitons are also discussed.

nlin.SI↗

arXiv2020

Painlevé property, local and nonlocal symmetries and symmetry reductions for a (2+1)-dimensional integrable KdV equation

The Painlevé property for a (2+1)-dimensional Korteweg-de Vries (KdV) extension, the combined KP3 (Kadomtsev- Petviashvili) and KP4 (cKP3-4) is proved by using Kruskal's simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac-Moody-Virasoro algebra. The local point symmetries are used to find the related group invariant reductions including a new Lax integrable model with a fourth order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.

nlin.SI↗

arXiv2020

A novel (2+1)-dimensional integrable KdV equation with peculiar solution structures

The celebrated (1+1)-dimensional Korteweg de-Vries (KdV) equation and its (2+1)-dimensional extention, the Kadomtsev-Petviashvili (KP) equation, are two of the most important models in physical science. The KP hierarchy is explicitly written out by means of the linearized operator of the KP equation. A novel (2+1)-dimensional KdV extension, the cKP3-4 equation, is obtained by combining the third member (KP3, the usual KP equation) and the fourth member (KP4) of the KP hierarchy. The integrability of the cKP3-4 equation is guaranteed by the existence of the Lax pair and dual Lax pair. The cKP3-4 system can be bilinearized by using Hirota's bilinear operators after introducing an additional auxiliary variable. Exact solutions of the cKP3-4 equation possess some peculiar and interesting properties which are not valid for the KP3 and KP4 equations. For instance, the soliton molecules and the missing D'Alembert type solutions (the arbitrary travelling waves moving in one direction with a fixed model dependent velocity) including periodic kink molecules, periodic kink-antikink molecules, few cycle solitons and envelope solitons are existed for the cKP3-4 equation but not for the separated KP3 equation and the KP4 equation.

nlin.SI↗

arXiv2019

Soliton molecules and asymmetric solitons in fluid systems via velocity resonance

Soliton molecules may be formed in some possible mechanisms in both theoretical and experimental aspects. In this letter, we introduce a new possible mechanism, the velocity resonant, to form soliton molecules. Under the resonant mechanism, two solitons may be formed to a kink-antikink molecule, an asymmetric soliton, a two-peak soliton and/or a far away bounded molecule depended on the selections of the wave numbers and the distance between two solitons of the molecule. The results are exhibited via three well known fifth order integrable systems which approximately solve a general fluid model that also appeared in many other physical fields. The interactions among multiple soliton molecules for these three integrable systems are elastic.

nlin.SI↗

arXiv2019

Exact parity and time reversal symmetry invariant and symmetry breaking solutions for a nonlocal KP system

A nonlocal Alice-Bob Kadomtsev-Petviashivili (ABKP) system with shifted-parities ($\hat{P}_s^x$ and $\hat{P}_s^y$ parities with shifts for the space variables $x$ and $y$) and delayed time reversal ($\hat{T}_d$, time reversal with a delay) symmetries is investigated. Some types of $\hat{P}_s^y\hat{P}_s^x\hat{T}_d$ invariant solutions including multiple soliton solutions, Painlevé reductions and soliton and $p$-wave interaction solutions are obtained via $\hat{P}_s^y\hat{P}_s^x\hat{T}_d$ symmetry and the solutions of the usual local KP equation. Some special $\hat{P}_s^y\hat{P}_s^x\hat{T}_d$ symmetry breaking multi-soliton solutions and cnoidal wave solutions are found from the $\hat{P}_s^y\hat{P}_s^x\hat{T}_d$ symmetry reduction of a coupled local KP system.

nlin.SI↗

arXiv2019

Multi-place nonlocal systems

Two-place nonlocal systems have attracted many scientists' attentions. In this paper, two-place non-localities are extended to multi-place non-localities. Especially, various two-place and four-place nonlocal nonlinear Schrodinger (NLS) systems and Kadomtsev-Petviashvili (KP) equations are systematically obtained from the discrete symmetry reductions of the coupled local systems. The Lax pairs for the two-place and four-place nonlocal NLS and KP equations are explicitly given. Some types of exact solutions especially the multiple soliton solutions for two-place and four-place KP equations are investigated by means of the group symmetric-antisymmetric separation approach.

nlin.SI↗

arXiv2018

Prohibitions caused by nonlocality for Alice-Bob Boussinesq-KdV type systems

It is found that two different celebrate models, the Korteweg de-Vrise (KdV) equation and the Boussinesq equation, are linked to a same model equation but with different nonlocalities. The model equation is called the Alice-Bob KdV (ABKdV) equation which was derived from the usual KdV equation via the so-called consistent correlated bang (CCB) companied by the shifted parity (SP) and delayed time reversal (DTR). The same model can be called as the Alice-Bob Boussinesq (ABB) system if the nonlocality is changed as only one of SP and DTR. For the ABB systems, with help of the bilinear approach and recasting the multi-soliton solutions of the usual Boussinesq equation to an equivalent novel form, the multi-soliton solutions with even numbers and the head on interactions are obtained. However, the multi-soliton solutions with odd numbers and the multi-soliton solutions with even numbers but with pursuant interactions are prohibited. For the ABKdV equation, the multi-soliton solutions exhibit many more structures because an arbitrary odd function of $x+t$ can be introduced as background waves of the usual KdV equation.

nlin.SI↗

arXiv2018

Nanopteron solution of the Korteweg-de Vries equation

The nanopteron, which is a permanent but weakly nonlocal soliton, has been an interesting topic in numerical study for many decades. However, analytical solution of such a special soliton is rarely considered. In this Letter, we study the explicit nanopteron solution of the Korteweg-de Vries (KdV) equation. Starting from the soliton-cnoidal wave solution of the KdV equation, the nanopteron structure is shown to exist. It is found that for the suitable choice of the wave parameters, the soliton core of the soliton-cnoidal wave trends to be a classical soliton of the KdV equation and the surrounded cnoidal periodic wave appears as small amplitude sinusoidal variations on both sides of the main core. Some interesting features of the wave propagation are revealed. In addition to the elastic interaction, it is surprising that the phase shift of the cnoidal periodic wave after the interaction with the soliton core is always half of its wavelength, and this conclusion is universal to soliton-cnoidal wave interactions.

nlin.SI↗

arXiv2017

A novel type of rogue waves with predictability in nonlinear physics

Rogue waves named by oceanographers are ubiquitous in nature and appear in a variety of different contexts such as water waves, liquid Helium, nonlinear optics, microwave cavities, etc. In this letter, we propose a novel type of exact (2+1)-dimensional rogue waves which may be found in many physical fields described by integrable and nonintegrable models. This type of rogue waves are closely related to invisible lumps. Usually, a lump is an algebraically localized wave in space but visible at any time. Whence a lump induces a soliton, the lump will become invisible before or after a fixed time. If a bounded twin soliton is induced by a lump, the lump will become a rogue wave (or instanton) and can only be visible at an instant time. Because of the existence of the induced visible solitons, the rogue wave may be predictable in some senses. For instance, the height, the position and the arrival time of the rogue wave can be predictable. The idea is illustrated by the celebrate (2+1)-dimensional Kadomtsev-Petviashvili equation in its extended form.

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arXiv2017

Coherent structure of Alice-Bob modified Korteweg de-Vries Equation

To describe two-place events, Alice-Bob systems have been established by means of the shifted parity and delayed time reversal in Ref. [1]. In this paper, we mainly study exact solutions of the integrable Alice-Bob modified Korteweg de-Vries (AB-mKdV) system. The general Nth Darboux transformation for the AB-mKdV equation are constructed. By using the Darboux transformation, some types of shifted parity and time reversal symmetry breaking solutions including one-soliton, two-soliton and rogue wave solutions are explicitly obtained. In addition to the similar solutions of the mKdV equation (group invariant solutions), there are abundant new localized structures for the AB-mKdV systems.

nlin.SI↗

arXiv2017

Alice-Bob Physics: Coherent Solutions of Nonlocal KdV Systems

In natural and social science, many events happened at different space-times may be closely correlated. Two events, $A$ (Alice) and $B$ (Bob) are defined correlated if one event is determined by another, say, $B=\hat{f}A$ for suitable $\hat{f}$ operators. Taking KdV and coupled KdV systems as examples, we can find some types of models (AB-KdV systems) to exhibit the existence on the correlated solutions linked with two events. The idea of this report is valid not only for physical problems related to KdV systems but also for problems described by arbitrary continuous or discrete models. The parity and time reversal symmetries are extended to shifted parity and delayed time reversal. The new symmetries are found to be useful not only to establish AB-systems but also to find group invariant solutions of numerous AB-systems. A new elegant form of the $N$-soliton solutions of the KdV equation and then the AB-KdV systems is obtained. A concrete AB-KdV system derived from the nonlinear inviscid dissipative and barotropic vorticity equation in a $β$-plane channel is applied to the two correlated monople blocking events which is responsible for the snow disaster in the winter of 2007/2008 happened in Southern China.

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arXiv2016

Alice-Bob systems, $P_s$-$T_d$-$C$ principles and multi-soliton solutions

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 (C), shifted-parity ($P_s$, parity with a shift), delayed time reversal ($T_d$, time reversal with a delay) and their possible combinations such as PT, PC, $P_sC$, $P_sT_d$ and $P_sT_dC$ etc. can be successively used. Especially, some special types of $P_s$-$T_d$-$C$ group invariant multi-soliton solutions for the KdV-KP-Toda type, mKdV-sG type, NLS type and discrete $H_1$ type AB systems are explicitly constructed.

nlin.SI↗

arXiv2013

Consistent Riccati Expansion and Solvability

A consistent Riccati expansion (CRE) is proposed for solving nonlinear systems with the help of a Riccati equation. A system is defined to be CRE solvable if it has a CRE. Various integrable systems are CRE solvable. Furthermore, it is also revealed that many CRE solvable systems, including the Korteweg de-Vries, Kadomtsev-Patviashvili, nonlinear Schrdinger and sine-Gordon equations, possess a common determining equation which describes interactions between a soliton and a cnoidal wave.

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arXiv2013

Bosonization, Singularity Analysis, Nonlocal Symmetry Reductions and Exact Solutions of Supersymmetric KdV Equation

Assuming that there exist at least two fermionic parameters, the classical N= 1 supersymmetric Korteweg-de Vries (SKdV) system can be transformed to some coupled bosonic systems. The boson fields in the bosonized SKdV (BSKdV) systems are defined on even Grassmann algebra. Due to the intrusion of other Grassmann parameters, the BSKdV systems are different from the usual non-supersymmetric integrable systems, and many more abundant solution structures can be unearthed. With the help of the singularity analysis, the Painlevé property of the BSKdV system is proved and a Bäcklund transformation (BT) is found. The BT related nonlocal symmetry, we call it as residual symmetry, is used to find symmetry reduction solutions of the BSKdV system. Hinted from the symmetry reduction solutions, a more generalized but much simpler method is established to find exact solutions of the BSKdV and then the SKdV systems, which actually can be applied to any fermionic systems.

nlin.SI↗

arXiv2013

Interactions Between Solitons and Other Nonlinear Schrödinger Waves

The Nonlinear Schrödinger (NLS) equation is widely used in everywhere of natural science. Various nonlinear excitations of the NLS equation have been found by many methods. However, except for the soliton-soliton interactions, it is very difficult to find interaction solutions between different types of nonlinear excitations. In this paper, three very simple and powerful methods, the symmetry reduction method, the truncated Painlevé analysis and the generalized tanh function expansion approach, are further developed to find interaction solutions between solitons and other types of NLS waves. Especially, the soliton-cnoidal wave interaction solutions are explicitly studied in terms of the Jacobi elliptic functions and the third type of incomplete elliptic integrals. In addition to the new method and new solutions of the NLS equation, the results can unearth some new physics. The solitons may be decelerated/accelerated through the interactions of soliton with background waves which may be utilized to study tsunami waves and fiber soliton communications; the static/moving optical lattices may be automatically excited in all mediums described by the NLS systems; solitons elastically interact with non-soliton background waves, and the elastic interaction property with only phase shifts provides a new mechanism to produce a controllable routing switch that is applicable in optical information and optical communications.

nlin.SI↗

arXiv2012

Nonlocal symmetries related to Bäcklund transformation and their applications

Starting from nonlocal symmetries related to Bäcklund transformation (BT), many interesting results can be obtained. Taking the well known potential KdV (pKdV) equation as an example, a new type of nonlocal symmetry in elegant and compact form which comes from BT is presented and used to make researches in the following three subjects: two sets of negative pKdV hierarchies and their corresponding bilinear forms are constructed; the nonlocal symmetry is localized by introduction of suitable and simple auxiliary dependent variables to generate new solutions from old ones and to consider some novel group invariant solutions; some other models both in finite dimensions and infinite dimensions are generated by comprising the original BT and evolution under new nonlocal symmetry. The finite-dimensional models are completely integrable in Liouville sense, which are shown equivalent to the results given through the nonlinearization method for Lax pair.

math-ph↗

arXiv2011

Bosonization of Supersymmetric KdV equation

Bosonization approach to the classical supersymmetric systems is presented. By introducing the multi-fermionic parameters in the expansions of the superfields, the $\mathcal {N}=1$ supersymmetric KdV (sKdV) equations are transformed to a system of coupled bosonic equations. The method can be applied to any fermionic systems. By solving the coupled bosonic equations, some novel types of exact solutions can be explicitly obtained. Especially, the richness of the localized excitations of the supersymmetric integrable system are discovered. The rich multi-soliton solutions obtained here have not yet been obtained by using other methods. Unfortunately, the traditional known multi-soliton solutions can also not be obtained by the bosonization approach of this paper. Some open problems on the bosonization of the supersymmetric integrable models are proposed in the both classical and quantum levels.

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arXiv2009

Approximate homotopy symmetry method and homotopy series solutions to the six-order boussinesq equation

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the six-order boussinesq equation. We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders, educing the related homotopy series solutions. The convergence region of homotopy series solutions can be adjusted by the auxiliary parameter. Series solutions and similarity reduction equations from approximate symmetry method can be retrieved from approximate homotopy symmetry method.

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